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The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how they impact markets. They recently published a piece on China's new neutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science they analyze.
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Where senior editors argue through the news with world leaders and policy makers in twice weekly long format shows. Basically an extremely high quality podcast. Whether it's scientific innovation or shifting global politics, The Economist provides comprehensive coverage beyond headlines. As a toe listener, you get a special discount. Head over to economist.com slash TOE to subscribe. That's economist.com slash TOE for your discount.
Welcome to the Iceberg of String Theory, a technical edition. The Iceberg format is one where you initially explore preparatory, surface-level concepts, then progress ever more into the intricacies of a topic, which tend to be known only to a specialized few, until eventually you arrive at the
obscure dark frontiers of the deepest layers of the field. On the Special Theories of Everything podcast, we're going to be exploring string theory like you've never seen it before. You'll learn more about the hinterlands of this field in the next two hours than you will watching say 20 hours of Michio Kaku documentaries or Neil deGrasse Tyson rants.
Why? Because you'll be shown the actual math instead of hand-wavy, metaphoric explanations that leave you slack-jawed, deracinated from the equations, and even misinformed. My name's Kurt Jaimungal, and on Theories of Everything, I use my background in mathematical physics from the University of Toronto to explore unifications of gravity with the Standard Model and have also become interested in fundamental laws in general as they relate to explanations for some of the largest
philosophical questions that we have, such as what is consciousness? How does it arise?
In other words, it's a peregrination into the all-encompassing nature of the universe. We'll cover the abstruse math of string theory, black holes, as well as other toe frameworks like geometric unity and loop quantum gravity. This episode took a combined 300 hours across four different editors and several rewrites on my part. It's the most labor that's gone into any single theories of everything video.
Layer 1 Types of String Theory
In string theory, there are five so-called consistent formulations or flavors. There's Type I, there's Type IIa, Type IIb, Heterotic SO32, and Heterotic E8 x E8. Type I string theory is characterized by open and closed strings with the gauge group SO32, coming from something called the Chan-Paton factors at the endpoints of the open strings.
Type IIa and Type IIb are both closed string theories, with Type IIa being non-kyro and Type IIb actually being kyro. The heterotic string theories are based on a hybrid of 26-dimensional bosonic string theory and a 10-dimensional superstring theory, also resulting in closed strings. Heterotic actually means hybrid. You can always sound clever to someone studying string theory by saying, oh, do you study heterotic strings? They'll respect you exactly 3% more.
open and closed strings. As mentioned before, there are broadly two types of strings, open strings, which have endpoints, and then there's closed strings forming loops. This formula on screen is specific to closed strings and accounts for additional properties such as the winding number w and momentum n in a compactified space.
Compactified spaces are something that we'll explore later, so don't worry if this terminology confuses you. R is the compactification radius, and alpha prime is called the Regislope. These will come up over and over. By the way, the Regislope is related to the so-called string tension. All of these we'll discuss in detail later. M Theory
The five flavors of string theory are related through something called dualities, such as T-duality connecting type IIa with type IIb, and then there's S-duality linking type I with heterotic SO32. The fact of these dualities is what spurred the idea of M-theory, which is an 11-dimensional unifying framework encompassing all five string theories,
rather than a 10-dimensional theory. It does so by introducing a new type of brain called a membrane, which we'll talk more about later. By the way, when someone says that string theory is in 10 dimensions, they actually mean 9 plus 1, so 9 spatial dimensions and 1 time dimension, and when they say it's 11-dimensional, they mean 10 plus 1. The reason is they're usually talking about space-time dimensions as a whole.
Apparently the M in M-Theory stands for Matrix or Membrane or Mystery or Mother, but I think it stands for an upside-down W for Witten, much like how the W of Wario is an upside-down M for Mario. Left and Right Moving Strings
String modes are characterized by their oscillations along the worldsheet and are described by the Polyakov action on screen. Notice I keep saying mode and not vibration. That's because no string theorist talks about vibrations unless they're being condescending to a lay public. Generally they speak about modes. Or even spectra, which are distinct states of a string, each with their own quantum number like energy, charge, mass, spin, winding number.
If you introduce light cone coordinates, which you'll see on screen as sigma plus and minus, again on the world sheet, then you can separate that Polyakov action into left and right moving components, leading to the Fourier expansions on screen. It may sound confusing, but this is akin to decomposing a complex function into a real and imaginary part. The energy momentum tensor, TAB, also decomposes into left and right moving components, T plus plus and say T minus minus.
which generates something called the Verasoro algebra. This allows us to classify, or label, our states into conformal weights. Personally, I like to denote the right-moving conformal dimension with a tilde, as there's already one too many h-bars in physics. There's another subtlety here of matching left and right mode numbers in order to preserve Lorentz covariance, but the iceberg must go on. Gravitons.
Gravitons are the hypothesized massless spin two particles said to be responsible for gravitation. They come from rank two tensor field perturbations H mu nu. Now all of that's a mouthful, but I could have just said it's a massless spin two particle. Why? Because there are theorems by Weinberg and others that suggest the particle associated with gravity would have those properties. And furthermore, any particle that's massless, chargeless and spin two would be the particle of gravity. Thus their equivalent.
You get this by linearizing the Einstein field equations around a flat background metric, giving the equation on screen. This is the aforementioned perturbation. It should be mentioned that no one has observed a graviton, and furthermore, we have great reasons to believe that we never will, even in principle. This is a point that Freeman-Dyson makes. Thus, it's unclear if the graviton is even a scientific concept in the Popperian sense. Dualities in string theory.
You'll hear T and S dualities discussed frequently. T dualities are transformations like R moving to something like being proportional to the inverse of R, where R represents the compactification radius, and alpha prime is the Regi slope. This connects type IIa and type IIb superstring theories. How so? It maps a type IIa theory on a circle of radius R to a type IIb theory on a circle of radius alpha prime over R, and vice versa.
S-duality, on the other hand, explores the equivalence between weak and strong couplings in string theory. It's particularly evident in the SL2Z invariance of type 2B string theory, which acts on a complex parameter combining the string coupling GS and certain Raymond-Raymond fields. By the way, I've heard this pronounced Raymond, I've heard this pronounced Raymond, I'm just gonna stick with Raymond. This makes it only slightly more difficult than taking the inverse of GS.
This S-duality links type I string theory with heterotic SO32 theory, which then gives insights into non-perturbative string dynamics, since strong couplings are useful for non-perturbative studies and weak ones for perturbative. The other S and T-dualities are shown on screen.
S-duality hinges on incorporating elements like D-brains and oriental folds. Basically, you can think of both of these dualities as inversions. One inverts the coupling strength, and the other inverts the radius of the extra dimensions after compactifying. Heterotic Strings
Confession, I deceived you earlier out of kindness and love when I told you that there are five flavors of string theory that are 10-dimensional. There are actually several more than that. One of them, even the original string theory, is 26-dimensional and only described bosons, not fermions. And most of the matter that we see is fermionic, where the bosons are there to allow interactions between them. Heterotic string theory combines left-moving bosonic modes from the 26-dimensional bosonic string theory.
with right-moving fermionic modes from a 10-dimensional superstring theory. This abomination is gotten to by compactifying 16 additional dimensions in the left-moving sector on an internal lattice, resulting in two consistent heterotic theories, the SL32 and the E8 cross E8. But what do we mean by consistent here? What do we mean by super here? Is a superstring a string that's been bitten by a radioactive spider? We'll explore that in one of the deeper layers. The short answer is yes.
With more detail, let's use some notation on screen here to represent the root lattice. So here's VE8 of the E8 Lie algebra, and the heterotic theories are defined by their lattices through this construction.
You'll notice here that 26 equals 10 plus 8 plus 8, though you'll also notice that 26 does not equal 10 plus 32. The reason is that you don't deal with the group SO32 directly. You don't even deal with its 16 dimensional root lattice. Instead, you deal with the weight lattice of spin 32 modded out by Z2. Regislope
The Regi slope, denoted by A', is a fundamental parameter in string theory that relates the mass squared of a string to its angular momentum J through the linear Regi trajectory written on screen. Sluskin covers this in the first lecture on string theory at Stanford, and the link to that is in the description. This trajectory represents the spectrum of excited string states as a relationship between mass and angular momentum.
Personally, I think that the word trajectory is misleading since it implies that something is moving through space, but rather this is a plot of an observed pattern of quantum numbers. The Regi slope is inversely proportional to string tension, T, with A' equaling something proportional to the inverse of T.
For those quantum field theorists interested in scattering amplitudes, the Regi trajectory comes from the analytic continuation of the amplitude into the complex angular momentum plane, where the physical region corresponds to the poles of the amplitude. This means considering angular momentum as a complex number, rather than just an integer or a half integer as a standard.
World Sheet Symmetry The world sheet of a string is the two-dimensional surface that a string sweeps out in space-time shown on screen here.
The basic symmetries of the worldsheet include reparametrization invariance and then something called Weyl symmetry. Be careful not to call it Weyl symmetry. If you do, string theorists will respect you exactly 3% less.
Reparametrization invariance means you can choose whichever coordinates you like on the world sheet and it won't affect the physical predictions. Vial symmetry is the rescaling of the world sheet metric, which is a feature in conformal field theories or CFTs. CFTs are something that we'll explore next. Recall the Polyakov action here. This little guy is invariant under both reparametrization and Vial transformations. Conformal symmetry and the Polyakov action.
Conformal symmetry means when you scale the metric, you preserve angles. This means that while your volume can change, like the volume of a circle, the shape doesn't, like the shape of the circle is the same. The fact of this symmetry allows us to simplify calculations in that juicy polyakov action before. The G here is the regular spacetime metric that we know and love, and the H is the worldsheet metric.
This symmetry leads to a vanishing of the trace of the angular momentum tensor, yielding the verisaural constraints, which are important when talking about the so-called string quantization. Conformal symmetry also allows us to decompose into holomorphic and anti-holomorphic correlators, thus reducing calculations into far simpler one-dimensional CFTs.
Conformal symmetry classifies string states by conformal weights and ghost numbers. Ghosts are particles that are supposed to be unable to be detected, but are necessary for calculations. The physical states are determined by the BRST cohomology, satisfying the following conditions for the BRST charge Q, ensuring invariance under the worldsheet symmetry transformations.
Layer 2
Here, I'd like to make a note. Depending on your background, much of this math may sound unintelligible. It may sound like gibberish. That's okay. It's important not that you drink from the fire hose, but rather that you merely get wet. In other words, don't feel dismayed if you don't understand Spanish from the get-go. Rather immerse yourself in Spain, for instance, and that, along with a bit of practice, will advance you.
Even John von Neumann said the point of math isn't to understand it, but rather to get used to it. You would think that Fields' medalist Richard Borchards would need only a single book to understand commutative algebra, but instead he had to learn about it not from one source, not from two, not three, but eight.
Universe on a Brain Theory
Our universe could exist as a three-brain on a higher dimensional bulk, with standard model particle physics confined to the brain, while gravity extends into the extra dimensions. In string theory, d-brains, also known as derrishly brains, serve as endpoints for open strings. The action for a d-brain is given by the Dirac-Born-Infeld action, shown on screen here, where T is the brain tension, and gamma is the induced metric, and the calligraphic F is the field strength tensor.
However, more general brains do exist, such as Newman Brains, which allow strings to move off the brain. The Randall-Sundrum model exemplifies the brain-world scenario, with two three-brains embedded in a 5D anti-decider spacetime, where one of these brains represents our universe. The RS metric is given on screen, where k is the adias curvature scale, r is the compactification radius, and phi is the extra-dimension coordinate.
The RS model addresses the hierarchy problem by localizing gravity near the standard model brain or the quote-unquote visible brain, resulting in a large hierarchy without fine-tuning.
In this framework, the Planck scale is transformed into the TeV scale by the warp factor given by this decaying exponential on screen, which gives a so-called natural explanation for the large disparity between the two. By the way, I pronounce the Dirac equation Dirac and not Dirac equation because I just can't help but think about Dwayne Johnson writing a hyperbolic PDE. String Cosmology and Inflation
String cosmology is a framework for investigating inflationary models. Compactification schemes such as Calabi-Yau manifold, orbifold, and flux compactifications, all of which we'll talk about later, impact virtually every cosmological quantity. How? The moduli field from these compactifications influence the dynamics of inflation in string-inspired scenarios, like the large volume scenario with Kaler moduli and the radial dilaton.
Dilatants we'll talk about later. And if you're interested in the low energy effective action that's on screen, the inflationary potential is V. Cosmic strings. String theory predicts cosmic strings, F and D term inflation, and axiom monodromy models. Contrary to what people say that string theory has no predictions, these actually do yield testable predictions on the tensor to scalar ratio R, on the scalar spectral index M, and on non-Gaussianicities.
The tricky part is that the predictions vary, meaning they're not falsifiable. Cosmic strings are essentially a thin line stretching across the universe, which may have formed during phase transitions in the early universe. Think of them as cracks in space of concentrations of energy. Actually, cosmic strings may have been found recently, but this doesn't mean string theory is correct. Why? Because despite the name, cosmic strings are predicted by several other theories, not just string theory. String gas.
An alternative to inflation is something called string gas cosmology, which focuses on the thermodynamic properties of string gases and the Hagedorn temperature, usually denoted by TH. If you see my tutorial masterclass on undergrad physics in two hours, which is linked in the description, then you'll see why I'm fond of this tilde notation rather than the approximate notation.
There's a problem in cosmology called the horizon problem, which is why the CMB is so uniform, as well as the flatness problem, which is why our universe is basically flat. String gas cosmology attempts to address both at once with a quasi-static Hagedorn phase. For fractal-like scale invariant spectra of fluctuations, the specific dynamics and interactions of the string gas during these Hagedorn phases are important. Verisaur algebra, symmetry algebras and infinite generators.
The Virasora algebra is a central extension of the Witt algebra, which is the algebra of infinitesimal conformal transformations in two dimensions. The commutation relations are given on screen here, with LM being the generators, C being the central charge, and of course, M and N are integers. This infinite dimensional algebra encodes the symmetries of the world sheet under conformal transformations, reflecting the structure of the two dimensional conformal field theories, or CFTs.
The algebra's representations are characterized by the eigenvalues of L0, known as the conformal weights, delta. In string theory, the central charge is related to the spacetime dimension D via this formula, and this, by the way, only applies in certain contexts like bosonic string theory. Otherwise, there are other relations.
The infinite generators of the Virasora algebra, indexed by LM, are used in the construction of vertex operators, which describe the interactions of strings and are subject to something called the operator product expansion in CFT, which we will expand on more later. Quantum Yang-Baxter equation.
The quantum Yang-Baxter equation is an equation in integrable systems, specifically in quantum integrable models, generalizing the classical Yang-Baxter equation, which shows up in soliton theory. Given by this unruly formula on screen, it comprises intertwiners, which is what those Rs are over there. And these intertwiners are invertible linear operators, which act on tensor products of quantum spaces. Each of those lambdas denotes a spectral parameter.
The quantum Yang-Baxter equations are seen in statistical mechanics, quantum groups, and knot theory. Regarding statistical mechanics, it enables the construction of integrable lattice models, such as the six vertex model via the algebraic beta ansatz, offering exact solutions for correlation functions and their thermodynamic properties. Actually, Edward Frankel talked about the beta ansatz in this podcast on this channel, Theories of Everything here, among other topics like consciousness and the failure of string theory. The link is in the description.
In quantum group theory, the quantum Yang-Baxter equation results in the discovery of quantum deformations of Lie algebras called Drinfeld-Gymbo quantum groups. It's denoted here by this U with usually a Q is underneath and in brackets is the Lie algebra G. So not the group G, but the Lie algebra of the group.
It has wide applications and conformal field theory. The solutions to these quantum Yang-Baxter equations are known as R matrices and are necessary for constructing invariance of knots and links such as the Jones polynomial and the HomFly polynomial, which generalizes the Jones polynomial.
By the way, correlators are a physicist's fancy way of saying greens functions, and that's a mathematician's fancy way of saying solutions to inhomogeneous differential equations. And that's just a pretentious way of saying responses to disturbances in a field. Stress energy tensor and conformal weight.
In string theory, the stress-energy tensor TAB encapsulates the energy and momentum density on the worldsheet and can be obtained by varying the Polyakov action with respect to the worldsheet metric h.
The requirement of conformal symmetry leads to the traceless condition, which in turn gives rise to the varisora constraints required for string quantization. As mentioned previously, the stress energy tensor can be decomposed into holomorphic and anti-holomorphic parts with the complex world sheet coordinates z and then z-bar. Conformal weights here, h and h tilde, characterize the fields in string theory, determining their transformation behavior under these conformal transformations. The Green-Schwarz mechanism.
The Green-Schwarz mechanism is something that resolves anomalies in type 1 and heterotic superstring theories. Anomalies happen when you have classical symmetries, like even gauge symmetries and diffeomorphism invariance, when they're preserved at the classical level, but then they're violated at the quantum one. Before the Green-Schwarz mechanism, the anomaly was represented by a non-vanishing gauge variation of the effective action. So here lambda is the gauge transformation parameter.
The mechanism demonstrates that specific combinations of space-time and world-sheet anomalies cancel, ensuring the theory's consistency. The observation is given by, if you take the whole integral of the two-form field B here, so calibrating field, and x8 is the eight-form characteristic class of the gauge bundle, which is what was introduced by Green and Schwartz, integrate all of that over the ten-dimensional space-time.
After the Green-Schwarz mechanism is applied, the anomaly vanishes and we have that that variation before is now finally equal to zero. This mechanism ensures local supersymmetry in 10-dimensional spacetime while imposing constraints on the gauge group and the spacetime dimensions. First string revolution.
The first string revolution occurred during the mid-1980s and was primarily ignited by the discovery of the Green-Schwarz anomaly cancellation mechanism in type I string theory, specifically with that gauge group mentioned, SL32. It was subsequently extended to the chiral-heterotic E8 cross-E8 string theories.
These developments demonstrated the absence of these anomalies, which are inconsistencies that come about when gauge symmetries, such as electromagnetism and diffeomorphism symmetries related to gravity, are not preserved in a quantized theory, but are there in the classical one.
In string theory, the Green-Schwarz mechanism employs that two-form B field called the Kalb-Rehman field like we talked about before, where its field strength is H. It's a three-form. The anomaly cancellation condition is expressed as, if you take the trace, well, you'll see the expression over here, and F denotes the field strength of the gauge fields, and R represents the Riemann curvature tensor in the context of gravity.
This mechanism, especially in the presence of sources or complex configurations, showcases that dH is generally not zero, unlike in the vacuum scenarios where dH can be zero. Ed Whitten, by the way, thinks that the first string revolution should be called the second string revolution, because according to Ed, the first one was the discovery of string theory. But I think that's just semantics, depending on if you're considering the word revolution, applying to the revolution of physics or revolutionizing string theory itself.
Louisville integrability concerns the existence of sufficient independent conserved quantities in involution for a dynamical system ensuring complete integrability.
The key aspect of Louisville integrability is a lax pair representation given by this formula on screen where L is a linear operator depending on a spectral parameter again like lambda here and M and N are matrices containing the system's dynamical information. The compatibility condition of the lax pair it's written on screen is the derivative of the L with respect to time being equal to some commutation relations. This guarantees the conservation of the spectral invariance making the system integrable.
To put this in simpler terms, this gives a well-behaved system evolving predictably due to the presence of these conserved quantities. The Veneziano amplitude. The amplitude here, based on Euler's beta function, represents early steps towards string theory. Historically, it was discovered in 1968.
How does this relate to the strong nuclear force? Well, initially it was applied to meson scattering. It employs Mandelstam variables, so s and t, for squared energy and momentum transfer respectively, with alpha prime as Reggi slope. The beta function has elegant analytic properties, so for instance, poles at non-positive integers and symmetries, like you can switch the factors that go into the beta function.
Examining the physical region of its poles reveals the resonance mass spectrum, akin to determining the frequency distribution of a vibrating string, a concept that started the string theory journey. String Theory Background Fields In string theory, background fields define space-time geometry and string interactions while strings propagate.
So the metric tensor G encodes space-time curvature as usual, determining distance between points and playing a role in general relativity, of course. Don't ask me why string theorists capitalize this G, whereas in every other context I know it's a lowercase g. There's even another capital G in the context of M theory, namely the field strength of the C field.
The Antisymmetric Two-Tensor Field, B, is known as the Kalbramian Field, generalizes the electromagnetic vector potential, and contributes to the strings coupling to a two-form field, affecting the world sheet action. The Dilaton Field, Phi, is a scalar field, and it sets the string coupling constant via this formula, which is usually just the exponential of Phi. It controls the strength of the string interactions. Sometimes you'll hear people say that string theory comes down to a single parameter, and it's usually this that they're referring to.
The low energy effective action for the string is given by this formula on screen where R is the Ricci scalar and H is the derivative of the Calbrahman field and G is the metric tensor's determinant. A choice of these background fields affects the compactification schemes as well as d-brain configurations. Thereby, it impacts the derived low energy physics and phenomenological predictions in string theory. In other words, different choices here yield different physics. Flux Compactifications
Flux compactifications in string theory involve background fluxes that stabilize moduli, addressing the so-called moduli stabilization problem.
These fluxes are quantized according to the flux quantization condition, which is if you integrate over the entire field strength of the gauge field with a compact cycle within an internal manifold, then you get N. Considering the GVW or the Gaka-Vafa-Wittin superpotential, W, where H is the Nebu-Schwarz-Nebu-Schwarz three-form fluxes, so NS-NS three-form, and tau is an axiodilatron.
Also, this sigma is the holomorphic three-form of the internal manifold. So, how do flux compactifications stabilize vacuum expectation values? They freeze the geometric moduli, such as the size and the shape of the extra dimensions for scalar fields in the effective four-dimensional theory, which you can think of as a shape controller for these extra dimensions. This stabilization is used to obtain the dissitter vacua, which is needed because we live in a dissitter space, not an anti-dissitter one.
Layer 3 At this point, congratulations! You now know more than 9 out of 10 people who say that they either like or dislike String Theory. Shirk's Anti-Gravity Joel Shirk is one of the founders of String Theory, who unfortunately died unexpectedly in tragic circumstances only months after the supergravity workshop at Stony Brook in 1979.
The workshop proceedings were dedicated to his memory with a statement that Shirk, who was diabetic, had been trapped somewhere without his insulin and went into a diabetic coma. He was only 33 years old.
A year prior to his death, Shirk published a little-known paper titled, Anti-Gravity, a crazy idea. The concept of anti-gravity emerges from the introduction of a massless vector field, denoted as A mu with a superscript L, referred to as the anti-graviton.
The antigraviton couples to a conserved current, J, associated with the quarks and leptons' unclad mechanical masses. This is in contrast with the graviton, which interacts with their actual masses. A force between two atoms can be expressed as F equals this formula on screen, where M and M0 are the real and unclad masses respectively, and G is the gravitational constant. You may be wondering, doesn't this notion of antigravity clash with the equivalence principle?
It seems to, but this clash can be resolved if a scalar field acquires a non-zero vacuum expectation value similar to how SU2 cross SU1 breaks down into U1. This causes the L field to acquire a mass term, which changes the potential into one with a different minimum. Schurck showed that this anti-gravity is a quality of any extended supersymmetric gravitational model.
The Swampland
The Swampland Conjecture originates from Vafa's work in 2005. It posits criteria to differentiate consistent low-energy effective field theories with a quantum gravity completion, especially from string theory, from seemingly consistent EFTs that don't.
In other words, we have different solutions to string theory. We don't know which one is correct. We know where we want to get to, namely the Standard Model plus General Relativity. You may say, Kurt, the question is, well, which of the possible string theory solutions, also known as vacua, get you there? And I'd say that's a wonderful question. You're so bright. The ones that don't get you there are part of the Swampland. Now, Swampland sounds like a negative word,
but actually the larger the Swampland the better because you'll be able to narrow down the space of possible solutions in string theory. Two central conjectures in the Swampland arena are the weak gravity conjecture and the distance conjecture. The weak one says for consistent quantum gravity and consistent by the way here means free from unwanted features like non-unitarity causality violation and unphysical singularities
In other words, the weak gravity conjecture implies that particles with a specific charge to mass ratio are needed to avoid inconsistencies in quantum gravity.
The distance conjecture, on the other hand, says that if we move in field space by some distance, let's say delta phi, that the EFT, the effective field theory, breaks down at a scale proportional to what you see on screen with a constant alpha. Why? Because infinite towers of states become exponentially light. Now, an infinite tower of states is a term you'll hear plenty, and it means an unbounded series of particles that become progressively lighter as one moves further in field space.
This is fantastic and fanatical, because it means moving sufficiently far in field space leads to the emergence of new physics. The further we explore, the more physics we have to account for. Technically speaking, the Swampland Conjecture isn't just about, well, which vacua lead to the Standard Model plus General Relativity, but it's also about determining the general properties that any consistent quantum gravity theory must have. The Transplankian Censorship Conjecture.
Metastable means something is stable for a period of time, but it's not the most stable possible. That is, it has a higher energy than the true stable state.
You may have heard something called the cosmic censorship hypothesis of Roger Penrose, which states that singularities, such as those occurring in the collapse of massive objects that form black holes, are always hidden from an external observer by an event horizon, so they're censored.
Well, the trans-Planckian censorship conjecture, on the other hand, is another censorship principle that has connections to the Swampland criteria by providing constraints on the observable universe in theories of quantum gravity. Which constraints? Constraints on the initial conditions of our observable universe, specifically stating that the physical processes occurring at distances smaller than the Planck length
Moonshine and String Theory
Moonshine refers to the unexpected connections between finite group representations, modular functions, and vertex operator algebras. You don't need to know what any of those are. All that's important is that they were at least once thought to be part of different fields of mathematics. The most famous example is what's called the monstrous moonshine conjecture, which links the largest sporadic simple group, the monster group, to a modular function called the J-function, which is given by this formula on screen.
The conjecture, proven by Bortrads using vertex operator algebras and their associated characters, states that the coefficients Cn in the J-function expansion encodes the dimensions of the irreducible representations of the monster group. So what the heck does this have to do with string theory? It turns out that certain CFTs
When compactified on the torus, give partition functions with modular invariants and characters, encoding group representation data. To translate that a tad, formally speaking, you'll see a formula on screen, and this is for any ABCD belonging to SL2Z. Umbral moonshine, on the other hand, relates something called Nyamir lattices to Mateo and other sporadic groups. I don't know how to pronounce these names. I'm a self-studier. I've only read these.
Entropic gravity postulates that gravity is induced from the statistical tendency of systems to maximize entropy described by the formula on screen here. In this description,
Exotic Dualities
There are more dualities than just T and S, young Padawan. Each of these are large enough that we'll explore later. There's U-Duality, there's Mirror Symmetry, more on that soon, ADS-CFT, there's Montan and Olive Duality or Electric Magnetic Duality, there's K3 Vibration Duality, there's Open and Closed String Duality, there's F3 slash Heterotic Duality,
So let's start with U-duality. What this does is combine T-duality and S-duality in M-theory, placing them in a single duality group in one dimension higher.
Mirror symmetry, a type of t-duality, relates these Callib-Yau manifold with different Hodge numbers. At least this is how it was initially formulated. Hodge decomposition is something we'll explore in a podcast shortly on this channel with Professor Eva Miranda, so subscribe if you're interested in geometric quantization. What you do in mirror symmetry is you exchange what's called the Caylor structure or the symplectic structure, and it has applications in enumerative geometry, which we'll talk about again later.
Montan in all of duality is an S-duality in supersymmetric gauge theories. This relates magnetic and electric charges via the exchange of coupling constants. This is also known as electric-magnetic duality, which is not to be confused with electromagnetic duality, even though some people accidentally say that.
K-3 vibrations duality is about the relationship between K-3 surfaces and elliptic vibrations, where an elliptic vibration is a morphism from a variety X, let's say, to a base B, such that almost all of the fibers are elliptic curves. Open and closed string duality describes the equivalence between open strings with boundary conditions determined by D-branes and closed strings in the presence of Raymond-Raymond fluxes.
F-theory slash heterotic duality connects F-theory, a 12-dimensional framework extending type IIb string theory, to heterotic strings via compactification on elliptically-fibred Calib-Yau manifolds. Mirror symmetry.
Mirror symmetry, this is a deep topic. Mirror symmetry is a duality relating two Calabi-Yau manifolds, M and W, interchanging their complex and scalar structure. The mirror map on screen here is bi-directional and relates the complex moduli, say phi of M, to the symplectic moduli, say psi of W, where this F here is the pre-potential and T, A are the symplectic parameters.
In topological string theory, the A model and B model are topological fields derived from the original string theory by focusing on its topological properties associated with the K-Laring complex structure respectively. The A model computes the Gromov-Witten invariance, and these little guys capture information about holomorphic curves in M, while B computes what are called periods of the holomorphic 3-0 form on W.
The Gaffa-Khamar-Vafa invariance, on the other hand, are their younger, snappier sister, which reformulate the Gromov-Witten invariance in integer numbers. Mirror symmetry connects the A model on M and the B model on B and vice versa. What this does is enable computations of one model's observables using the other model's techniques. Turns out there are like 10 to the 10 examples of distinct data points of CY3 manifolds, so that's Kali-Biao3 manifolds, making it one of the largest data sets in all of math, if not the largest.
Mirror symmetry itself can be its own iceberg. Speaking of which, I have several other ideas for other iceberg podcasts, like the iceberg of consciousness theories or the iceberg of theories of truth. If you have suggestions, then leave them below in the comment section. Extra dimensions and compactification. See why three manifolds are what are being compactified in string theory. Whenever you hear about extra dimensions, they're usually referring to these guys.
There's something else called a Joyce manifold, a subclass of Calabi-Yau manifold, with exceptional holonomy, so G2, smooth, they're compact, they're Riemannian, they're seven-dimensional, and they have a non-degenerate three-form phi, which is invariant under G2. These come up in M theory. Every time you have extra dimensions, you have to answer the question about why we don't see them. One answer is that, hey, they're just too small, they're compactified.
The problem is that not only are there several different possible structures for these extra dimensions, but there are several different ways you can compactify. Each of these spawn different physics. So far none of them have been found to be even remotely resembling our world.
By the way, it's also false to say that string theory doesn't operate in four dimensions. It does. There is a string theory of exactly four dimensions. The problem is that those four dimensions are all spatial dimensions, or all temporal dimensions, or you can also have two
space and then two times thus they're disregarded what i'm wondering though is that is there some way to wick rotate one of those extra dimensions one of those four into something from the euclidean case to a minkowski space much like peter white does in his euclidean twister unification which is explored here on this podcast
If you want to know more about dark dimensions, which suggests that dark matter is associated with these extra dimensions, then watch this video by Sabine Hassenfelder, linked in the description. In fact, if you want to know about almost any physics topic, just Google Sabine and that physics term, it's always a useful, though polemical, starting point. Conifold Transitions. Conifolds may sound like a type of manifold or a variety, but they actually refer to the singularities on a variety.
Conifolds help us understand topology changes in string compactifications as they involve transitions between distinct Kalibi-Yau manifolds. When a CY3 develops a conical singularity, then this transition commences.
This can be resolved either through something called a small resolution or a deformation, both of which result in a new Calabi-Yau manifold. In a small resolution, the conifold point transitions into a projective cycle of finite size, smoothing it out, while in a deformation, the conifold point is replaced by a non-vanishing three-form flux. Governed by Picard-Lefchet's monodromy, the periods of holomorphic three-forms transform through this process. Conifold transitions can be described by the exchange of massless
Instant Tons are topologically non-trivial solutions to the anti-self-dual Yang-Mills equations, where F is the field strength tensor and the tilde F is the Hodge dual in four-dimensional Euclidean space, and we're talking about classical Yang-Mills equations here. Okay, so all of that is a mouthful, but you can think of them as what extremizes the action in certain Yang-Mills theories.
or, to translate that a tad, what are physical solutions? These solutions are characterized by their topological charge, or instanton number, k. Donaldson invariants are topological invariants of smooth, compact, oriented 4-manifolds that were put forward by Simon Donaldson, a Fields medalist, in the 1980s. The construction of these invariants involves counting the number of instantons on a 4-manifold m, modulo gauge transformations, subject to certain constraints on their characteristic classes.
Characteristic classes are invariants of vector bundles. There exists an extension of these Donaldson invariants, which are useful for physicists, called the Seeberg-Witten invariants, which are used to describe the low energy effective action of n equals 2 supersymmetric Yang-Mills theories. In string theory, tachyon condensation is a process involving tachyons, particles with imaginary mass, which can destabilize the vacuum state and trigger infinite transitions to a lower energy state.
This is well studied in the context of open-string tachyons attached to d-brains, where the tachyon potential has the form on screen here. The tachyon condensation drives the system toward a stable configuration, effectively removing the d-brains from the spectrum and reducing the energy of the system. Sen's conjecture, which we'll talk about later, states that the endpoint of tachyon condensation corresponds to the annihilation of the d-brain, resulting in a closed-string vacuum.
Supersymmetry.
supersymmetry is a symmetry between balsonic and fermionic degrees of freedom governed by the supersymmetric algebra with the compatibility between the q's on screen here and they're the supercharge operators those alphas with the dots are spinner indices and the p represents the spacetime momentum operator it's not so intimidating this implies that for every balsonic particle there exists a fermionic superpartner and vice versa
Historically, the concept of supersymmetry was independently discovered by three groups in about the 1970s, so early 1970s by Galfan and Lichtman, Ramon and Neveu and Schwartz. In string theory, supersymmetry is a consequence of the cancellation of worldsheet anomalies we talked about earlier, and that also avoids tachyonic instabilities. Broken supersymmetry is said to be imperative in addressing the so-called hierarchy problem, controlling the Higgs boson mass and providing viable candidates for dark matter.
Extended supersymmetry. Extended supersymmetry theories are super interesting. The ordinary supersymmetry that you hear about on pop-side channels is actually n equals 1 supersymmetry, but there are other extended versions with n greater than 1. This just means that it has more generators, and thus more superpartners, and thus more particles. For instance, the n equals 2 superconformal algebra given on screen here, where GR is the superconformal generator and LR the verisora generators,
Due to the additional constraints imposed by supersymmetric generators, the number of free parameters is actually reduced, increasing the predictive power, which is like minimizing the overfitting. It sounds like because we have many more particles being predicted, that it's much more of a broad theory in terms of its predictions.
Extended SUSE leads to smaller massless particle content and further cancellation of anomalies. When you extend your supersymmetry past n equals 1, you get as a benefit more control over non-perturbative effects
and enhanced stability of the vacuum, essential for constructing consistent and stable string vacua, and phenomenologically viable models. For 10-dimensional superstring theories, we usually have either that n equals 1 or n equals 2, though you can also have differing amounts of supersymmetry on the left and the right modes, like we talked about in the heterotic case earlier. Now, you may be wondering about higher values of n
Low Energy Effective Gravity
In string theory, the low-energy effective action governs the dynamics of massless fields and connects familiar gravitational physics to the underlying string theoretical framework. Formally, the effective action is described as follows, where G denotes the spacetime metric and phi is the dilaton and H is the Neville-Schwarz three-form field strength. The dilaton field introduces the string coupling via the formula on screen, which modulates the strength of string interactions.
N equals two quantum field theories.
Extended supersymmetry, and supersymmetry in general, isn't just for string theory, but for quantum field theory. In the n equals 2 supersymmetric quantum field theory, topological invariants, like we mentioned before, there's Donaldson and Seeberg-Witten invariants, have massive roles to play. Donaldson invariants emerge from the moduli space of anti-self-dual connections in twisted supersymmetric Yang-Mills theory, while a certain twisting procedure aligns the Lorentz and R symmetry groups, resulting in a topological theory.
In the context of n equals 2 supersymmetric quantum field theories, this twisting process refers to the modification of the supercharges such that it becomes a scalar under Lorentz transformations. The partition function of the twisted n equals 2 super Yang-Mills theory localizes on the moduli of anti-self-dual connections, and the observables are given by the correlation functions of operators corresponding to cohomology classes.
Seiberg-Witten invariants, which generalize Donaldson invariants, come about from the low energy effective action of n equals 2 super Yang-Mills theory. This is governed by the Seiberg-Witten curve and the pre-potential, which encode the modulized space of vacua. These invariants are computationally more tractable and can be expressed as integrals over differential forms. Interestingly, there's a correspondence called the Seiberg-Witten-Donaldson correspondence, which relates these two types of invariants.
These invariants have connections to string theory, notably in type IIa and heterotic string compactifications, where n equals 2 quantum field theories appear on d-brain world volumes. Multiverse of the string landscape
The string theory landscape refers to the vast array of 10 to the 500, sometimes as quoted, possible vacua resulting from string theory's extradimensional compactifications, such as on collibial manifolds and even through other techniques like flux compactifications. These vacua lead to a multitude of different gauge groups, particle content, and cosmological constants for the low-energy effective field theories.
Historically, the term landscape was first used by Lee Smolin in the Life of the Cosmos book. Each vacuum represents a possible universe with its corresponding physical laws resulting in a multiverse concept. It's unknown if each of these universes exist. Are we just one of the 10 to the 500 universes? The fermionic string
The fermionic string action is given on screen here, where the gammas are the world-sheet gamma matrices, and the nabla denotes the world-sheet covariant derivative. This action is invariant under supersymmetry transformations, and thus we say it's supersymmetric. In the 1970s, this guy named Pierre Raymond, and then this other guy named John Schwartz, and then this other guy named André Neveux,
developed the Raymond-Nevus-Schwartz formulation, the RNS formulation. This implements the GSO projection, which is something that removes tachyonic and unphysical states from the spectrum. Operator product expansion. This allows the computation of correlation functions by expressing the product of two operators in proximity as a weighted sum of operators at a single point.
Here, phi denotes the primary fields, c represents the OPE coefficients, h signifies the conformal weights, as usual, and z and w denote the world sheet coordinates. The OPE significantly simplifies the calculations of amplitudes for processes in string theory by utilizing the conformal structure of the world sheet. In string theory, vertex operators correspond to string mode creation or annihilation, and their OPEs encode information about string interactions.
For instance, in balsonic string theory, the OPE of two tachyon vertex operators is given by this formula, which relates the interaction amplitude of two tachyons. Imagine each operator as a character in a story. When there are two characters, so operators interact, which means they come close on the world sheet,
Holographic Theories
One fateful year in 1997, Maldeseyna put forward a conjecture known as the ADS-CFT correspondence, which states that there's a duality between gravitational theories on an anti-de Sitter space and conformal field theories on the boundary of those spaces, generally expressed as the partition functions of each equaling one another. You can think of the partition function as a way of saying, hey, this function contains all the information about the system.
This duality provides an extremely powerful tool for studying strongly coupled gauge theories. How? By mapping them to weakly coupled gravitational theories, and vice versa. Sometimes you get a formula connecting the ADS radius, r, with the strong coupling g, the number of colors, n, and the Regislope alpha prime is given by r to the fourth proportional to a product of all of them, with alpha being squared.
This is why the ADS-CFT correspondence is said to be quote-unquote more accurate in the large n limit, where the classical gravity approximation is valid. It's also another reason why the theorists aren't terribly concerned about it being an ADS space and not a DS one, so a desider space. A desider space wouldn't have a boundary like this, at least not necessarily, but if we're taking n to infinity anyhow, then the radius goes there as well.
The hope is that there will eventually be some translation or application to decider space, thus describing our universe. For clarity, the n here corresponds to the gauge group rank, so usually it's SU2, which is n equals 2, SU3 is n equals 3, and when someone says that they're considering the large n limit, what that means is to consider numbers of n, so integers, sorry, natural numbers of n, which are far larger than say 2 or 3, even all the way up to
Celestial Holography
Celestial holography is one of the most beautiful sounding terms in all of physics. It studies encoding scattering amplitudes in asymptotically flat spacetimes onto a celestial sphere at null infinity. In other words, it's another way of looking at holography in string theory that isn't just ADS-CFT.
An integral component in this approach is the celestial spheres parameterization by conformal coordinates, omega and omega tilde, with the Mellon transformation associating bulk amplitudes with celestial correlators, given by this formula on screen here, where delta represents the conformal weights and H denotes the dimension of the local operators. The vertex operators V in the worldsheet CFT correlate to local operators on the celestial sphere within string theory, with their conformal weights defining the string states masses and spins.
Celestial holography can be thought of as a Rosetta stone between scattering amplitudes, conformal symmetry, and string theory. Historically, celestial holography emerged as an outcome of investigating the symmetries of soft theorems, which is actually a hilarious term, meaning the study of particles with momentum approaching zero.
The Mellon Transform applied to scattering amplitudes is the connection between bulk physics and conformal structures in a similar manner to how the Fourier Transform unearths connection between, say, time and frequency.
Celestial holography generalizes the BMS symmetry, which was researched by Strominger, which itself goes back to the Bondi-Metzner-Sachs group in 1962. Strominger studied this symmetry in about 2013 or so, and celestial holography can be seen as an extension of this work. The celestial sphere refers to scry plus or minus, or the future and past null light cones.
In a recent video by Sabrina Pasteurski, she contrasts celestial CFT with ADS4CFT3. The primary difference between BMS CFT and celestial holography is that the latter focuses on encoding scattering amplitudes and asymptotically flat spacetime onto a celestial sphere at null infinity, while BMS CFT is more concerned with the symmetries of soft theorems.
Super Currents
In Type II string theories, supercurrents encode worldsheet supersymmetric transformations. The supercurrent, G plus or minus, is given by this formula here, where the psi's are the worldsheet fermions, and the x's represent the spacetime coordinates, and H is the Neville-Schwartz three-form field string. These supercurrents satisfy the superconformal algebra, including the verisora algebra for the energy-momentum tensor, and the U1 current, J, with an additional anti-commutation relation.
Why is string theory so successful at producing results in other seemingly unrelated areas of math? Why is it so fruitful that entire new fields of mathematics are spawned? This is a puzzle because this usually happens with physical theories that have evidence associated with them, like quantum mechanics,
with his study of infinite-dimensional Hilbert spaces and quantum cohomology and general relativity and quantum field theory. Part of the answer is sociological, but we don't know how much of the relative pure mathematical success of string theory is because of historical reasons of, say, power and arrogance, such as those outlined by Eric Weinstein, Lee Smolin, and Peter White, or how much of this is because string theory is indeed striking at the heart of physical reality.
Layer 4 Defining String Theory
So, what is a string theory exactly? This isn't something that's asked in most string theory courses. You learn motivations, starting with the Reggi slope and then how Feynman diagram singularities can be smoothed out because you've now moved from one dimension to two dimensions. And then you start to explore more and more mathematical consequences. But few people stop to ask like, hey, when I hand you a theory, how do you know if it's a string theory?
Is it the presence of a Nambu-Gado action or that Polyakov one? Is it somehow that the tension parameter shows up? Is it any theory with extended objects even if they're more than one dimensionally extended? Sometimes this question becomes so general that it will lead even the creators of string theory to call any quantum field theory a string theory.
Actually, it would be far more accurate to say that a string theory is an example of a type of quantum field theory, where you either have strings or brains. By the way, it's unclear even what a quantum field theory is, and you can see the talks by Natty Seidberg, Dan Freed, and Nima Arkani Hamed. Those are in the description as well as they're on screen right now. The Second String Revolution
The second string revolution highlighted the non-perturbative aspects of string theory and led to major advancements. So what happened? In 1995, there was the proposal of the existence of another theory called M theory, an 11 dimensional framework which would encompass all five major string theories. Sometimes people say it will quote unquote unite them, but it's more accurate to say it encompasses them or relates them.
M theory relates to type 2a string theory via compactification on a circle with radius r. So you follow what's on screen, this formula where L11 is the 11-dimensional Planck length and LS is the string length.
M-theory, when compactified on a Z2 orbifold, also connects it to heterotic E8 x E8. The inception of M-theory can be traced back to Witten and Horová's attempts to understand the strong coupling limit of Type IIa string theory. This ignited a spark in both the physics and math community, from which our current flame is a descendant of. The pre-Big Bang scenario cosmological model
In string cosmology, the pre-Big Bang scenario suggests that time predates the conventional Big Bang, with a contracting phase followed by a diluting phase and then a bounce, leading to the observed expanding universe. There are several theories on cosmogony, something I may do an iceberg on, so that is theories of how the universe came to be and where it's going. Some suggest that time emerged from space, some suggest that both emerged from something non-spacetime-like, such as Hawking and Hartle, but today, here, we have something different.
Here it's suggested that time existed even prior to the Big Bang. This framework emerges from the low energy effective action of string theory, so given by this formula on screen, where phi is the dilaton field, H is the antisymmetric tensor field strength, and V is the dilaton potential. A key feature is scale factor duality, and also given by the transformations on screen, with A being the scale factor and eta being the conformal time.
The pre-Big Bang model describes a universe evolving from a weakly-coupled, highly-dilute state, so dilaton-driven inflation, to a strongly-coupled, hot, dense state before transitioning to the standard Big Bang epoch. Dilute, in this case, by the way, means what you think it means, namely sparse and cool matter rather than dense and hot matter.
Actually, Gabriel Venziano, the founding father of string theory, was urged by the legendary Stephen Hawking himself to consider the cosmological implications of string theory during a 1986 visit to Boston University, laying the groundwork for future developments in string cosmology. Hagedorn's Universe
Is there a maximum temperature? This is an interesting question because we think that there's a minimum temperature, namely absolute zero. So is there some finite version of absolute infinite temperature? Well, in string theory, the Hagedorn temperature, TH, signifies just this. At this temperature, a phase transition occurs, characterized by the prolific production of huge strings, called long strings, actually.
The Hagedorn temperature is given by TH equals the inverse of 2 pi times the square root of the Regi slope. You can see by adjusting the string tension, TH can be made lower than even the Planck temperature. The Hagedorn universe could be a candidate for the state of the universe before the Big Bang. In this scenario, the universe would be in a highly energetic string-dominated phase,
Interestingly, in the 1960s, Ralph Hagedorn proposed the concept of the Hagedorn temperature in the context of the statistical bootstrap model applied to hadrons before the development of string theory. Non-commutative geometry and string theory.
Non-commutative geometry, coming from the Moyle product, so the Moyle star product, replaces the usual point-wise multiplication of functions with a non-commutative product defined as follows, where theta is an anti-symmetric tensor representing the non-commutativity, and those derivatives are derivatives with respect to x and x' respectively.
In string theory, this geometry emerges due to a constant, Neveu-Schwarz B field. Historically, non-commutative geometry was developed in the 1940s by Joseph Moyle and later popularized by Elaine Konis. The Cyberg-Witten map relates the commutative and non-commutative fields by introducing correction terms, called theta corrections. The discovery of non-commutative geometry and its connection to string theory reveals unintuitive structures potentially underlying the universe.
2-0 super conformal field theory. Now this is super interesting, at least to me. If you know what a category is, then you know that you can generalize them to two categories and three categories all the way up to infinity categories. 2-0 theory is related to two categories, which involve objects, so states, morphisms, so think transformations, and two morphisms, which are higher level structures, arrows between arrows.
2-0 theory was proposed in the mid-90s, centered around the dynamics of strings within M5 brains and their interactions with M2 brains.
Specifically, it deals with strings on the boundary of M5 brains that form the edges of M2 brains. 2-0 theory employs differential forms, which will be used for representing fields, and homological algebra, for studying properties like boundary and symmetry. Mathematically, you can use algebraic structures called L-infinity algebras, which generalize Lie algebras to accept any number of inputs, rather than the standard 2.
2-0 theory is considered to be a candidate for the most symmetric field theory, potentially exhibiting a form of maximal supersymmetry. For more, watch this brief lecture by Christian Siemen, who's a professor of mathematical physics at Harriet Watt University.
F-theory is a 12-dimensional, non-perturbative framework, again, there's that ill-defined word, which generalizes type IIb string theory with varying string couplings and compactifying on something called elliptically-fibred Calabi-Yau fourfolds, not threefolds, this time it's Cy4. Vafa, the founder of this theory, also linked F-theory compactifications to M-theory compactifications on elliptically-fibred Calabi-Yau manifolds.
this elliptic vibration is given by the weierstrass equation on screen here but this time f and g are sections of a line bundle given by k minus four and k minus six on the base manifold b and kb is the canonical bundle on the base manifold
Historically, the discovery of F-theory in the 1990s by Kamranwafa and collaborators was a pivotal step toward unifying different string theories and understanding their non-perturbative behavior. Apparently the F stands for father, but I think that the F stands for F.U. Witten, my theory is better. The quantum hall effect in string theory.
The quantum Hall effect is a phenomenon observed in two-dimensional electron systems subjected to extreme magnetic fields, which leads to the quantization of the Hall conductance, where nu is the filling factor.
This effect's connection to string theory comes through d-brains, specifically through d2-brains with an external magnetic field. Why? It's because you can understand the world volume action, which is a generalization of the world sheets action, for a d2-brain by including a Chern-Simons term. Integrating this term over the brain results in a Hall conductance expression similar to the quantum Hall effects formula.
Type IIb string theory compactification on a six-dimensional manifold with a non-trivial three-flux form can give a four-dimensional low-energy effective theory resembling the quantum hall effect, where again the filling factor is related to the quantized three-form field flux. In other words, string theory gives another perspective on the quantum hall effect. Not invariance and Chern-Simon's theory.
Knot invariants, including the Jones polynomial, serve as tools to classify and distinguish knots. It seems like knots are trivial, but that's, haha, not the case. The Jones polynomial is given by this formula on screen, where k denotes the knot, and this absolute value of k represents the number of crossings.
While R is the R matrix stemming from the quantum group, so UQ of SL2C, and the UQ stands for the quantum deformation of the universal enveloping algebra. When Q equals 1, you get the regular universal enveloping algebra.
The connection between knot invariance and the three-dimensional Chern-Simons theory, which is a topological quantum field theory with the action given on screen, is partially what's responsible for Ed Whitten getting the Fields Medal, a medal which has been analogized to the Nobel Prize of math. In this action, A denotes the gauge connection, and M represents a three-manifold, and K is the level.
Through the Wilson loop operator given on screen here, which takes into account a loop in M and a gauge group representation R, we secure a link between not invariance and Chern-Simons theory. But the question is how? The vacuum expectation value of the Wilson loop operator remains invariant under ambient isotopy and can be determined via the path integral formulation of Chern-Simons theory. Now, this is remarkable.
Who would have thought that a quantization procedure, like the path integral, would have anything to do with knots? The Wilson loop can be used to detect knots and links through the lens of quantum field theory, analogously to how electric charges and magnetic monopoles interact in electromagnetism. You may even say string theorists tie themselves into knots over this. I'm here all evening, people. String Field Theory
String field theory is a background-independent approach to studying string dynamics, with its action formulated as following. Keep in mind, one of the largest criticisms of string theory from Lee Smolin in the early days is that string theory lacks a background-independent formulation. In general relativity, spacetime is the background and it has dynamics to it in a two-way street with matter slash energy.
That is, matter affects space-time and vice versa, as you've heard. When someone says that string theory is background dependent, they mean that most often what you do in string theory is you specify a background rather than you calculate one, and even when you provide one, it doesn't have the same sort of dynamics to it that one would think a quantum theory of gravity should have.
Loop quantum gravity instantiates such dynamics from the get-go, but it lacks quantum field theory, whereas string theory instantiates quantum field theory from the get-go, but lacks such dynamics. The string field, on the other hand, represents an infinite component field that encodes all string excitations, while the BRST charge operator Q makes a reappearance.
After quantization, an interaction term, here, is introduced, resulting in the string field theory potential. This potential allows us to analyze non-perturbative effects, such as the aforementioned tachyon condensation. The homotopy algebraic structure, known as the A-infinity algebra, encodes the associative star product, as well as higher homotopy products. The problem with string theory, compared to, say, regular string theory, is that it's primarily developed for bosonic strings and not superstrings,
This is partially why it's not been so popular. By the way, do you know who one of the pioneers of string field theory is? Michio Kaku! Yes, the dubious Mr. Quantum Supremacy.
BPS states allow us to understand non-perturbative aspects of string theory. These are states that preserve a portion of supersymmetry, commonly half, and have a mass formula with some bound, where m is the mass, q denotes charges, and gs represents the string coupling constant.
BPS states provide three major benefits. So number one, they're resistant to quantum corrections since their supersymmetry preservation constrains loop corrections. This ensures the stability of properties like masses and interactions despite both perturbative and non-perturbative effects. Number two, there are dual relations mapping BPS states in one theory to those in another. So Prasad and Sommerfield's original work connected BPS states to soliton solutions in around 1975.
3. They count microscopic states, allowing us to calculate the black hole entropy through the Strominger-Waffel formula given on screen, where S is the Bekenstein-Hawking entropy as we know and love and C is the central charge of the CFT and Q1 and Q5 and N are black hole associated charges.
In other words, BPS states serve as stepping stones for understanding the topography of string theory, not topology. That's a common malapropism. As you study string theory and quantum field theory at the more theoretical level, BPS states are everywhere. Sen's conjecture Sen's conjecture, proposed by Ashok Sen, concerns the behavior of these BPS states, particularly in type 2 string theories, as the string coupling gets varied.
This conjecture posits that these bps states with non-zero mass remain stable even as you vary the coupling constant. The technicality here is that you have to assume that there are no other bps states with the same quantum numbers passing through masslessness during the process. This condition is expressed mathematically as follows, with m denoting the mass of the bps states.
Twister String Theory and Cosmological Models
Penrose's twister theory can be used to compute scattering amplitudes in gauge and gravity theories, with a twister denoted a z with a superscript alpha, which conforms to the incidence relation given on screen here. Specifically, the twister z alpha encodes information about light rays in spacetime and this incident relation relates these twisters to specific points in flat spacetime, so Minkowski spacetime.
In twister string theory, the action is given on screen here, where the twisters connection to space time comes from something called the Penrose transform, which maps twisters to space time fields.
This transform is defined as something that takes holomorphic sheaf cohomology classes on twister space and maps them to solutions of massless field equations on Michalski spacetime, often expressed as follows, where the calligraphic O represents holomorphic functions on twister space and the latter part represents massless fields in spacetime.
You can liken the Penrose transform to a change of basis in linear algebra, except here we're transforming from one representation of a system, so twisters, to another spacetime fields, under a certain set of rules. Constructing De Sitter vacua in type IIb string theory, using flux compactifications, stabilizes moduli and yields a positive cosmological constant.
The use of twister theory in constructing de Sitter vacua comes from the formulation and solution of supersymmetric constraints within these cosmological models, particularly in relation to the complex geometry of the compactified dimensions, often represented as transformations in the complex structure of collibial manifolds. By the way, if you can think of other applications of twisters to string theory, then let me know. Maybe there's one for non-geometric flux compactifications. Maybe we'll write a paper together.
Yangians. The Yangian Algebra, a class of Hopf Algebras, generalizes the concept of Lie Algebras and has connections to so-called integrable systems. Formally, the Yangian Algebra is associated with a Lie Algebra and its defining relations are given on screen where j are the generators and f denotes the structure constants of the underlying Lie Algebra g and kappa stands for the level of the associated affine Kakmudi Algebra.
Think of Yangians as a natural way to connect algebraic structures with integrable systems. Within string theory, Yangians come up in the ever-popular ADS-CFT correspondence, this time connecting Type IIb string theory on a five-dimensional sphere background to N equals 4 super Yang-Mills theory
The Yangian symmetry of integrable spin chain models emerges in the planar limit of n equals 4 theory.
Spin chains are the one-dimensional version of the Ising model, and planar limits mean considering only the leading order term 1 over n in the expansion where n is the rank of the gauge group Sun. Note, one of these n's is calligraphic, and the other is not because, confusingly, they refer to different facets.
Don't shoot the messenger. Physics is abound with such pedagogical bewilderments. Anyhow, Yangian symmetry contributes to constructing scattering amplitudes, especially within the Grassmannian formulation. Here, amplitudes emerge as integrals over the Grassmannian manifold, and the Yangian symmetry imposes constraints on the integrand, streamlining the computations.
An example of a Grassmannian integral in the context of scattering amplitudes is given on screen where calligraphic A is the scattering amplitude for N particles and the Grassmannian is GR and we also have an integration measure D and a volume VOL. The PFs are faffions of certain matrices associated with the geometry of the problem. Don't worry if this looks like gibberish, we'll talk more about this when we get to the amplitude hedron. BMN matrix model.
Now we get to one of the big boys, the BMN matrix model. This guy's a proposal for a non-perturbative formulation of Big Daddy M Theory itself.
with action given on screen. Here, Xi are the Hermitian matrices representing transverse coordinates of D0 brains and the Xi are fermionic matrices related to the supersymmetric extension of the model. The action characterizes D0 brain dynamics in a specific plane wave background connected to the Penrose limits of ADS4 x S7 and ADS7 x S4 geometries.
Inspired by Berenstein, Maldacena, and Nostas' Seminal 2002 paper, which within the large end limit, the BMN model has a phase structure with different phases corresponding to different M-theory geometries.
Rigorously, this means that the BMN matrix model in the large N limit can be used to study M theory by analyzing its phase structure described by the eigenvalue distribution of matrices. These are related to the distribution of D0 brains though in transverse space.
The action can be derived from the M-theory supermembrane action via something called truncation, which I won't get into here. Put simply though, truncation means what it sounds like. It's the process of simplifying something larger to something smaller. In this case, truncation means to take a full string theory and consider only a subset of its models or degrees of freedom, making calculations more manageable. The BMN model provides a strong weak coupling duality, which you remember means S-duality.
This time it's between the gauge theory on the D0 brains and M theory in the plane wave background. This generalizes the ADS-CFT correspondence. BFSS matrix model.
This is the older, scrawnier cousin of the BMN. The BFSS provides a non-perturbative definition of M-theory by considering quantum mechanical system of N-coincident D0 brains, whose actions are described by the action on screen here, where the XIs again represent 9 Hermitian matrices,
associated with the transverse coordinates of the D0 brain, and theta is a 16-component Majorana vial spinner introducing fermionic degrees of freedom. And g is the coupling constant, of course. I say scrawnier because this model has difficulties with the definition of the ground state. It's also formulated in flat spacetime. The action displays UN gauge symmetries and local symmetries, with the positive definite bosonic potential coming from the commutator and the fermionic term representing supersymmetry.
In the large N limit, the BFS matrix model is conjectured to encompass M theory in something called the infinite momentum frame, ascribing the eigenvalues of the X matrices to the D zero brain positions in transverse space. Actually, if you like cone gauge quantized the M2 brain and add a turn Simon's mass term to the BFS, then you almost get to the previous BMN model. We talk more about both the BMN and the BFS models here in this podcast with string theorist Stefan Alexander.
IKKT Matrix Model
This is the creepy, awkward, super-genius neighbor of both of the previous two. This model is a non-perturbative definition of, again, type IIb string theory, where spacetime emerges from the dynamics of n by n Hermitian matrices. The action is given on screen, and the gamma term here are the 10-dimensional version of the gamma matrices that you learn in undergrad for quantum mechanics. Bet you didn't know you could generalize them to virtually any dimension! The model's action has Lorentz symmetry and type IIb supersymmetry.
Sometimes people say about physics theories that it has manifest Lorentz supersymmetry. I don't like the term manifest because it's an equivocal term much like saying the quote-unquote following exercise is trivial. The IKKT model was developed when its creators were investigating de-instanton contributions to type 2b superstring theory. The non-commutative geometry of spacetime is represented by commutators here which imply that the spacetime points no longer are sharply defined. Does this mean spacetime is doomed?
Well, that's something we talk about with Peter Voigt here in this podcast about string theory and spacetime. The large and limit of IKKT matrix model is conjectured to capture the full non perturbative dynamics of type two B superstring theory in 10 dimensions. So far, this hasn't been proven. And if anyone knows how, let me know again, we'll write a paper together. The low energy effective action of the model reproduces the 10 dimensional type two supergravity action supporting its connection to type two B superstring theory.
Kovanov Homology
Chern-Simon action is a tool, a powerful tool, in the study of topological invariance and gauge theories. Unfortunately, it's restricted to exactly three dimensions by construction. Kovanoff's extension seeks to generalize the theory to four dimensions using a connection B on a gerb with the action as follows on a four-dimensional manifold N. Kovanoff homology is a categorification of the Jones polynomial.
Historically, Chern-Simon's theory emerged in the late 1970s through the works of Xingxuan Chern and James Harris Simon, whose collaboration gave rise to new insights in various branches of mathematical physics. The key challenge is now to explore whether the structures of three-dimensional Chern-Simon's theory, such as Naughton Variance and Wilson Loops, can be successfully captured in a four-dimensional extension. We talk about Chern-Simon and Kovenov Homology here in this podcast with Dror Barnettin.
Black holes and higher compositional laws. In the Stu model of string theory, so the STU model, which is a specific model of compactification involving certain fields called STU, we explore the connection between extremal black holes, so those of maximum charge, and by Garva's higher compositional laws, which generalize the classical compositional laws of quadratic forms to higher degrees. U-duality orbits of these black holes are characterized by their charge vectors in tensor Z2, tensor Z2.
By the way, that's orbits in the group theoretic sense rather than in the planetary sense. It's hypothesized that these group orbits correspond to equivalence classes of something called Bargava's cubes, which are numerical representations of algebraic structures containing triples of balanced-oriented ideals, which is a specific way of structuring certain subsets of rings, in rings of discriminant D.
It may be that black hole microstates correspond to narrow class group classes. The narrow class group is a concept from algebraic number theory. It's a generalization of the ideal class group, which is a fundamental group associated with a ring that measures the failure of unique factorization of that ring. The Bekenstein-Hawking formula written on screen here with delta mirroring d escapes a full explanation, so who knows?
Fun fact, Bargava is a genius mathematician who won the field medal and grew up less than an hour away from me. Shout out to my fellow Torontonians. P.S. the STU black hole is a supersymmetric extremal solution in n equals 2, d equals 4 supergravity characterized by three independent complex scalars STU.
Its charge configuration is given by the integer matrix on screen here, where the electric and magnetic charges are denoted Q and P respectively, and the charge cube obtained via the tensor product Q tensor Q tensor Q yields a 6 by 6 by 6 integer tensor analogous to Bargava's cube of integers, which governs higher compositional laws describing the action of the arithmetic group SL2Z on integral and re-quadratic forms.
Higher Dimensional Non-Geometric Backgrounds What does non-geometric mean?
How can you have a non-geometric space? Recall that a geometric space is one where you can have a globally well-defined notion of a metric, and that your space obeys the usual differential geometric rules, such as compatibility of coordinate patches and a defined notion of parallel transport. There are some spaces that don't have these qualities, yet are spaces in their own right. Maybe they have a metric, for instance, but it's only local, not global. Maybe they have non-commutative or non-associative spatial aspects.
In string theory, we explore R-spaces and their R-fluxes and T-folds coming from non-trivial H-fluxes. The R-spaces and their associated R-fluxes are related to the geometric fluxes that come about in the process of compactification. They can be understood as generalizations of torsion in the underlying geometry,
and they play a role in determining the effective low-energy field theory. On the other hand, T-folds are a class of non-geometric backgrounds that come from compactifying string theory on manifolds with non-trivial H fluxes.
These H fluxes are associated with the three-form field strength H of the Nevo-Schwarz-Nevo-Schwarz two-form potential B. The non-trivial H fluxes can lead to interesting topological features, for example, non-geometric monodromies and non-commutative geometry, each of whom have implications for the structure of the compactified theory.
The non-geometric Q flux is characterized by the formula on screen. So what role does Q flux play? Well, it bestows our spaces their non-commutative and non-associative nature. Now the tricky part is how do you patch in a compatible fashion these local coordinate charts? How do we go about addressing this with something called double field theory that embeds doubled spacetime? It merges geometric
Algebraic K-theory
K-theory is a way to study topological invariance of vector bundles classified by growth in D groups K. The connections between K-theory and string theory come about when you classify D-brains. The RRs of the Ramon-Ramon field strength F exhibit quantization of the form, being members of the equivalence class, also the cohomology class,
where X denotes the space-time manifold commonly related to k-theory classes. There's a great introduction to k-theory and co-bordism theory in this lecture here. This is an advanced topic. Even defining what k-theory is, what a growth-indique group is, is going to take several minutes. Type IIb string theory classifies d-brains by k0 through even-rank k-theory classes, while type IIa does so by k1 via odd-rank k-theory classes.
In simpler terms, K-theory allows us to discern the difference between objects that can be continuously deformed into one another, much like various D-brain configurations. These ideas, as applied to physics, came about through the work of Atiya, Witten, and Harava, among several others. For the string theoretic context we're interested in, specific supersymmetric theories, like the Witten index, a topological invariant counting BPS states,
corresponds to the K-theory Euler class, where the dimension here represents the rank of the K-theory group. The Atiyah-Herzabrotch spectral sequence establishes the link between K-theory and ordinary cohomology, and is an active research to this day, trying to understand how D-brains behave. I'm thinking of doing an iceberg on algebraic geometry. If you would like to collaborate on it, please let me know in the comments. W-strings. W-algebras extend the verisaur algebra,
By incorporating higher spin currents. To account for these currents, the BRST charge has to be modified, where T is the stress energy tensor and CN are the ghost fields. UN are the higher spin currents and the contour integration is over some curve C. As usual, physical states are required to satisfy the BRST condition. W strings are a different type of string, coming from W algebras, but they have issues. Some of them being that they have negative norm states,
and problems with unitarity. The exact connection between alternatives of string theory, including tachyonic, baryonic, non-geometric backgrounds, and fractional strings in W-algebras isn't currently clear. The stress energy tensor in the context of W-algebras provides geometric information about the world sheet. This is at the frontier of research, even though its inception dates back to the 1980s. The failure of string theory.
The failure of string theory is something that's been talked about for decades before it was cool to dunk on string theory. Now it's all in vogue. Hey, I hate string theory. There were just four initial primary critics, Eric Weinstein, Lee Smolin, Peter Wojt, and Sabine Hassenfelder. Though you can lump several other physics and math professors there as well, they just weren't so vocal. Okay, so what is meant when people say that string theory has failed?
Okay, number one, there's a lack of experimental evidence. String theory has not provided any testable predictions that could be verified or falsified through experiments, which is a fundamental requirement for scientific theory. This is technically false. It's provided predictions that are just far out of the range of what we can currently test. Number two, a lack of connection to the standard model.
Throughout this whole iceberg so far, I was careful to say the word quantum field theory rather than the standard model. And that's because despite the hype, string theory is far from being a unification of the standard model with gravity.
Rather, it makes compatible gravity with quantum field theory. This is decidedly different. Quantum field theory is a general framework and it suffers from its own issues of inequivalent representations and not being rigorous, at least enough for the mathematicians. And the standard model itself is a far cry from being the unique quantum field theory. Number three, non-unique solutions.
Recall, there are various possible vacua that outnumber the particles in the observable universe. The academic and research environment has often favored string theory, leading to pressure on physicists to conform to this framework, at the expense of exploring other ideas.
Number five. In terms of data showing string theories decline as a litmus test, you can see the failure by the lack of Wikipedia entries in the history section of string theory, where every decade prior warranted its own section before the last two decades. Despite thousands and thousands of more people working on string theory than ever before, it only has a single entry. Furthermore, it's the smallest entry.
Number six, another data point you can use as a rough heuristic is that browsing Ed Whitten's publication record on Google Scholar, you can notice a decrease in string-related articles with time. Number seven, even Ed Whitten's collaborator Edward Frankel discusses the failure of the original promise of string theory to provide a unique theory of everything, going unacknowledged
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Layer 5 Non-BPS Brains
In m-theory compactifications on Kali-Biao threefolds, non-BPS brains are intriguing objects. They're extremal solutions, meaning that they saturate the so-called Bogomolnyi bound, and that means basically that they have a minimum amount of energy given a fixed set of quantum numbers like charge.
Some of these non-BPS brains are called non-BPS attractors and have a connection to something called the weak gravity conjecture, which I'll explain shortly, though we did talk about it earlier in the Swampland program. Unlike BPS brains or BPS states, which preserve half the supersymmetry, non-BPS brains preserve even less.
resulting in non-vanishing central charges and non-trivial scalar potentials. This quote-unquote attractor mechanism is given by the effective potential formula on screen here and governs the behavior of moduli fields ZI at the horizon of extremal black holes. Z denotes the central charge and capital DI signifies the Caylor covariant derivative and capital GI is the gauge kinetic function. An intuitive way that at least I understand non-BPS attractors
is that they can be thought of as objects that attract scalar fields to specific values in the moduli space near the horizon, stabilizing them and breaking any remaining supersymmetry. By the way, the weak gravity conjecture is the statement that gravity will always be the weakest force in any consistent quantum theory of gravity.
Black Hole-Cudit Correspondence
In recent work by Rios, extremal black holes in 5D and 6D are investigated within the framework of string theory, making use of n equals 8 and n equals 2 supergravity correspondences to find a connection between quantum states and spacetime geometry.
The key idea here is to consider extremal black holes as qudits, so a higher dimensional generalization of qubits or qutrits. And we do this through the lens of Hopf vibrations and Jordan Algebras. To be precise, Rios demonstrated that rank 1 elements in Jordan Algebras of degree 2 and 3 can be associated with qubits and qutrits respectively.
In particular, Q bits are formalized by H2O, while Q trits are formalized by H3O, and the O is the octonions. What's cool is that when you take into account U-duality groups, these transformations can be understood as quantum information theory's SLOOCC operations, acting on charge vectors Q by GQ, where G denotes the U-duality group.
So what does this mean? It means that extremal black holes can be viewed as a cosmic quantum circuit and their entropic and dynamic properties may potentially be emulated by quantum algorithms. See the paper here for more information. Dilaton and genus expansion.
The dilaton field is one of the most important fields in string theory, denoted by this phi, which is a scalar field responsible for the so-called genus expansion of string amplitudes. This genus expansion is like expanding in Feynman diagrams, except for strings.
It also determines the strength of string interactions with the coupling constant given as the exponential. The dilaton field equation expressed on screen here shows the relationship between the dilaton field and the curvature R, the Regis slope as usual, and the three-form as usual H. In the genus expansion,
string amplitudes are classified according to the topology of their world sheet surfaces with the genus G representing the number of handles or even equivalently holes in the world sheet. This can be interpreted as a perturbative series in the string coupling where each term is proportional to G to the 2G minus 2 and the first G there it's difficult to say is the string coupling constants in G
Geometric Quantization
Now geometric quantization is a method for constructing quantum theories from classical systems. So you start by identifying the classical system's phase space with a complex line bundle, dubbed the quote-unquote pre-quantum line bundle. Associating this bundle entails a covariant derivative, though you have to make use of the K-layer potential.
and the resulting curvature aligns with the phase space's symplectic form. Quantum states are viewed as sections of this line bundle, satisfying Q psi, where psi represents a section of the pre-quantum line bundle, and Q is a quantum operator derived from the classical Hamiltonian.
Specifically though, the quantum states are better described as sections of a quotient bundle obtained by dividing the pre-quantum line bundle by the chosen polarization. This is a large topic that I'll be exploring more on an upcoming podcast with Eva Miranda, so feel free to subscribe to see it as many students are only taught the Feynman path integral or canonical quantization as quantization. The Langlands Program.
The Langlands program is a broad set of conjectures in number theory and representation theory that's at the forefront of research and math, some of the most pure of pure math. That's why it's surprising that it has connections to physics, particularly in two-dimensional quantum field theories and 4D gauge theories.
See, there's something called the Langlands Correspondence. This relates automorphic representations of a reductive algebraic group G, which you can think of as the group of symmetries of a number field, to representations of its dual group G-check, which is actually another reductive algebraic group. Now, the Langlands program is so broad that it has various sub-programs like the Geometric Langlands Correspondence,
which is a version of the Langlands correspondence for curves over algebraically closed fields. This has physics quote-unquote applications, and I say that lightly because it's not clear if you can call them even applications, and it shows connections between the action of the chiral algebra on the space of conformal blocks.
This connects the representations of the loop group G hat to the space of D modules on the moduli stack G hat check, so the dual group local systems, which provides a deeper insight into the action of the chiral algebra on the space of conformal blocks. Within the scope of electric magnetic duality, the four dimensional n equals four super Yang-Mills theory provides an example of s duality, which has a specific connection to the Langlands correspondence
through the identification of the electric and magnetic gauge groups with the Langlands dual groups. I talk more about both string theory and the Langlands program here in this podcast with Edward Frankel. There's also this lecture by Ed Witten on gauge theory, geometric Langlands and all that. Link to all resources are in the description. Modular forms and string partition functions.
Modular forms originated from the work of Gauss, Riemann, and Klein. They're complex analytic functions with specific transformation properties under discrete subgroups of SL to R, usually SL to Z. In string theory, the world-cheap conformal theory's partition function, Z, must be invariant under modular transformations for consistency.
This restricts allowed compactification lattices and conformal field theories. Modular forms like the elliptic genus ZEG represent BPS states contributing to black hole microscopic degeneracy. Black hole entropy can be derived from Fourier series coefficients in these modular forms, connecting the mathematical structure and the physical properties of black holes in string theory.
Actually, Andrew Strominger and Kamran Vafa established this connection in about 1996. Interview with both of those, so Strominger and Vafa will be coming up on toe on this topic as well as on the topic of modular bootstraps and CFTs. String Sigma Models with West Zumino Witten Terms. So the WZW term is represented by the integral here, where K is the level and A is the gauge field. It acts as a topological invariant and quantizes H flux.
If you think this looks like the Chern-Simons term, you are correct. They both originate from the same structure, a three-form constructed from the gauge field. The Chern-Simons term is typically found in 3D topological field theories and is represented by the integral of a three-form, just like the WZW term. Both involve the gauge field, their exterior derivatives, and then the wedges. However, there are differences in their coefficients and the overall context in which they appear.
The WZW term is relevant for string sigma models and conformal field theories while the Chern-Simons term plays a role in the topological field theories and is associated again with the invariance and linking numbers in Jones polynomial. The WZW term can be thought of as a curvature term necessary to maintain the consistency of string theory with the central charge formula being corrected to what's on screen here where H check is the dual coxeter number of the Lie group G.
This combined action here retains conformal invariance as long as the background fields comply with the equations of motion and the WZ terms satisfy something called the Polyakov-W consistency condition. Black hole string transitions.
In a paper recently published by Maldacena and Witten in 2023, they look into the connection between black holes and highly excited strings. Actually, this revisits the self-gravitating string solutions by Horowitz and Polczynski made decades earlier. Their analysis of the linear sigma models for the heterotic strings demonstrates a smooth transition from the Horowitz-Polczynski solutions to black holes,
a connection hindered in type 2 superstring theories by differing supersymmetric indices. The entropy S of charged black hole solutions derived from these string solutions through generating techniques adheres to the relation given on screen here, where Q and P represent the charges and S0 is the entropy of the neutral solution. This is super exciting because it shows a new avenue for connecting black hole entropy, quantum states, and string theory.
JT Gravity and Black Holes We've talked about the ADS-CFT duality before, denoted here, at least up to a Legendre transformation. Something we haven't mentioned before is that there's actually a simplified two-dimensional dilaton gravity model called JT Gravity.
If I could pronounce the author's names, I would, but I can't, so I'll show it on screen. This time, we have a correspondence of ADSC2 and CFT1 that is a two-dimensional dilaton gravity model defined by the action given here where H is the induced metric on the boundary and K is the extrinsic curvature. So why is JT gravity important? Because its equations of motion are trivial in the bulk, but they are non-trivial at the boundary, meaning that we have an elementary context
for exploring holography. Its solutions encompass ADS2 black holes whose entropy can be associated with the dilaton field. Now the asymptotic behavior, again this means far away from some region of interest, of the dilaton field provides a practical regularization to quantify thermodynamic properties. Machine learning. This is a new field that has exploded in interest in the past decade.
Recall the vast landscape of string theory. Researchers are seeing how the heck can neural networks tackle the parameter space, so the 10 to the 500 proposed vacua. A notable application involves the exploration of these CY manifolds, like we mentioned before, where a machine learning algorithm predicts Hodge numbers from the input adjacency matrix of the quiver diagram of the toric diagram.
This leads to a regression problem formulated as follows, with A being the adjacency matrix and F is the learned function. Yang He, along with several others, were pioneers in applying machine learning to this field in 2017. As for F-theory compactifications, machine learning deduces the gauge group and matter content from the singularity structure of an elliptically-fibrid kalibi-yau 4-fold given as input.
Chiral Factorization Algebras
Chiral factorization algebras, which are spearheaded now by Emily Cliff, are a rigorous method for investigating quantum field theories. You've heard that quantum field theory suffers from the problem of being rigorously defined. This is only partially true. There are actually several rigorous formulations. It's just that none of them capture the full breadth of quantum field theory. A chiral algebra is a vertex operator algebra, V, that fulfills the operator product expansion relation on screen here.
for both v of w and v of z that are members of this vertex operator algebra.
This association has singularities with simple poles at most, embodying locality in chiral conformal field theories. So, how do factorization algebras fit into this? Well, firstly, they're a generalization of vertex operator algebras, and secondly, they provide a systematic, ground-up methodology to develop conformal field theories. For a VOA-V, the related factorization algebra F of V
assigns a state space to each interval i in R. Factorization maps are such that we have this relation here, which almost looks like an exponential property for disjoint unions i1 and i2 within i, and these preserve the OPEs. By exploiting the world-cheap conformal symmetry, holomorphic and anti-holomorphic factorization of correlators is achievable, thus reducing calculations to one-dimensional conformal field theories. Geometric Unity
When thinking about string theory, it's useful to think about alternatives. Usually, loop quantum gravity is proposed as the primary contender, but that's only a contender in the quantum gravity stage, not on the toe unification stage. That is to say, it's not clear how loop quantum gravity is a unification of general relativity and the standard model.
That whole Toll Unification stage is a decidedly different stage than the Quantum Gravity one, and there are not many combatants on it. Wolfram is one such combatant, Peter Wojt is another, Garrett Lisi is another, Eric Weinstein is another, with his Geometric Unity approach.
Usually Eric explains it as a theory where the four-dimensional space-time that we know and love is not fundamental but rather emergent, but I think that's doing geometric unity a disservice. One of the reasons that I like geometric unity is because it takes seriously as a primitive a four-dimensional manifold which is then used to construct other unfamiliar structures and familiar ones. Geometric unity is quite intricate and can well have its own iceberg.
But what other structures? Well, the Observer's, for instance, which is characterized by a triple X4, Y14, and embedding into a higher dimensional Riemannian monofold. These embeddings are local Riemannian and induce a metric on X4, thus generating a normal bundle. At some point, you choose a signature, which then gives the so-called chimeric space Y7,7.
The main principle bundle in GU is as follows, where the first guy is the double cover of the frame bundle of the chimeric bundle, H is the unitary group of 64,64, and this row, this variation on row, is the representation of the spin group on complex Dirac spinners.
From this, you get what looks like space-time spinners and internal quantum numbers. There are other arguments for recovering bosonic particles as well. Often in the discussion of toes is the discussion of Grand Unified Theories, or GUTs, but just so you know, GUTs aren't toes. However, there's one GUT called the SU10 model, or the Georgi-Glashow model. There's also the SPIN10 Georgi model, and there's a SPIN4 x SPIN6 Patissala model.
These all have significance in GU, with the number 10 here being related to the 10 degrees of freedom in the four-dimensional Romanian metric. Geometric unity is quite intricate and can well have its own iceberg. Non-critical strings Non-critical strings deviate from the critical dimension, which is 10 as we know for superstrings and then 26 for bosonic strings.
They're related to the cancellation of conformal anomalies, which is a different type of anomaly we haven't discussed. To study non-critical strings, random matrices are usually introduced. Consider the random matrix ensemble on screen here, where M is an N by N Hermitian matrix, V of M is a potential function, and lambda is a coupling constant.
This ensemble is a discretized world-sheet action for non-critical strings with M representing discretized world-sheet fields and V of M encapsulating string interactions. In the 1980s, the study of non-critical strings using random matrices led to the discovery of the double scaling limit by Bresen, Edzicson, Parisi,
The scaling behavior of the matrix model near critical points exposes properties of non-critical strings such as string susceptibility, which is determined by the specific heat exponent alpha via the relation that gamma equals 2 minus alpha.
To maintain conformal invariance in non-critical strings, Louisville theory is used, and we do so by introducing a Louisville field, phi that couples to the world sheet curvature, effectively compensating for the deviation from the critical dimension. By the way, Louisville theory is something you use to maintain conformal invariance when working with dimensions different from the critical dimension. Type 0A and 0B.
Tachyonic states are characterized by an imaginary mass and faster than light propagation, though this only happens if you interpret as a particle and if you interpret the coupling constant as being mass. In bosonic string theory, for instance, the mass squared of a string state is given as follows, where n is the excitation level and a is the normal ordering constant.
It turns out that in addition to the five flavors of string theory that you know and love, there are several more, two of them being these type 0a and type 0b, but these are characterized by these flagecious tachyons, as well as because they describe only bosons, thus you hear little about them. Fractional Strings and Non-Integer Conformal Weights
Fractional strings are strings characterized by non-integer mode numbers. This means that the strings' vibrational modes don't conform to simple harmonic patterns. Conventional conformal weights result from the normal ordering of the verisora generators L0 and then L bar 0, with integer conformal weights tied to the quantization oscillators.
However, for fractional strings, we have non-integer conformal weights which defies typical quantization. We have to re-examine string spectra and world sheet symmetries because of these, if we're to take them seriously. The modified conformal weights can be discerned through the formula here for H, where K squared signifies the spacetime momentum,
and M stands for the fractional mode number, and alpha prime is the, again, the Reggi slope, the traditional verisora constraints are impacted, culminating in the updated conditions, which are on screen here. Now here's the question, what the heck could a fraction of a harmonic mean? It's not clear to me how to visualize them. Fractional strings aren't studied anywhere near as much as regular strings, which is, again, why you haven't heard of them.
When people talk about the five flavors, always keep in mind, we're talking about vanilla, chocolate, strawberry, mint, and cookie dough. But those aren't the only flavors. There's also, hey, there's peanut butter cup, something that the toe logo looks like, by the way. Unconventional twisted heterotic string theory.
Unconventional twisted heterotic string theory is a different approach than usual to heterotic string theory has been proposed by introducing twisted boundary conditions using a twist operator omega as an automorphism of the world sheet satisfying that omega squared equals one.
which acts on the left moving sector by modifying the oscillators alpha mu n to omega for all n and mu. The twisted action is given by simply a sum of both the right and the left one, though the left one now has a twistedness in it. So the twisted left moving action is derived by replacing the conventional oscillators with their twisted counterparts. By choosing specific twist operators, different massless spectra and gauge groups can be obtained.
The Schurck-Schwarz mechanism is a historical example of a twisted string theory applied to the breaking of supersymmetry, and this requires compactifying an extra dimension with a twist. To ensure world-cheap conformal symmetry and consistency, the choice of this twisting must commute with the BRST charge, allowing quantization of the twisted heterotic strings via the familiar BRST cohomology. Monstrous M-theory in 26 plus 1 dimensions
In monstrous M-theory, a recent extension of the standard M-theory to 26 plus 1 dimensions by Chester, Rios and Morani, the massless spectrum of M-theory is shown to have connections to the so-called monster group.
This is what we discussed earlier in the Monstrous Moonshine Conjecture. The deep origins or motivation for the decomposition of the Greece algebra into 98280, direct summed with 98304 and then we have 1, was unknown to Conway. Mirani realized that one of these middle factors, 98304, is a would-be quote-unquote gravitational, so a spin one and a half field.
which is typically found in supergravity. The 98280 was understood to be half the leech lattice, possibly like a Z2 orbifold or the identification of positive roots. The 1 is, of course, the dilaton. The new approach suggests an n equals 1 spectrum in these 27 dimensions or an n equals 2 spectrum in 26 dimensions, so 25 space dimensions.
Analyzing M-theory for n equals 1, so minimal supergravity, in this spatially odd dimensional setting, isn't simple. The moduli space geometry is linked up with the monster group's complexified elements, analogous to the vertex operator algebra representations.
This fusion of the largest sporadic groups representation theory with high energy physics potentially reveals new symmetries in spacetime and I'm excited to see where this research goes, especially as I personally don't know of many applications of the Greece algebra to physics. By the way, if you're wondering about how did they get around Nomm's theorem, they found a way with nested brain worlds. So this no-go theorem applies only when you reduce down to three plus one dimensions.
Double field theory is a t-duality approach to string theory, where you augment spacetime by doubling its dimension, combining the winding and momentum modes of strings into double coordinates, where x tilde and x represent signifying winding and momentum modes, respectively. The DFT framework involves a double metric,
that sees the conventional metric and the B field as equal. The DFT action is the following, where phi represents the dilaton as usual and r is the Ricci scalar in the double dimension geometry and 2D is the doubled space-time coordinate count, where D is the initial count.
Although the DFT action respects generalized diffeomorphisms, incorporating transformations that blend both Xi and tilde Xi, a stringent constraint must still be instituted for consistency and to retrieve the standard string theory by curtailing these degrees of freedom to the initial dimension count, though this is still being debated today. DFT serves as a geometric method to grasp T-duality, its so-called unifies diverse string theories under a shared framework.
Loop Quantum Gravity LQG is a non-perturbative, background-independent quantum gravity framework reconciling quantum mechanics and general relativity.
It employs Ashtakar variables representing the gravitational field via an SU2 connection A and its conjugate E. The last one is called a densitized triad. Spin networks are graphs with vertices labeled with an intertwiner I and edges by irreducible representations J of SU2 form the foundation of loop quantum gravity.
Just like the motivation for string theory, loop is also quite simple mathematically speaking, and also, like string theory, its humble beginnings, belie its subsequent tortuous flowering. Loop quantum gravity creates the Hilbert space basis of gravitational field quantum states, each representing a quantized three geometry for the three spatial dimensions. Discrete spectra come about for area A and volume V operators.
Transition amplitudes between spin networks originate from spin-foam evaluation, modifying the famous quantum field theoretic path integral technique. Loop quantum gravity was developed or discovered, depending on your philosophical framework, in the 80s by Ashtakar, Rovelli, and Smolin,
Plenty of work was also done in the 90s by John Byas as well. To this day, it's seen as an antagonist to string theory, but Lee Smolin told me in a recent podcast just last week that string theory and loop quantum gravity are two sides of the same coin. Layer six. Quantum entanglement.
One of the most astounding subjects in modern PopSci is quantum entanglement, with its ostensible faster than light signaling. Let's explore this by starting with entropy.
If we take the von Neumann entropy, where rho is the reduced density matrix, then the holographic entropy, which includes the Ryu-Taki-Yanagi formula as a special case, connects the entanglement entropy with the area of a minimal surface gamma. This relationship gave rise to the so-called ER equals EPR conjecture or heuristic, whatever you want to call it. But what does this mean? It suggests that entangled pairs of particles are equivalent to wormholes.
Now, if that wasn't remarkable enough, it has the further implication that space-time geometry itself emerges from the entanglement structure of underlying quantum states. But what about that firewall argument? That one that suggests a breakdown of the equivalence principle at the black hole event horizon due to maximal entanglement?
The firewall argument, well, it was proposed by four researchers named Almherry, Mirov, Polchinsky, and Sully, abbreviated as Amps, raised concerns about the validity of ER equals EPR.
According to Amps, a black hole that's maximally entangled with another system, for instance Hawking radiation, can't also be entangled with its own interior, as that would violate the so-called monogamy of entanglement principle. Consequently, the smooth spacetime structure near the horizon, as predicted by general relativity, would break down, and the observer would experience a firewall instead.
This argument has led to this huge debate among physicists, with some proposing possible resolutions such as the soft hair proposal by Hawking, Perry and Strominger, or the idea of state dependence, which states that the experience of an observer falling into a black hole depends on the specific quantum states of the system. This is all fascinating and highly speculative. Let me know if you'd like me to do an iceberg on black holes.
A rigorous analysis of string field theory in the context of non-commutative geometry necessitates the introduction of the Moyel star product into the action given on screen here. The Moyel star product is defined by the following where A is the algebra of the functions on the phase space. Here this theta is a constant anti-symmetric matrix that characterizes the non-commutativity of spacetime coordinates and f and g are again the functions on phase space.
The Moilstar product is an associative but non-commutative product that generalizes the usual pointwise product of functions on phase space in the context of non-commutative geometry. What the Moilstar product does effectively is to deform the commutation relations of the spacetime coordinates and the corresponding fields. This leads to a modification of the usual commutation relations, propagators, and interaction vertices.
In non-commutative spacetime, coordinates satisfy the following algebra. This is supposed to capture some of the fuzziness of spacetime at the string scale. Non-commutative geometry, though, has its roots with mathematicians like Elaine Konis, John von Neumann, and Marie Gerstenharber, all of whom explored it in different contexts before it found its application in string theory.
The appearance of the star product in the scalar field's kinetic term, mass term, and interaction term actually comes from the Seeberg-Witten map that we talked about before, which in turn comes from the open string low energy effective action of the noncommutative scalar field. Quantum Groups and String Theory
Quantum groups are denoted as follows with this U and the subscript Q of a Lie algebra G and what they are non-commutative deformations of the universal enveloping algebra of the Lie algebra with a deformation parameter Q. Now the defining relation for certain generators is AB equals QBA.
The R matrix, which satisfies the Yang-Baxter equation, encodes the non-commutativity with the defining relation on screen here. Importantly, quantum groups retain the structure of Hopf algebras, allowing a description of both algebra and co-algebra actions.
In the limit when q goes to 1, quantum groups reduce to their classical counterparts, both in terms of Lie algebras and Lie groups. By the way, Hopf algebras are algebraic structures that simultaneously generalize groups, associative algebras, and Lie algebras. How so?
They have two algebra maps, so a co-product here, which encodes the algebraic structure, and a co-unit, which encodes the identity element of the group-like structure. Hop algebras also possess something called an antipode map, which provides something like the inverse of the group-like elements, and they satisfy this relation on screen here. Their relevance to string theory originates from integrable systems and conformal field theories through the underlying world sheet CFT and its quantum group symmetry.
The connection to braid groups comes from the R matrix, which describes the braid properties of tensor categories associated with those quantum groups. For rational CFTs, the fusion rules, which are given by the Verlinde formula, can be derived using quantum group representations relating conformal weights of primary fields to representation labels.
The pair with the lovely names Drinfield and Jimbo independently introduced quantum groups in the 1980s primarily to investigate integrable systems. Drinfield was also mentioned in the book with Edward Frankel and again the Edward Frankel podcast is on screen here. Love and Math is the book. Exceptional Field Theory
This is a geometrical scaffold housing varied representations of string theory and 11-dimensional supergravity, employing the terminology of exceptional Lie groups and their corresponding geometry, which are the exceptional in the name exceptional field theory. Its action is on screen here, where G is the EFT metric, D is the dilaton, and H is a measure of the three-form field string. With capital D adopting different values in this exceptional field theory, multidimensional flexibility is apparent.
The exceptional Lie groups transform into global symmetry groups, resulting in exceptional geometries. Now, while EFTs don't unite all string theories, it explores them as specific sectors corresponding to unique solutions of the EFT equations of motion. Amplitude Hedron
The Amplituhedron is something that Donald Hoffman readily brings up, so it's useful to have an explanation here. Donald has been interviewed several times on this channel before, once solo, with the technical exploration of his theories, another with Joscha Bach, one with John Vervecky, another with Bernardo Kastrup and Susan Schneider, and yet another one with Philip Goff.
The topics usually center around consciousness, though here we'll talk about Nima Arkhani Hamed's amplitude hydron. What this is, is a specific type of convex polytope within RK that encodes scattering amplitudes in N equals 4 supersymmetric Yang-Mills theory. This is realized by a relationship with the positive Grassmannian. This means it's a space of K by N matrices with positive minors.
Mathematically, the amplituhedron is induced from a mapping of the positive Grassmannians under a specific positive map given on screen here, where the map is defined by taking the positive Grassmannian to the amplituhedron via a linear map as follows, with the constraint that all k plus 1 minors of C are non-negative.
Scattering amplitudes can then be computed via integration over the canonical form of the amplitude hedron, providing a way that avoids some complexities of some Feynman diagrams. The amplitude hedron is connected to string theory through that good old celebrated AdS-CFT correspondence relating N equals 4 super Yang-Mills theories to type 2b super string theories in an AdS5 cross S5 background.
It should be specified that it's the scattering amplitudes rather than the amplituhedron itself that connects to this correspondence, with the convex polytope being this calculational tool like a middleman. In 2013, the amplituhedron was introduced by Nima Arkani-Hamed and his collaborator Jora Slav,
Partially inspired by the study of ancient math objects called a sociahedra. These date back to the 1960s and appear in various branches of mathematics including algebraic topology and combinatorics. The amplitude hedron is appealing because it suggests that we might not need fields.
Double copy theory.
The double copy theory establishes a remarkable correspondence between gauge and gravity theories through something called KLT relations, where gravity amplitudes can be expressed as the square of Yang-Mills gauge theory amplitudes. I like this phrase poetically, but for me it should be expressed a bit more rigorously because, at least for myself, when I hear that the Dirac equation is the square root of the Klein-Gordon equation, or that spinors are the square root of some other structure, personally just confuses me more until I see the math.
When we say that gravity amplitudes are the quote-unquote square of the Yang-Mills amplitudes, we mean that the gravity scattering amplitude can be obtained as a product of two Yang-Mills scattering amplitudes with a modified kinematic substitution given on screen here. This corresponds to the closed string amplitude being constructed from the open string amplitude in the KLT relations. This, by the way, links closed and open string amplitudes.
The color kinematics duality requires that the kinematic numerators satisfy the same Jacobi identities as the color factors. Following this duality, if we have CA equals CB plus CC, then NA is defined as NB plus NC. This allows us to express the graviton scattering amplitude as the square of gluon scattering via something called the BCJ double copy construction. This encompasses the KLT relations which were discovered in the late 1980s.
I forgot to mention that we also have to enforce momentum conservation given on screen here. Now this entire double copy theory is interesting to me because it produces a significant reduction in computational complexity for scattering amplitudes and gravity theories while still drawing connections between gauge and gravity theories similar in spirit to what the amplitude hedron did. You plane integral.
The low-energy effective action for the Type IIb string theory on K3 surfaces relies heavily on the evaluation of U-plane integrals. Recall that K3 surfaces are smooth, compact, complex two-dimensional manifolds with a trivial canonical bundle and a holonomy group SU2, and they're important because of their role in supersymmetry, mirror symmetry, and Calabi-Yau manifolds in compactification.
In this context, the U-plane is the moduli space parameterized by the complex coupling constant, where this theta represents the Raymond-Raymond scalar field and gs denotes the string coupling constant. The BPS states describe the spectrum of stable configurations in the theory. By the way, I've heard other names for the U-plane like the S-duality orbit or the Coulomb branch or the Seiberg-Whitten moduli space and the moduli of vacua.
The integrand takes the form of an exponential multiplied by d and f, where the integer n and the degeneracies d of n specify the bps spectrum, and the modular forms f of k capture the automorphic properties. To evaluate U-plane integrals, you have to use something called the Rademacher expansion. Now, I probably butchered that, and at first I thought that was the same Rademeister as in the moves, but it's something different.
This Rademacher expansion expresses the modular forms as a sum of Poincare series, which allows us to isolate pertinent information from the integrand as follows, where S represents the Clustermann sum and S is a modular parameter. The U-plane integrals are connected with the Mach modular forms, a class of non-holomorphic modular forms, generalizing the classical Eisenstein series, not Einstein, but Eisenstein, and that links number theory and geometry to string theory.
M Theories and Multiple Dimensions of Time. We usually talk about 10 plus 1 dimensions of space-time or 3 plus 1, etc. There's always this plus 1 at the end. This means it's one-dimensional. However, there is work by bars that has two dimensions of time. But what does this mean mathematically? So mathematically, the concept of multiple time dimensions are captured by extending the metric tensor to include extra temporal components
Or you may see it as x squared plus y squared plus z squared minus t squared. It just has extra minuses after it. In Barr's work, he introduces a second spacetime coordinate, t prime, described by a d plus two dimensional spacetime. You can take this even further to discuss 3D time in the same way that we discuss 3D space.
How? In the context of extending super Yang-Mills theories through exceptional periodicities, this recent work by Rias, by Chester, by Morani, they consider the super algebra in D equals 27 plus 3 dimensions.
The descending dimensional sequence from a super algebra in D equals 27 plus 3 to 26 plus 1 reduces the dimensions directly along an 11 dimensional brain world volume yielding an N equals 1 super algebra in D equals 11 plus 3 which upon successive dimensional truncation aligns with the N equals 1 super algebra in D equals 10 plus 1 and D equals 11 plus 1 as well as type 2A to B strings.
This suggests an 11-dimensional brain world volume origin for string dualities in both M and F theory, though this time with signature 11-3. Wilson surfaces and loop space connections
These guides provide insights into the underlying symmetries and structures of M-theory. In particular, Wilson surfaces generalize the concept of Wilson loops, expressed as follows, which represent the parallel transport of particles in gauge fields. In M-theory, Wilson surfaces describe higher dimensional extensions and interactions with M-brains, such as M2-brains coupled to the 3-form potential C3, and M5-brains coupled to the 6-form potential C6.
Wilson surfaces are expressed similarly as before, where Cn are the n-form potentials and sigma is a p-dimensional sub-manifold. The loop space connections, denoted by alpha, are introduced as calligraphic alpha equals a plus b2 plus c3 plus so and so on, where Roman a is the usual gauge connection and b2 and Cn are higher-form connections.
Loop space connections build on Wilson loops. How? They extend parallel transport to deal with extended objects looping through space. This helps us understand the non-perturbative features of M theory as well as it's meant to reveal more dualities. Arithmetic Geometry
In arithmetic geometry, one studies algebraic varieties over number fields and zeta functions, like the Haase-Wei zeta function. These zeta functions contain information about the distribution of rational points and other geometric invariants, such as those talked about by one of the Millennium Prize problems, the Birch-Swinerton-Dyer conjecture. Though, this conjecture refers specifically to the rank of an elliptic curve group and the order of vanishing of its associated L function.
in String Theory Compactifications and the Arithmetic Properties of Zeta Functions. This is heavily related to the discoveries in mirror symmetry in 1991 by physicists and mathematicians Candelas, Assa, Green, and Parks. See this talk here about the Langlands and arithmetic quantum field theory, though this is not about string theory. Categorical Symmetries
In higher category theory, categorical symmetries come from an abstraction of traditional symmetries represented by group actions. Let's focus on two groups. So mathematically, a two group is viewed as a strict monoidal category, with all objects and morphisms being invertible. In symbols, a two group is a collection of objects and morphisms and multiplications and inversions and identities, where there are only two objects, so G0 and G1.
Categorical symmetries come up in string theory through higher gauge theories, which describe extended objects like D-brains and M-brains, and these D-brains can be associated with gerbs, they can be associated with two categorical generalizations of line bundles, and twisted versions of ordinary bundles. Their transition functions are described as elements of the automorphism 2 group of the principal U1 bundle, not just BU1.
Recall that a BU1 is defined as the classifying space of U1 bundles, so in other words BU1 is the same as U1, except you mod out by contractible spaces on which U1 acts freely. Historically, categorical symmetries originated from John Biases and Jane Dolan's study of higher-dimensional algebras in the 1990s.
M5 brains have categorical symmetries. Why? Their self-dual three-form is governed by a three-categorical structure, specifically through the two connection components as follows. The three-form field strength H is induced by a two connection on a gerb, with A being a one-form connection and B is a two-form connection, such that the field strength can be expressed as follows, with F equals dA. The role of three algebras is also useful in describing world-sheet dynamics.
Higher spin gravity.
If you listen to this podcast, you'll hear me say often that it's not so clear gravity is merely the curvature of spacetime. Yes, you heard that right. You can formulate the exact predictions of general relativity, but with a model of zero curvature with torsion, non-zero torsion, that's Einstein Cartan. You can also assume that there's no curvature and there's no torsion, but there is something called non-matricity. That's something called symmetric teleparallel gravity. Something else I like to explore are higher spin gravitons.
Higher spin gravity theories are characterized by massless fields with spin greater than 2, such as Vasilyev's higher spin gravity in ADS-4. The action for these theories has a form similar to the above, where H and phi represent the higher spin fields. These theories possess infinite dimensional gauge symmetries, but so does general relativity, given you ordinarily consider the diffeomorphism group. So how is this different than usual?
The difference lies in the types of gauge transformations and the structure of the gauge fields. In higher spin gravity, gauge transformations are associated with tensor fields of higher rank, so S-1, while general relativity involves vector fields. Therefore, it exhibits that conjecture duality with certain large n CFTs with higher spin symmetries, such as the O-n vector model CFT. Historically,
Franz Dahl's work during the late 1970s and early 1980s laid the foundation for higher spin gravity, notably with his equation for massless fields of arbitrary spin. In some ways, you can think of this as allowing for more ways to quote-unquote wiggle in space-time rather than the regular two degrees of freedom of ordinary gravity theories. The Atiya-Singer Index Theorem
This index theorem is a landmark result in differential geometry and topology. What it does is compute something called the analytical index of elliptic differential operators, and by doing so, shows the connection between the topology of a manifold and the solutions of partial differential equations on it. An analytical index is the difference between the dimensions of the kernel and the co-kernel of an elliptic operator.
and elliptic differential operators are linear partial differential operators that satisfy a certain condition called the ellipticity condition which guarantees the existence of solutions and good estimates for their behavior expressed as follows for large psi where p is called the principal symbol of the operator
meaning the highest order homogeneous part of the operator in local coordinates, and psi is a point in the cotangent bundle. Because of this, elliptic operators have favorable properties such as the existence of smooth solutions and well-posedness. In string theory, the theorem has found application in establishing anomaly cancellation conditions when applied to the elliptic Dirac operator on the worldsheet. The index is associated with the topological invariance of the worldsheet like the Euler characteristic
and the Herzabrutsch signature, through the following expression, where A is the A-roof and L is the L-genus of the manifold X, and CH is the churn character of the relevant bundle. The A-roof is defined as the Pfaffian of the curvature form, divided by the Pfaffian of the tangent bundle, so this expression on screen here, and the Pfaffian is a polynomial function associated with a skew symmetric matrix, such that the square of the Pfaffian equals the determinant of the matrix.
Modulize Stabilization
Recently, a paper was published by Bassiori, which provides non-perturbative terms in the superpotential and the combined effects of logarithmic loop corrections and two non-perturbative superpotential, Caylor moduli-dependent terms. How so? They, the authors, derive the following effective potential, which takes into account both the perturbative and non-perturbative contributions, where a, b, c, psi, and eta are coefficients that depend on various parameters of the theory.
So what does this mean, Kurt?
Well, my friend, the result shows that fluxes exist for large and even moderate volume compactifications, which defines a decider space and stabilizes moduli fields.
So, why is this important, Kurt? Well, this is an important finding because it demonstrates the existence of stable, decider vacua in type 2b string theory, which was previously known to be extremely challenging. The obtained effective potential appears to be promising for cosmological applications, such as cosmological inflation models, understanding dark energy, and the universe's expansion.
Dark energy.
Dark energy is about the expansion of the universe. Some think it's as simple as, well, it's just the cosmological constant, and others think it has to do with more mysterious modifications of the laws. The study of string cosmology is about examining string theory's implications on the universe's evolution, including dark energy and accelerated expansion. Let's consider the low energy effect of action,
By now you should be familiar with these symbols, but for those who skipped around and want a refresher, that capital G is the metric, the dilaton field is phi, the NSNS3 form strength is H, and F is the RRP form field strength, and the lone G is the determinant of the metric.
By compactifying extra dimensions to four-dimensional spacetime, you get a 4D action and a scalar potential which is affected by these fields. This leads to something called a quintessence-like dark energy scenario. Quintessence is a scalar field with a potential responsible for the accelerated expansion of the universe
Is string theory the flashlight we need to illuminate the dark corners of the universe? Ambatwister String Theory
You've heard of twisters, but have you heard of ambitwisters? What are they? Well, they generalize twisters by considering the complexified phase space of null geodesics instead of Minkowski spacetime. The ambitwister space is a huge space that contains twister space as a subspace. Ambitwister string theory is a framework that uses both twister and ambitwister spaces to describe scattering amplitudes of massless particles.
The worldsheet action is expressed as follows, with A and P being auxiliary fields related to the twister variables. Conformal symmetry, which of course is present in conventional string theory, is also there in ambitwister strings.
What's the difference? Their target space comprises the space of complex null geodesics rather than just regular space-time. The CHY formula gives a compact representation for tree-level amplitudes of massless particles expressed as integrals over the moduli space of punctured Riemann spheres.
This can be understood as a considerably efficient method of representing many particle interaction outcomes. Sir Roger Penrose's pioneering work on twister theory in the 1960s laid the groundwork for ambitwister strings to emerge decades later. Although ambitwister string theory simplifies scattering amplitudes, encoding soft limit
Non Archimedean Geometry
There's another field called the P-adic numbers. So, P-adic numbers are defined as equivalence classes of Cauchy sequences of rational numbers converging with respect to something called the P-adic norm. Now, just as there's non-Euclidean geometry, there's also something called non-Archimedean geometry. The P-adic numbers, denoted as Q with the subscript P,
form the completion of the rational numbers Q with respect to the Pyatik valuation, augmenting Q by incorporating something like digits and infinite amount of digits to the left, rather than to the right, as we're conventionally used to. Pyatik string theory was originated by Volovich in the 1980s, and it embeds the string worldsheet into Pyatik spacetime using the adapted Polykov action. Notice the Pyatik norm here. This allows invariance under Pyatik reparameterizations and Weyl transformations.
Piatek string amplitudes have factorization properties similar to their Archimedean counterparts, allowing for Piatek analogues of Veneziano and Verasoro-Chapiro amplitudes.
Tachyonic condensation occurs in the peatic setting, giving a non-perturbative description of D-brains. The Adelic product formula, associating products of amplitudes with certain topological invariants, hints at connections between peatic and Archimedean string theories, although this remains wonderfully speculative. Another physical theory that involves the peatic numbers is the so-called invariant set theory by Tim Palmer, which suggests that the universe evolves on a fractal attractor.
Inumerative Geometry
Topological string theory has applications in enumerative geometry, particularly through the use of Gromov-Witten invariance. Now, those are those correlation functions that count the number of holomorphic curves within a Kolob-Yau manifold weighted by their genus G and homology class C that we talked about approximately an hour ago. These invariants are computed in the A model, so the symplectic one, and the B model, so the complex one, for topological string theories.
They give information on the modular space of Kali-B.Yau manifolds, Yukawa couplings, and string theory compactifications, and what's important for this topic, the intersection number for counting problems in enumerative geometry. In other words, rational curves on a quintic threefold. Gromov-Witten invariance generalized classical intersection theory. Symplectic modular symmetry in heterotic string vacua.
Ishiguro, Kabayashi, and Otsuka recently examined the unification of flavor, CP, and U1 symmetries coming from symplectic modular symmetry in the context of heterotic string theory on Kolibiao 3-folds. They found that these symmetries can be unified into the symplectic group's modular symmetries of Kolibiao 3-folds with H being the number of moduli fields. Together with the Z2-CP symmetry, they're enhanced to this group here, which is the generalized symplectic modular symmetry.
They have S3, S4, T' and S9 non-abelian flavor symmetries on explicit toroidal orbifold with and without resolutions on Z2 and S4 flavor symmetries on three parameter examples of collibial threefolds.
Layer 7.
Congratulations on making it this far. Now we're in the deepest layer in one of the most thorny subjects, not only in physics, not only in math, but in all fields imaginable. It's useful to understand the math of string theory, even if string theory ends up missing the mark, because the problems being addressed here are problems at the heart of the physical universe. However, of course, you shouldn't mistake in the physical universe as being synonymous with reality. This is a point that Hilary Putnam makes.
Despite this, understanding string theory gives you a bedrock at the fount of reality, the reality that can be established mathematically and logically. Let's get on with the iceberg. This is a grueling problem in physics. We often assume that there is such a correspondence, which is just yet to be found rigorously, but even defining it rigorously is formidable.
Further, there are nine major problems. Number one, mapping between gravitational and field theory configurations. The issue is to find an exact dictionary that conjoins gravitational states with the states of the boundary conformal field theory.
When you have configurations with less symmetry, it's not clear how to do this. Number two, ADS space as a regulator for flat space physics. The use of ADS space as a regulator to extrapolate to flat space physics involves taking the limit where the ADS radius of curvature R goes to infinity. This process, while keeping the local physics unchanged, isn't fully developed, especially in understanding how the ADS boundary conditions translate to flat space observables.
Number three, holography in light-like boundaries. Understanding holography for light-like boundaries, as in the case of Minkowski spacetime, differs significantly from time-like boundaries typical of ADS CFT. The existence and nature of large end limits for theories that aren't gauge theories and for theories with less or no supersymmetry isn't anywhere as developed either. Number five, sub-ADS locality.
How do you understand the emergence of bulk physics at scales smaller than the ADS radius? The solvable models we have currently of holography don't capture the locality expected from gravity in the bulk, which should be evident at scales much smaller than this radius. Number six, time evolution.
The bulk reconstruction techniques developed so far primarily address static or equilibrium situations. The dynamical evolution of non-trivial states, particularly those involving black hole formation and thermalization, aren't well understood.
Bulk operators must be dressed gravitationally to be gauge invariant, but the precise nature of this dressing in context with significant back reaction isn't fully understood as well. Dressing in this context, by the way, means incorporating the influence of gravitational fields generated by the operator itself on its definition, ensuring to feel morphism and variance. This is particularly relevant for operators that couple strongly to gravity. Number eight, entanglement wedge reconstruction.
So the conjecture that the boundary subregion R is dual to the entanglement wedge W, rather than to the causal wedge CR, raises the question about the reconstruction of these bulk operators. The entanglement wedge can extend beyond the causal wedge, potentially including regions behind horizons, which complicates the understanding of bulk locality and the encoding of bulk information in the boundary theory.
By the way, the entanglement wedge refers to the region of spacetime in the bulk that can be reconstructed from boundary subregion entanglement, while the causal wedge is the bulk region causally connected to that boundary subregion. And lastly, number nine, black hole interior. The description of the black hole interior is still an open problem in ADS-CFT. What is the existence of firewalls? What is the fate of an infalling observer? We don't know.
fuzzballs and the microstructure of black holes. The fuzzball proposal in string theory suggests that black holes possess a microstructure composed of stringy excitations or fuzzballs which replace the classical event horizon as well as the singularity. This stems from the correspondence between black holes and d-brain bound states. It's an attempt to describe the near horizon geometry using the dual conformal field theory.
To translate that a tad, the Fuzzball conjecture replaces the mysterious core as well as the edge of black holes with information storing strings. The Bekenstein-Hawking formula agrees with the degeneracy of these Fuzzball states, accounting for the microstates that generate the black hole entropy. The Fuzzball conjecture was first proposed by string theorist Mathur and his collaborators in 2002. You'll hear this term plenty, microstructures and microstates.
To be specific, the microstructure generally refers to the arrangement of string excitations that comprise the black hole's interior, while microstates are these distinct field configurations that these excitations can take, each one corresponding to a unique quantum state. Essentially, they represent the different ways that strings can
vibrate or be bound together within the fuzzball giving rise to the black holes entropy now how do you generalize these fuzzballs not only to non-extremal black holes but to other broader classes of black holes this is an open problem also what's the exact mechanism for retrieving information from these fuzzballs we don't know but the answer to these can help resolve the black hole information paradox so good luck
Achieving background independence in string theory remains a large unsolved problem, but it's not as unsolved as it was a decade ago. There's more and more progress about background independence results in certain scenarios. For instance, this recent lecture a few months ago by Ed Whitten. But why is this such a confounding conundrum? Well, it's because string theory's perturbative roots demand a predefined background.
However, Kurt, what if you incorporate the Poisson tensor derived from the Polyakov action? Does that not allow for curved backgrounds? Not exactly. Accommodating dynamic backgrounds is different than merely curved backgrounds. It requires a non-perturbative foundation for string theory. But Kurt, what about matrix models or the generalizations like tensor models or higher spin holography? Great point!
You are on it today. The issue is extending those results to a more general setting. And just so you know, a universally accepted non-perturbative definition remains unfound. This was one of the major critiques of one of the earlier Lee Smolin books of string theory. By the way, a podcast with Lee Smolin was just released about a week ago. Check the description or click subscribe to get notified. Pure Spinner Formalism
In superstring theory, an alternative to traditional Raymond Neville Schwartz and Green-Schwartz formalisms exist and they're called pure spinner formalism. So, what makes this PSF different? The formalism employs what are called pure spinners, which are a special class of spinners being self-dual and annihilated by a maximal isotropic subset of gamma matrices N.
The formalism also simplifies calculations, especially for higher loop amplitude, using the simpler BRST charge given on screen. Now this baby girl is less complicated than her counterpart in RNS formalism. The pure spinner space can be constructed as a quotient of the common spinner space that you know and love, by the maximal isotropic subspace, represented mathematically here.
where D signifies a Dirac spinner and N is the null subspace. These spinners enable a covariant quantization of the super string, eliminating the oddities of the picture-changing operators as well as ghost fields. Nathan Berkovitz birthed the pure spinner formalism in his pursuit for more symmetric solutions to super string theory constraints. Waterfall fields and hybrid inflation.
In new work published in just 2022, which by the way is only a blink of the eye in this field, Antonietas, Lacombe, and Leon Torres presented a cosmological inflation scenario within the framework of type 2b flux compactifications. What makes their work different? They used three magnetized D7 brain stack.
The inflation is associated with a metastable decider vacuum and the inflation is identified with the volume modulus. The authors propose that the inflation ends due to a waterfall field, which drive the evolution of the universe from a nearby saddle point toward a global minimum with tunable vacuum energy.
This tunable vacuum energy could potentially describe the current state of our universe. The authors detail their model, including the implementation of what's called hybrid inflation, also the analysis of open string spectrums, and the dynamics of the waterfalls on this decider vacuum and inflation. The authors conclude that their model successfully implements the main principles of hybrid inflation.
The introduction of these waterfall fields in this model is a pioneering mechanism for driving the universe's evolution from a metastable decider vacua to a global minimum, potentially even explaining dark energy. String Net Condensation and Emergent Spacetime
This is a mechanism in topological quantum field theory. String nets suggest that space-time isn't fundamental but comes from something pre-geometric in condensed matter systems. Elementary excitations in a lattice such as spins and qubits form string-like structures that, when condensed, lead to phase transitions.
The ground state of a topologically ordered system is described by a superposition of string net configurations with the string net wave function given here where L denotes the string label on edge E and delta is the branching rule at vertex V. The emergent spacetime geometry is a result of collective string net behavior. So you may ask, where does the metric come into play? The metric emerges from interactions between string nets and their corresponding tensor networks.
The low-energy excitations resemble particles in a 3-plus-1-dimensional spacetime as the emergent gauge fields in gravity are realized via fusion and braiding of anionic excitations in the system. Anions are exotic quasi-particles in two-dimensional systems. The emergent gauge group structure relies on anion fusion rules, while emergent gravity stems from topological entanglement entropy.
Whether this is how the world works or not, this gives new tools for those studying the building blocks of space time. Eclectic flavor groups. This is a brand new area of research. The best resource I found was this 2020 open access article on screen here. Eclectic flavor groups combine traditional discrete flavor symmetries with modular flavor symmetries.
They analyze a model based on the Delta 54 traditional flavor group and the finite modular group Sigma Prime 3, resulting in the eclectic flavor group given on screen here. Keep in mind that it's called eclectic and not electric. I made this mistake at least 10 times when writing the script because of pesky muscle memory. This scheme is highly predictive, constraining the representations and modular weights of matter fields and hence the structure of the scalar potential and super potential.
the superpotential and scalar potential transform under the eclectic flavor group such that they combine to an invariant action. Discrete R-symmetries emerge intrinsically from the eclectic flavor groups and this model's predictive power is showcased by the severe restrictions on the possible group representations and modular weights for matter fields, which in turn control the superpotential and scalar potential structures.
The Caylor potential is Hermitian and modular invariant with leading contributions given by the standard form and additional terms suppressed by the volume of the orbifold sector. Because of the connection between R symmetries and modular transformations within these eclectic flavor groups, this research may provide insight into discrete symmetries in string compactifications. O Minimal Structures
Originally introduced by Louvain d'Andries in the 80s, these O-minimal structures are a way of simplifying the topology of semi-algebraic sets. The key idea is to break down any definable set in an O-minimal structure into a finite number of cells, so basic building blocks like intervals and their higher dimensional analogs. You can do so by following the cell decomposition theorem.
In string theory, considering the moduli space of Kali-Biao manifolds, more explicitly on screen here, where CYN represents the set of all Kali-Biao n-folds in O-minimal structures calligraphic O and M sub-calligraphic O denote the corresponding moduli space. This is brand new research and the best paper I found is by Grimm on taming the landscape of effective theories, that is, using O-minimal structures to explicate the Swampland.
String Universality String Universality is the conjecture that every consistent quantum gravity theory corresponds to the vacuum of some string theory or string theory compactification. It's based on the fairly braggadocious belief that string theory encompasses all possible quantum theories of gravity, at least within certain conditions like a fixed number of dimensions and certain amounts of supersymmetry.
We can symbolically represent this conjecture as follows, where QG is the space of all consistent quantum gravity theories and the calligraphic STV is the space of all string theory vacua and this is a surjective map. This conjecture is part of a broader set of ideas known as the Swampland program that we talked about earlier. In fact, string universality is seen as the endpoint
of the Swampland program, where string theory is the ultimate quantum theory of gravity. But, you may ask, what about loop quantum gravity? Recall, loop quantum gravity is a non-perturbative and background-independent approach, which attempts to quantize gravity directly by focusing on the geometric and topological aspects of spacetime. Importantly, it does not rely on supersymmetry, which is a key ingredient in many of the string theoretic constructions.
Now, advocates of string universality would just argue that, hey, loop quantum gravity is not a complete nor consistent quantum theory of gravity, or some may say it will eventually be subsumed by string theory anyhow. This is a point that Ed Witten made in a recent book called Conversations on Quantum Gravity. But what does it mean to have a consistent quantum theory of gravity?
I find it helpful to replace the word consistent with non-pathological, because to me consistency has a particular mathematical logic meaning, and quantum field theorists don't use the word consistency in this sense. The pathologies that I refer to could be violating any one of the following, so unitarity, which you can think of as conserving probability, causality is another one which you can think of as no faster than light propagation of information or communication.
String Theory and the Search for Aliens
String theory's extra-compactified dimensions raise questions, such as whether unconventional biochemistry, including potentially higher-dimensional life, may exist. Let's clarify that this connection is extremely speculative, far from what can be tested currently scientifically. At least, we think so. Now, there is the case to be made, as Lee Smolin does, that we may already possess data to answer such questions, and it's staring us right in the face we just lack the theoretic understanding to interpret the data.
Brains can be seen as generalizations of strings, as you well know, given that you're now at Layer 7, serving not only as the boundaries where these strings terminate, but also as fundamental, multi-dimensional structures in their own right. Could other advanced civilizations be making use of these spaces either for faster-than-light travel, or constructing wormholes for slower-than-light travel but vast-distance travel?
Or even as places for their own existence. Can you manipulate local vacuum states to create pocket universes? Interestingly, Alexander Westfall, a string theorist, gave a talk 10 years ago to SETI, the academic organization behind the search for extraterrestrial life.
It was about the string theory landscape that we talked about near the beginning of this iceberg. Each quote-unquote bubble universe in this multiverse may have different fundamental properties leading to a proliferation of possibilities for the emergence of life. There may even be avenues for communication. String Consciousness You've heard of Penrose and Hameroff's idea that the same mechanism responsible for quantum gravity
is twinly responsible for consciousness. It's known as orchestrated objective reduction and we've covered it here on this podcast with Hameroff himself. If this is the case, and if it's also the case that we have string universality, which connects all quantum gravities to string theory, then it's not so far-fetched to conjoin string theory and consciousness. Questions of consciousness such as the hard problem and the so-called problem of other
I'm skeptical because it's going to become part of physics. Yet, of course, whatever you think about consciousness, it's an important part of us. I don't know how we perceive anything including physics.
And that has to do, I think, with the mysteries that bother a lot of people about quantum mechanics and its applications to the universe. So quantum mechanics kind of has an all-embracing property that, to completely make sense, it has to be applied to everything in sight, including ultimately the observer. But trying to apply quantum mechanics to ourselves makes us extremely uncomfortable, especially because of our consciousness, which seems to clash with that idea.
Consider Carl Jung. In one sense, what Carl was doing was psychology but in another sense, what he was doing was attempting a rudimentary form of the physics of the mind. That is, what are the natural laws that govern the psyche?
You may say, hey, well, they're not mathematical, and so they don't count as the same sort of laws, and that's exactly right. What's also true is that before Newton and before Kepler, before anyone who placed mathematics at the fount of the world, there were hundreds of years of philosophizing with imprecise language and models of the times about nature, such as Thales, a pre-Socratic Greek philosopher, who suggested that water is the origin of all things and the lodestone has a soul.
The Search for a Final Theory
Is the unification of general relativity with the standard model the last stumbling block in the reductive search for regularities at the sustentation of the world? Do we have to solve every major physics problem such as the matter-antimatter asymmetry? Do we live in a privileged place in the universe? Should the final theory, if it's meant to be a theory of indeed everything in the literal sense, explain consciousness or purpose?
Should a final theory be able to explain even itself? What does it even mean to explain? How essential is mathematical beauty or simplicity in guiding us? Where does the direction of time fit in? Not to mention initial conditions and boundary values. Would a final theory also tell us which interpretation of quantum mechanics is correct?
Is the notion of causality to be redefined, even abandoned? Is it the case that the true theory of everything is by definition unfalsifiable, and thus the final theory is one that lies outside the purview of Popperian science? What about what lies outside in principle observation, like singularities? What about observation itself? Where do you fit in?
These questions are ones that date back decades, even millennia. We simply don't know. I certainly don't know. But on this channel, Theories of Everything, each of these are explored in extreme detail, as rigorously as we can. The universe is just waiting for someone like you to take a crack at it.
Alright, congratulations. That was a strenuous exercise. I'm sure at least it was for myself. String theory is a fascinating and deep rabbit hole. Personally, I loved learning about string theory. The past few months that I've spent working on this video has invigorated me, even if I'm not sold when people say that string theory has elicited new math and that's some justification or testament to it being on a more correct
I don't buy that, but I have found it incredibly fun, absolutely loved it. It's wonderfully engrossing in the same way that some people find listening to Beethoven is engrossing. Now I'd say I'm a neophyte in this all and if someone wants to collaborate with me then please comment the word collab, c-o-l-l-a-b. This way I can control f
and find others who want to work on icebergs I have several ideas for instance the extraterrestrial iceberg explained or the free will iceberg or the iceberg on theories of time or the consciousness iceberg or the iceberg of entropy or the iceberg of causality several several ideas your comments below will help me prioritize because these take months to make literal months I would like to thank at this point
All the editors, there were four of them. So that's Prajwal, Colin, Akshay, and most of all, Zach. Thank you, thank you so much. A combined hundreds of hours, 400, I believe, by the time this is done. And that's not including the hours that I put in myself in the editing and the writing and the rewriting and the voiceovers and then changing. And then, hey, I know it may seem that looking this good is just effortless for me. And it is, it is. I'll be honest.
There will be a correction section in the description because there are bound to be several notational mistakes, even verbal ones simply the omission
of a word or the addition of an extra syllable that shouldn't be there. Anyone who's edited a video for months knows that it can all just look like white noise at a certain point, like static. If you're confused, make sure to ask a question in the comments and I will respond or someone else will respond. There are other topics I wanted to cover here like Wolfram's theory. I ran out of time. I also wanted to do asymptotic safety and what it means to have negative dimensions of space.
I also wanted to cover string quantum field theory, which isn't exactly string theory, but for more on this, see the work of Lucas Cardoso. But just so you know, there is a whole podcast with Wilfrum on his theory of everything. It's on screen here. If you're interested, there are two, as Wilfrums appeared at least twice, actually three times on this channel.
There are four ways of supporting me. If you choose to, you should know that I do this out of pocket. There's no major funder. There's no connections that I have. Unfortunately, I get bitter about it because sometimes I look at other podcasters or other video creators who have friends who are in high places who connect them with other guests and connect them with other connections. And I'm just here lonely in Toronto like an umbratic weasel.
But if you would like to support theories of everything to make more content like this, then there are four ways. So there's PayPal for direct payments, like one time payments. There's crypto for the same reason. There's Patreon, which is monthly. And then now there's also you can join here on YouTube monthly. Thank you so much for staying with me for two hours, maybe two and a half. I'm unsure how long this will end up being, but it's been a blast. Take care.
Okay, now on to some brief channel updates. Stick around for the next minute as they may concern you. Firstly, thank you for watching, thank you for listening. There's now a website, curtjymongle.org, and that has a mailing list. The reason being that large platforms like YouTube, like Patreon, they can disable you for whatever reason, whenever they like.
That's just part of the terms of service. Now, a direct mailing list ensures that I have an untrammeled communication with you. Plus, soon I'll be releasing a one-page PDF of my top 10 toes. It's not as Quentin Tarantino as it sounds like. Secondly, if you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like helps YouTube push this content to more people like yourself
Plus, it helps out Kurt directly, aka me. I also found out last year that external links count plenty toward the algorithm, which means that whenever you share on Twitter, say on Facebook or even on Reddit, etc., it shows YouTube, hey, people are talking about this content outside of YouTube, which in turn greatly aids the distribution on YouTube.
Thirdly, there's a remarkably active discord and subreddit for theories of everything where people explicate toes, they disagree respectfully about theories and build as a community our own toe. Links to both are in the description. Fourthly, you should know this podcast is on iTunes, it's on Spotify, it's on all of the audio platforms. All you have to do is type in theories of everything and you'll find it. Personally, I gained from rewatching lectures and podcasts
I also read in
There's also crypto. There's also just joining on YouTube. Again, keep in mind it's support from the sponsors and you that allow me to work on toe full time. You also get early access to ad free episodes, whether it's audio or video. It's audio in the case of Patreon video in the case of YouTube. For instance, this episode that you're listening to right now was released a few days earlier. Every dollar helps far more than you think. Either way, your viewership is generosity enough. Thank you so much.
▶ View Full JSON Data (Word-Level Timestamps)
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"text": " The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how they impact markets. They recently published a piece on China's new neutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science they analyze."
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"text": " Where senior editors argue through the news with world leaders and policy makers in twice weekly long format shows. Basically an extremely high quality podcast. Whether it's scientific innovation or shifting global politics, The Economist provides comprehensive coverage beyond headlines. As a toe listener, you get a special discount. Head over to economist.com slash TOE to subscribe. That's economist.com slash TOE for your discount."
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"text": " Welcome to the Iceberg of String Theory, a technical edition. The Iceberg format is one where you initially explore preparatory, surface-level concepts, then progress ever more into the intricacies of a topic, which tend to be known only to a specialized few, until eventually you arrive at the"
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"text": " obscure dark frontiers of the deepest layers of the field. On the Special Theories of Everything podcast, we're going to be exploring string theory like you've never seen it before. You'll learn more about the hinterlands of this field in the next two hours than you will watching say 20 hours of Michio Kaku documentaries or Neil deGrasse Tyson rants."
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"text": " Why? Because you'll be shown the actual math instead of hand-wavy, metaphoric explanations that leave you slack-jawed, deracinated from the equations, and even misinformed. My name's Kurt Jaimungal, and on Theories of Everything, I use my background in mathematical physics from the University of Toronto to explore unifications of gravity with the Standard Model and have also become interested in fundamental laws in general as they relate to explanations for some of the largest"
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"text": " philosophical questions that we have, such as what is consciousness? How does it arise?"
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"text": " In other words, it's a peregrination into the all-encompassing nature of the universe. We'll cover the abstruse math of string theory, black holes, as well as other toe frameworks like geometric unity and loop quantum gravity. This episode took a combined 300 hours across four different editors and several rewrites on my part. It's the most labor that's gone into any single theories of everything video."
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"text": " Layer 1 Types of String Theory"
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"text": " In string theory, there are five so-called consistent formulations or flavors. There's Type I, there's Type IIa, Type IIb, Heterotic SO32, and Heterotic E8 x E8. Type I string theory is characterized by open and closed strings with the gauge group SO32, coming from something called the Chan-Paton factors at the endpoints of the open strings."
},
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"text": " Type IIa and Type IIb are both closed string theories, with Type IIa being non-kyro and Type IIb actually being kyro. The heterotic string theories are based on a hybrid of 26-dimensional bosonic string theory and a 10-dimensional superstring theory, also resulting in closed strings. Heterotic actually means hybrid. You can always sound clever to someone studying string theory by saying, oh, do you study heterotic strings? They'll respect you exactly 3% more."
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"text": " open and closed strings. As mentioned before, there are broadly two types of strings, open strings, which have endpoints, and then there's closed strings forming loops. This formula on screen is specific to closed strings and accounts for additional properties such as the winding number w and momentum n in a compactified space."
},
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"text": " Compactified spaces are something that we'll explore later, so don't worry if this terminology confuses you. R is the compactification radius, and alpha prime is called the Regislope. These will come up over and over. By the way, the Regislope is related to the so-called string tension. All of these we'll discuss in detail later. M Theory"
},
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"start_time": 292.244,
"text": " The five flavors of string theory are related through something called dualities, such as T-duality connecting type IIa with type IIb, and then there's S-duality linking type I with heterotic SO32. The fact of these dualities is what spurred the idea of M-theory, which is an 11-dimensional unifying framework encompassing all five string theories,"
},
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"start_time": 313.097,
"text": " rather than a 10-dimensional theory. It does so by introducing a new type of brain called a membrane, which we'll talk more about later. By the way, when someone says that string theory is in 10 dimensions, they actually mean 9 plus 1, so 9 spatial dimensions and 1 time dimension, and when they say it's 11-dimensional, they mean 10 plus 1. The reason is they're usually talking about space-time dimensions as a whole."
},
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"end_time": 349.36,
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"text": " Apparently the M in M-Theory stands for Matrix or Membrane or Mystery or Mother, but I think it stands for an upside-down W for Witten, much like how the W of Wario is an upside-down M for Mario. Left and Right Moving Strings"
},
{
"end_time": 375.52,
"index": 16,
"start_time": 350.435,
"text": " String modes are characterized by their oscillations along the worldsheet and are described by the Polyakov action on screen. Notice I keep saying mode and not vibration. That's because no string theorist talks about vibrations unless they're being condescending to a lay public. Generally they speak about modes. Or even spectra, which are distinct states of a string, each with their own quantum number like energy, charge, mass, spin, winding number."
},
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"text": " If you introduce light cone coordinates, which you'll see on screen as sigma plus and minus, again on the world sheet, then you can separate that Polyakov action into left and right moving components, leading to the Fourier expansions on screen. It may sound confusing, but this is akin to decomposing a complex function into a real and imaginary part. The energy momentum tensor, TAB, also decomposes into left and right moving components, T plus plus and say T minus minus."
},
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"end_time": 426.903,
"index": 18,
"start_time": 402.483,
"text": " which generates something called the Verasoro algebra. This allows us to classify, or label, our states into conformal weights. Personally, I like to denote the right-moving conformal dimension with a tilde, as there's already one too many h-bars in physics. There's another subtlety here of matching left and right mode numbers in order to preserve Lorentz covariance, but the iceberg must go on. Gravitons."
},
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"start_time": 428.422,
"text": " Gravitons are the hypothesized massless spin two particles said to be responsible for gravitation. They come from rank two tensor field perturbations H mu nu. Now all of that's a mouthful, but I could have just said it's a massless spin two particle. Why? Because there are theorems by Weinberg and others that suggest the particle associated with gravity would have those properties. And furthermore, any particle that's massless, chargeless and spin two would be the particle of gravity. Thus their equivalent."
},
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"text": " You get this by linearizing the Einstein field equations around a flat background metric, giving the equation on screen. This is the aforementioned perturbation. It should be mentioned that no one has observed a graviton, and furthermore, we have great reasons to believe that we never will, even in principle. This is a point that Freeman-Dyson makes. Thus, it's unclear if the graviton is even a scientific concept in the Popperian sense. Dualities in string theory."
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"text": " You'll hear T and S dualities discussed frequently. T dualities are transformations like R moving to something like being proportional to the inverse of R, where R represents the compactification radius, and alpha prime is the Regi slope. This connects type IIa and type IIb superstring theories. How so? It maps a type IIa theory on a circle of radius R to a type IIb theory on a circle of radius alpha prime over R, and vice versa."
},
{
"end_time": 539.189,
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"start_time": 514.206,
"text": " S-duality, on the other hand, explores the equivalence between weak and strong couplings in string theory. It's particularly evident in the SL2Z invariance of type 2B string theory, which acts on a complex parameter combining the string coupling GS and certain Raymond-Raymond fields. By the way, I've heard this pronounced Raymond, I've heard this pronounced Raymond, I'm just gonna stick with Raymond. This makes it only slightly more difficult than taking the inverse of GS."
},
{
"end_time": 555.606,
"index": 23,
"start_time": 539.189,
"text": " This S-duality links type I string theory with heterotic SO32 theory, which then gives insights into non-perturbative string dynamics, since strong couplings are useful for non-perturbative studies and weak ones for perturbative. The other S and T-dualities are shown on screen."
},
{
"end_time": 572.073,
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"text": " S-duality hinges on incorporating elements like D-brains and oriental folds. Basically, you can think of both of these dualities as inversions. One inverts the coupling strength, and the other inverts the radius of the extra dimensions after compactifying. Heterotic Strings"
},
{
"end_time": 601.954,
"index": 25,
"start_time": 573.456,
"text": " Confession, I deceived you earlier out of kindness and love when I told you that there are five flavors of string theory that are 10-dimensional. There are actually several more than that. One of them, even the original string theory, is 26-dimensional and only described bosons, not fermions. And most of the matter that we see is fermionic, where the bosons are there to allow interactions between them. Heterotic string theory combines left-moving bosonic modes from the 26-dimensional bosonic string theory."
},
{
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"index": 26,
"start_time": 602.227,
"text": " with right-moving fermionic modes from a 10-dimensional superstring theory. This abomination is gotten to by compactifying 16 additional dimensions in the left-moving sector on an internal lattice, resulting in two consistent heterotic theories, the SL32 and the E8 cross E8. But what do we mean by consistent here? What do we mean by super here? Is a superstring a string that's been bitten by a radioactive spider? We'll explore that in one of the deeper layers. The short answer is yes."
},
{
"end_time": 642.568,
"index": 27,
"start_time": 630.196,
"text": " With more detail, let's use some notation on screen here to represent the root lattice. So here's VE8 of the E8 Lie algebra, and the heterotic theories are defined by their lattices through this construction."
},
{
"end_time": 665.606,
"index": 28,
"start_time": 642.995,
"text": " You'll notice here that 26 equals 10 plus 8 plus 8, though you'll also notice that 26 does not equal 10 plus 32. The reason is that you don't deal with the group SO32 directly. You don't even deal with its 16 dimensional root lattice. Instead, you deal with the weight lattice of spin 32 modded out by Z2. Regislope"
},
{
"end_time": 689.07,
"index": 29,
"start_time": 667.381,
"text": " The Regi slope, denoted by A', is a fundamental parameter in string theory that relates the mass squared of a string to its angular momentum J through the linear Regi trajectory written on screen. Sluskin covers this in the first lecture on string theory at Stanford, and the link to that is in the description. This trajectory represents the spectrum of excited string states as a relationship between mass and angular momentum."
},
{
"end_time": 704.684,
"index": 30,
"start_time": 689.07,
"text": " Personally, I think that the word trajectory is misleading since it implies that something is moving through space, but rather this is a plot of an observed pattern of quantum numbers. The Regi slope is inversely proportional to string tension, T, with A' equaling something proportional to the inverse of T."
},
{
"end_time": 725.555,
"index": 31,
"start_time": 705.282,
"text": " For those quantum field theorists interested in scattering amplitudes, the Regi trajectory comes from the analytic continuation of the amplitude into the complex angular momentum plane, where the physical region corresponds to the poles of the amplitude. This means considering angular momentum as a complex number, rather than just an integer or a half integer as a standard."
},
{
"end_time": 748.422,
"index": 32,
"start_time": 725.555,
"text": " World Sheet Symmetry The world sheet of a string is the two-dimensional surface that a string sweeps out in space-time shown on screen here."
},
{
"end_time": 763.114,
"index": 33,
"start_time": 748.985,
"text": " The basic symmetries of the worldsheet include reparametrization invariance and then something called Weyl symmetry. Be careful not to call it Weyl symmetry. If you do, string theorists will respect you exactly 3% less."
},
{
"end_time": 791.118,
"index": 34,
"start_time": 763.66,
"text": " Reparametrization invariance means you can choose whichever coordinates you like on the world sheet and it won't affect the physical predictions. Vial symmetry is the rescaling of the world sheet metric, which is a feature in conformal field theories or CFTs. CFTs are something that we'll explore next. Recall the Polyakov action here. This little guy is invariant under both reparametrization and Vial transformations. Conformal symmetry and the Polyakov action."
},
{
"end_time": 813.353,
"index": 35,
"start_time": 792.79,
"text": " Conformal symmetry means when you scale the metric, you preserve angles. This means that while your volume can change, like the volume of a circle, the shape doesn't, like the shape of the circle is the same. The fact of this symmetry allows us to simplify calculations in that juicy polyakov action before. The G here is the regular spacetime metric that we know and love, and the H is the worldsheet metric."
},
{
"end_time": 837.637,
"index": 36,
"start_time": 813.353,
"text": " This symmetry leads to a vanishing of the trace of the angular momentum tensor, yielding the verisaural constraints, which are important when talking about the so-called string quantization. Conformal symmetry also allows us to decompose into holomorphic and anti-holomorphic correlators, thus reducing calculations into far simpler one-dimensional CFTs."
},
{
"end_time": 859.701,
"index": 37,
"start_time": 838.933,
"text": " Conformal symmetry classifies string states by conformal weights and ghost numbers. Ghosts are particles that are supposed to be unable to be detected, but are necessary for calculations. The physical states are determined by the BRST cohomology, satisfying the following conditions for the BRST charge Q, ensuring invariance under the worldsheet symmetry transformations."
},
{
"end_time": 875.862,
"index": 38,
"start_time": 859.701,
"text": " Layer 2"
},
{
"end_time": 900.247,
"index": 39,
"start_time": 877.125,
"text": " Here, I'd like to make a note. Depending on your background, much of this math may sound unintelligible. It may sound like gibberish. That's okay. It's important not that you drink from the fire hose, but rather that you merely get wet. In other words, don't feel dismayed if you don't understand Spanish from the get-go. Rather immerse yourself in Spain, for instance, and that, along with a bit of practice, will advance you."
},
{
"end_time": 916.715,
"index": 40,
"start_time": 900.247,
"text": " Even John von Neumann said the point of math isn't to understand it, but rather to get used to it. You would think that Fields' medalist Richard Borchards would need only a single book to understand commutative algebra, but instead he had to learn about it not from one source, not from two, not three, but eight."
},
{
"end_time": 941.596,
"index": 41,
"start_time": 916.715,
"text": " Universe on a Brain Theory"
},
{
"end_time": 970.845,
"index": 42,
"start_time": 942.995,
"text": " Our universe could exist as a three-brain on a higher dimensional bulk, with standard model particle physics confined to the brain, while gravity extends into the extra dimensions. In string theory, d-brains, also known as derrishly brains, serve as endpoints for open strings. The action for a d-brain is given by the Dirac-Born-Infeld action, shown on screen here, where T is the brain tension, and gamma is the induced metric, and the calligraphic F is the field strength tensor."
},
{
"end_time": 997.432,
"index": 43,
"start_time": 971.169,
"text": " However, more general brains do exist, such as Newman Brains, which allow strings to move off the brain. The Randall-Sundrum model exemplifies the brain-world scenario, with two three-brains embedded in a 5D anti-decider spacetime, where one of these brains represents our universe. The RS metric is given on screen, where k is the adias curvature scale, r is the compactification radius, and phi is the extra-dimension coordinate."
},
{
"end_time": 1007.927,
"index": 44,
"start_time": 997.875,
"text": " The RS model addresses the hierarchy problem by localizing gravity near the standard model brain or the quote-unquote visible brain, resulting in a large hierarchy without fine-tuning."
},
{
"end_time": 1034.309,
"index": 45,
"start_time": 1008.439,
"text": " In this framework, the Planck scale is transformed into the TeV scale by the warp factor given by this decaying exponential on screen, which gives a so-called natural explanation for the large disparity between the two. By the way, I pronounce the Dirac equation Dirac and not Dirac equation because I just can't help but think about Dwayne Johnson writing a hyperbolic PDE. String Cosmology and Inflation"
},
{
"end_time": 1063.114,
"index": 46,
"start_time": 1035.555,
"text": " String cosmology is a framework for investigating inflationary models. Compactification schemes such as Calabi-Yau manifold, orbifold, and flux compactifications, all of which we'll talk about later, impact virtually every cosmological quantity. How? The moduli field from these compactifications influence the dynamics of inflation in string-inspired scenarios, like the large volume scenario with Kaler moduli and the radial dilaton."
},
{
"end_time": 1091.954,
"index": 47,
"start_time": 1063.541,
"text": " Dilatants we'll talk about later. And if you're interested in the low energy effective action that's on screen, the inflationary potential is V. Cosmic strings. String theory predicts cosmic strings, F and D term inflation, and axiom monodromy models. Contrary to what people say that string theory has no predictions, these actually do yield testable predictions on the tensor to scalar ratio R, on the scalar spectral index M, and on non-Gaussianicities."
},
{
"end_time": 1121.561,
"index": 48,
"start_time": 1092.619,
"text": " The tricky part is that the predictions vary, meaning they're not falsifiable. Cosmic strings are essentially a thin line stretching across the universe, which may have formed during phase transitions in the early universe. Think of them as cracks in space of concentrations of energy. Actually, cosmic strings may have been found recently, but this doesn't mean string theory is correct. Why? Because despite the name, cosmic strings are predicted by several other theories, not just string theory. String gas."
},
{
"end_time": 1142.688,
"index": 49,
"start_time": 1123.08,
"text": " An alternative to inflation is something called string gas cosmology, which focuses on the thermodynamic properties of string gases and the Hagedorn temperature, usually denoted by TH. If you see my tutorial masterclass on undergrad physics in two hours, which is linked in the description, then you'll see why I'm fond of this tilde notation rather than the approximate notation."
},
{
"end_time": 1172.005,
"index": 50,
"start_time": 1143.302,
"text": " There's a problem in cosmology called the horizon problem, which is why the CMB is so uniform, as well as the flatness problem, which is why our universe is basically flat. String gas cosmology attempts to address both at once with a quasi-static Hagedorn phase. For fractal-like scale invariant spectra of fluctuations, the specific dynamics and interactions of the string gas during these Hagedorn phases are important. Verisaur algebra, symmetry algebras and infinite generators."
},
{
"end_time": 1199.616,
"index": 51,
"start_time": 1172.722,
"text": " The Virasora algebra is a central extension of the Witt algebra, which is the algebra of infinitesimal conformal transformations in two dimensions. The commutation relations are given on screen here, with LM being the generators, C being the central charge, and of course, M and N are integers. This infinite dimensional algebra encodes the symmetries of the world sheet under conformal transformations, reflecting the structure of the two dimensional conformal field theories, or CFTs."
},
{
"end_time": 1217.398,
"index": 52,
"start_time": 1199.974,
"text": " The algebra's representations are characterized by the eigenvalues of L0, known as the conformal weights, delta. In string theory, the central charge is related to the spacetime dimension D via this formula, and this, by the way, only applies in certain contexts like bosonic string theory. Otherwise, there are other relations."
},
{
"end_time": 1235.845,
"index": 53,
"start_time": 1217.688,
"text": " The infinite generators of the Virasora algebra, indexed by LM, are used in the construction of vertex operators, which describe the interactions of strings and are subject to something called the operator product expansion in CFT, which we will expand on more later. Quantum Yang-Baxter equation."
},
{
"end_time": 1263.336,
"index": 54,
"start_time": 1236.664,
"text": " The quantum Yang-Baxter equation is an equation in integrable systems, specifically in quantum integrable models, generalizing the classical Yang-Baxter equation, which shows up in soliton theory. Given by this unruly formula on screen, it comprises intertwiners, which is what those Rs are over there. And these intertwiners are invertible linear operators, which act on tensor products of quantum spaces. Each of those lambdas denotes a spectral parameter."
},
{
"end_time": 1293.268,
"index": 55,
"start_time": 1263.78,
"text": " The quantum Yang-Baxter equations are seen in statistical mechanics, quantum groups, and knot theory. Regarding statistical mechanics, it enables the construction of integrable lattice models, such as the six vertex model via the algebraic beta ansatz, offering exact solutions for correlation functions and their thermodynamic properties. Actually, Edward Frankel talked about the beta ansatz in this podcast on this channel, Theories of Everything here, among other topics like consciousness and the failure of string theory. The link is in the description."
},
{
"end_time": 1312.312,
"index": 56,
"start_time": 1293.831,
"text": " In quantum group theory, the quantum Yang-Baxter equation results in the discovery of quantum deformations of Lie algebras called Drinfeld-Gymbo quantum groups. It's denoted here by this U with usually a Q is underneath and in brackets is the Lie algebra G. So not the group G, but the Lie algebra of the group."
},
{
"end_time": 1328.456,
"index": 57,
"start_time": 1312.841,
"text": " It has wide applications and conformal field theory. The solutions to these quantum Yang-Baxter equations are known as R matrices and are necessary for constructing invariance of knots and links such as the Jones polynomial and the HomFly polynomial, which generalizes the Jones polynomial."
},
{
"end_time": 1350.265,
"index": 58,
"start_time": 1328.865,
"text": " By the way, correlators are a physicist's fancy way of saying greens functions, and that's a mathematician's fancy way of saying solutions to inhomogeneous differential equations. And that's just a pretentious way of saying responses to disturbances in a field. Stress energy tensor and conformal weight."
},
{
"end_time": 1361.903,
"index": 59,
"start_time": 1351.271,
"text": " In string theory, the stress-energy tensor TAB encapsulates the energy and momentum density on the worldsheet and can be obtained by varying the Polyakov action with respect to the worldsheet metric h."
},
{
"end_time": 1391.783,
"index": 60,
"start_time": 1362.381,
"text": " The requirement of conformal symmetry leads to the traceless condition, which in turn gives rise to the varisora constraints required for string quantization. As mentioned previously, the stress energy tensor can be decomposed into holomorphic and anti-holomorphic parts with the complex world sheet coordinates z and then z-bar. Conformal weights here, h and h tilde, characterize the fields in string theory, determining their transformation behavior under these conformal transformations. The Green-Schwarz mechanism."
},
{
"end_time": 1418.968,
"index": 61,
"start_time": 1392.961,
"text": " The Green-Schwarz mechanism is something that resolves anomalies in type 1 and heterotic superstring theories. Anomalies happen when you have classical symmetries, like even gauge symmetries and diffeomorphism invariance, when they're preserved at the classical level, but then they're violated at the quantum one. Before the Green-Schwarz mechanism, the anomaly was represented by a non-vanishing gauge variation of the effective action. So here lambda is the gauge transformation parameter."
},
{
"end_time": 1442.108,
"index": 62,
"start_time": 1419.377,
"text": " The mechanism demonstrates that specific combinations of space-time and world-sheet anomalies cancel, ensuring the theory's consistency. The observation is given by, if you take the whole integral of the two-form field B here, so calibrating field, and x8 is the eight-form characteristic class of the gauge bundle, which is what was introduced by Green and Schwartz, integrate all of that over the ten-dimensional space-time."
},
{
"end_time": 1461.22,
"index": 63,
"start_time": 1442.517,
"text": " After the Green-Schwarz mechanism is applied, the anomaly vanishes and we have that that variation before is now finally equal to zero. This mechanism ensures local supersymmetry in 10-dimensional spacetime while imposing constraints on the gauge group and the spacetime dimensions. First string revolution."
},
{
"end_time": 1479.428,
"index": 64,
"start_time": 1461.886,
"text": " The first string revolution occurred during the mid-1980s and was primarily ignited by the discovery of the Green-Schwarz anomaly cancellation mechanism in type I string theory, specifically with that gauge group mentioned, SL32. It was subsequently extended to the chiral-heterotic E8 cross-E8 string theories."
},
{
"end_time": 1494.548,
"index": 65,
"start_time": 1479.428,
"text": " These developments demonstrated the absence of these anomalies, which are inconsistencies that come about when gauge symmetries, such as electromagnetism and diffeomorphism symmetries related to gravity, are not preserved in a quantized theory, but are there in the classical one."
},
{
"end_time": 1515.606,
"index": 66,
"start_time": 1495.06,
"text": " In string theory, the Green-Schwarz mechanism employs that two-form B field called the Kalb-Rehman field like we talked about before, where its field strength is H. It's a three-form. The anomaly cancellation condition is expressed as, if you take the trace, well, you'll see the expression over here, and F denotes the field strength of the gauge fields, and R represents the Riemann curvature tensor in the context of gravity."
},
{
"end_time": 1545.401,
"index": 67,
"start_time": 1516.186,
"text": " This mechanism, especially in the presence of sources or complex configurations, showcases that dH is generally not zero, unlike in the vacuum scenarios where dH can be zero. Ed Whitten, by the way, thinks that the first string revolution should be called the second string revolution, because according to Ed, the first one was the discovery of string theory. But I think that's just semantics, depending on if you're considering the word revolution, applying to the revolution of physics or revolutionizing string theory itself."
},
{
"end_time": 1563.848,
"index": 68,
"start_time": 1545.691,
"text": " Louisville integrability concerns the existence of sufficient independent conserved quantities in involution for a dynamical system ensuring complete integrability."
},
{
"end_time": 1591.903,
"index": 69,
"start_time": 1564.036,
"text": " The key aspect of Louisville integrability is a lax pair representation given by this formula on screen where L is a linear operator depending on a spectral parameter again like lambda here and M and N are matrices containing the system's dynamical information. The compatibility condition of the lax pair it's written on screen is the derivative of the L with respect to time being equal to some commutation relations. This guarantees the conservation of the spectral invariance making the system integrable."
},
{
"end_time": 1612.056,
"index": 70,
"start_time": 1591.903,
"text": " To put this in simpler terms, this gives a well-behaved system evolving predictably due to the presence of these conserved quantities. The Veneziano amplitude. The amplitude here, based on Euler's beta function, represents early steps towards string theory. Historically, it was discovered in 1968."
},
{
"end_time": 1634.087,
"index": 71,
"start_time": 1612.381,
"text": " How does this relate to the strong nuclear force? Well, initially it was applied to meson scattering. It employs Mandelstam variables, so s and t, for squared energy and momentum transfer respectively, with alpha prime as Reggi slope. The beta function has elegant analytic properties, so for instance, poles at non-positive integers and symmetries, like you can switch the factors that go into the beta function."
},
{
"end_time": 1656.186,
"index": 72,
"start_time": 1634.343,
"text": " Examining the physical region of its poles reveals the resonance mass spectrum, akin to determining the frequency distribution of a vibrating string, a concept that started the string theory journey. String Theory Background Fields In string theory, background fields define space-time geometry and string interactions while strings propagate."
},
{
"end_time": 1676.049,
"index": 73,
"start_time": 1656.493,
"text": " So the metric tensor G encodes space-time curvature as usual, determining distance between points and playing a role in general relativity, of course. Don't ask me why string theorists capitalize this G, whereas in every other context I know it's a lowercase g. There's even another capital G in the context of M theory, namely the field strength of the C field."
},
{
"end_time": 1704.616,
"index": 74,
"start_time": 1676.049,
"text": " The Antisymmetric Two-Tensor Field, B, is known as the Kalbramian Field, generalizes the electromagnetic vector potential, and contributes to the strings coupling to a two-form field, affecting the world sheet action. The Dilaton Field, Phi, is a scalar field, and it sets the string coupling constant via this formula, which is usually just the exponential of Phi. It controls the strength of the string interactions. Sometimes you'll hear people say that string theory comes down to a single parameter, and it's usually this that they're referring to."
},
{
"end_time": 1734.753,
"index": 75,
"start_time": 1704.991,
"text": " The low energy effective action for the string is given by this formula on screen where R is the Ricci scalar and H is the derivative of the Calbrahman field and G is the metric tensor's determinant. A choice of these background fields affects the compactification schemes as well as d-brain configurations. Thereby, it impacts the derived low energy physics and phenomenological predictions in string theory. In other words, different choices here yield different physics. Flux Compactifications"
},
{
"end_time": 1744.906,
"index": 76,
"start_time": 1735.811,
"text": " Flux compactifications in string theory involve background fluxes that stabilize moduli, addressing the so-called moduli stabilization problem."
},
{
"end_time": 1768.097,
"index": 77,
"start_time": 1745.316,
"text": " These fluxes are quantized according to the flux quantization condition, which is if you integrate over the entire field strength of the gauge field with a compact cycle within an internal manifold, then you get N. Considering the GVW or the Gaka-Vafa-Wittin superpotential, W, where H is the Nebu-Schwarz-Nebu-Schwarz three-form fluxes, so NS-NS three-form, and tau is an axiodilatron."
},
{
"end_time": 1798.422,
"index": 78,
"start_time": 1768.524,
"text": " Also, this sigma is the holomorphic three-form of the internal manifold. So, how do flux compactifications stabilize vacuum expectation values? They freeze the geometric moduli, such as the size and the shape of the extra dimensions for scalar fields in the effective four-dimensional theory, which you can think of as a shape controller for these extra dimensions. This stabilization is used to obtain the dissitter vacua, which is needed because we live in a dissitter space, not an anti-dissitter one."
},
{
"end_time": 1829.428,
"index": 79,
"start_time": 1801.834,
"text": " Layer 3 At this point, congratulations! You now know more than 9 out of 10 people who say that they either like or dislike String Theory. Shirk's Anti-Gravity Joel Shirk is one of the founders of String Theory, who unfortunately died unexpectedly in tragic circumstances only months after the supergravity workshop at Stony Brook in 1979."
},
{
"end_time": 1843.677,
"index": 80,
"start_time": 1830.009,
"text": " The workshop proceedings were dedicated to his memory with a statement that Shirk, who was diabetic, had been trapped somewhere without his insulin and went into a diabetic coma. He was only 33 years old."
},
{
"end_time": 1860.998,
"index": 81,
"start_time": 1843.882,
"text": " A year prior to his death, Shirk published a little-known paper titled, Anti-Gravity, a crazy idea. The concept of anti-gravity emerges from the introduction of a massless vector field, denoted as A mu with a superscript L, referred to as the anti-graviton."
},
{
"end_time": 1890.213,
"index": 82,
"start_time": 1861.425,
"text": " The antigraviton couples to a conserved current, J, associated with the quarks and leptons' unclad mechanical masses. This is in contrast with the graviton, which interacts with their actual masses. A force between two atoms can be expressed as F equals this formula on screen, where M and M0 are the real and unclad masses respectively, and G is the gravitational constant. You may be wondering, doesn't this notion of antigravity clash with the equivalence principle?"
},
{
"end_time": 1913.848,
"index": 83,
"start_time": 1890.572,
"text": " It seems to, but this clash can be resolved if a scalar field acquires a non-zero vacuum expectation value similar to how SU2 cross SU1 breaks down into U1. This causes the L field to acquire a mass term, which changes the potential into one with a different minimum. Schurck showed that this anti-gravity is a quality of any extended supersymmetric gravitational model."
},
{
"end_time": 1934.002,
"index": 84,
"start_time": 1914.224,
"text": " The Swampland"
},
{
"end_time": 1951.869,
"index": 85,
"start_time": 1935.145,
"text": " The Swampland Conjecture originates from Vafa's work in 2005. It posits criteria to differentiate consistent low-energy effective field theories with a quantum gravity completion, especially from string theory, from seemingly consistent EFTs that don't."
},
{
"end_time": 1977.432,
"index": 86,
"start_time": 1951.869,
"text": " In other words, we have different solutions to string theory. We don't know which one is correct. We know where we want to get to, namely the Standard Model plus General Relativity. You may say, Kurt, the question is, well, which of the possible string theory solutions, also known as vacua, get you there? And I'd say that's a wonderful question. You're so bright. The ones that don't get you there are part of the Swampland. Now, Swampland sounds like a negative word,"
},
{
"end_time": 2002.875,
"index": 87,
"start_time": 1977.432,
"text": " but actually the larger the Swampland the better because you'll be able to narrow down the space of possible solutions in string theory. Two central conjectures in the Swampland arena are the weak gravity conjecture and the distance conjecture. The weak one says for consistent quantum gravity and consistent by the way here means free from unwanted features like non-unitarity causality violation and unphysical singularities"
},
{
"end_time": 2023.302,
"index": 88,
"start_time": 2002.875,
"text": " In other words, the weak gravity conjecture implies that particles with a specific charge to mass ratio are needed to avoid inconsistencies in quantum gravity."
},
{
"end_time": 2052.927,
"index": 89,
"start_time": 2023.609,
"text": " The distance conjecture, on the other hand, says that if we move in field space by some distance, let's say delta phi, that the EFT, the effective field theory, breaks down at a scale proportional to what you see on screen with a constant alpha. Why? Because infinite towers of states become exponentially light. Now, an infinite tower of states is a term you'll hear plenty, and it means an unbounded series of particles that become progressively lighter as one moves further in field space."
},
{
"end_time": 2082.688,
"index": 90,
"start_time": 2052.927,
"text": " This is fantastic and fanatical, because it means moving sufficiently far in field space leads to the emergence of new physics. The further we explore, the more physics we have to account for. Technically speaking, the Swampland Conjecture isn't just about, well, which vacua lead to the Standard Model plus General Relativity, but it's also about determining the general properties that any consistent quantum gravity theory must have. The Transplankian Censorship Conjecture."
},
{
"end_time": 2101.527,
"index": 91,
"start_time": 2083.541,
"text": " Metastable means something is stable for a period of time, but it's not the most stable possible. That is, it has a higher energy than the true stable state."
},
{
"end_time": 2116.647,
"index": 92,
"start_time": 2101.527,
"text": " You may have heard something called the cosmic censorship hypothesis of Roger Penrose, which states that singularities, such as those occurring in the collapse of massive objects that form black holes, are always hidden from an external observer by an event horizon, so they're censored."
},
{
"end_time": 2141.766,
"index": 93,
"start_time": 2116.92,
"text": " Well, the trans-Planckian censorship conjecture, on the other hand, is another censorship principle that has connections to the Swampland criteria by providing constraints on the observable universe in theories of quantum gravity. Which constraints? Constraints on the initial conditions of our observable universe, specifically stating that the physical processes occurring at distances smaller than the Planck length"
},
{
"end_time": 2162.108,
"index": 94,
"start_time": 2141.766,
"text": " Moonshine and String Theory"
},
{
"end_time": 2189.241,
"index": 95,
"start_time": 2162.756,
"text": " Moonshine refers to the unexpected connections between finite group representations, modular functions, and vertex operator algebras. You don't need to know what any of those are. All that's important is that they were at least once thought to be part of different fields of mathematics. The most famous example is what's called the monstrous moonshine conjecture, which links the largest sporadic simple group, the monster group, to a modular function called the J-function, which is given by this formula on screen."
},
{
"end_time": 2209.599,
"index": 96,
"start_time": 2189.804,
"text": " The conjecture, proven by Bortrads using vertex operator algebras and their associated characters, states that the coefficients Cn in the J-function expansion encodes the dimensions of the irreducible representations of the monster group. So what the heck does this have to do with string theory? It turns out that certain CFTs"
},
{
"end_time": 2234.565,
"index": 97,
"start_time": 2209.906,
"text": " When compactified on the torus, give partition functions with modular invariants and characters, encoding group representation data. To translate that a tad, formally speaking, you'll see a formula on screen, and this is for any ABCD belonging to SL2Z. Umbral moonshine, on the other hand, relates something called Nyamir lattices to Mateo and other sporadic groups. I don't know how to pronounce these names. I'm a self-studier. I've only read these."
},
{
"end_time": 2255.077,
"index": 98,
"start_time": 2235.009,
"text": " Entropic gravity postulates that gravity is induced from the statistical tendency of systems to maximize entropy described by the formula on screen here. In this description,"
},
{
"end_time": 2279.07,
"index": 99,
"start_time": 2255.247,
"text": " Exotic Dualities"
},
{
"end_time": 2301.084,
"index": 100,
"start_time": 2280.196,
"text": " There are more dualities than just T and S, young Padawan. Each of these are large enough that we'll explore later. There's U-Duality, there's Mirror Symmetry, more on that soon, ADS-CFT, there's Montan and Olive Duality or Electric Magnetic Duality, there's K3 Vibration Duality, there's Open and Closed String Duality, there's F3 slash Heterotic Duality,"
},
{
"end_time": 2310.486,
"index": 101,
"start_time": 2301.493,
"text": " So let's start with U-duality. What this does is combine T-duality and S-duality in M-theory, placing them in a single duality group in one dimension higher."
},
{
"end_time": 2337.09,
"index": 102,
"start_time": 2310.981,
"text": " Mirror symmetry, a type of t-duality, relates these Callib-Yau manifold with different Hodge numbers. At least this is how it was initially formulated. Hodge decomposition is something we'll explore in a podcast shortly on this channel with Professor Eva Miranda, so subscribe if you're interested in geometric quantization. What you do in mirror symmetry is you exchange what's called the Caylor structure or the symplectic structure, and it has applications in enumerative geometry, which we'll talk about again later."
},
{
"end_time": 2354.189,
"index": 103,
"start_time": 2337.09,
"text": " Montan in all of duality is an S-duality in supersymmetric gauge theories. This relates magnetic and electric charges via the exchange of coupling constants. This is also known as electric-magnetic duality, which is not to be confused with electromagnetic duality, even though some people accidentally say that."
},
{
"end_time": 2378.37,
"index": 104,
"start_time": 2354.548,
"text": " K-3 vibrations duality is about the relationship between K-3 surfaces and elliptic vibrations, where an elliptic vibration is a morphism from a variety X, let's say, to a base B, such that almost all of the fibers are elliptic curves. Open and closed string duality describes the equivalence between open strings with boundary conditions determined by D-branes and closed strings in the presence of Raymond-Raymond fluxes."
},
{
"end_time": 2394.309,
"index": 105,
"start_time": 2378.848,
"text": " F-theory slash heterotic duality connects F-theory, a 12-dimensional framework extending type IIb string theory, to heterotic strings via compactification on elliptically-fibred Calib-Yau manifolds. Mirror symmetry."
},
{
"end_time": 2419.497,
"index": 106,
"start_time": 2395.196,
"text": " Mirror symmetry, this is a deep topic. Mirror symmetry is a duality relating two Calabi-Yau manifolds, M and W, interchanging their complex and scalar structure. The mirror map on screen here is bi-directional and relates the complex moduli, say phi of M, to the symplectic moduli, say psi of W, where this F here is the pre-potential and T, A are the symplectic parameters."
},
{
"end_time": 2445.435,
"index": 107,
"start_time": 2419.753,
"text": " In topological string theory, the A model and B model are topological fields derived from the original string theory by focusing on its topological properties associated with the K-Laring complex structure respectively. The A model computes the Gromov-Witten invariance, and these little guys capture information about holomorphic curves in M, while B computes what are called periods of the holomorphic 3-0 form on W."
},
{
"end_time": 2475.435,
"index": 108,
"start_time": 2445.435,
"text": " The Gaffa-Khamar-Vafa invariance, on the other hand, are their younger, snappier sister, which reformulate the Gromov-Witten invariance in integer numbers. Mirror symmetry connects the A model on M and the B model on B and vice versa. What this does is enable computations of one model's observables using the other model's techniques. Turns out there are like 10 to the 10 examples of distinct data points of CY3 manifolds, so that's Kali-Biao3 manifolds, making it one of the largest data sets in all of math, if not the largest."
},
{
"end_time": 2501.834,
"index": 109,
"start_time": 2475.435,
"text": " Mirror symmetry itself can be its own iceberg. Speaking of which, I have several other ideas for other iceberg podcasts, like the iceberg of consciousness theories or the iceberg of theories of truth. If you have suggestions, then leave them below in the comment section. Extra dimensions and compactification. See why three manifolds are what are being compactified in string theory. Whenever you hear about extra dimensions, they're usually referring to these guys."
},
{
"end_time": 2529.138,
"index": 110,
"start_time": 2502.261,
"text": " There's something else called a Joyce manifold, a subclass of Calabi-Yau manifold, with exceptional holonomy, so G2, smooth, they're compact, they're Riemannian, they're seven-dimensional, and they have a non-degenerate three-form phi, which is invariant under G2. These come up in M theory. Every time you have extra dimensions, you have to answer the question about why we don't see them. One answer is that, hey, they're just too small, they're compactified."
},
{
"end_time": 2546.408,
"index": 111,
"start_time": 2529.718,
"text": " The problem is that not only are there several different possible structures for these extra dimensions, but there are several different ways you can compactify. Each of these spawn different physics. So far none of them have been found to be even remotely resembling our world."
},
{
"end_time": 2561.937,
"index": 112,
"start_time": 2546.408,
"text": " By the way, it's also false to say that string theory doesn't operate in four dimensions. It does. There is a string theory of exactly four dimensions. The problem is that those four dimensions are all spatial dimensions, or all temporal dimensions, or you can also have two"
},
{
"end_time": 2581.34,
"index": 113,
"start_time": 2561.937,
"text": " space and then two times thus they're disregarded what i'm wondering though is that is there some way to wick rotate one of those extra dimensions one of those four into something from the euclidean case to a minkowski space much like peter white does in his euclidean twister unification which is explored here on this podcast"
},
{
"end_time": 2610.828,
"index": 114,
"start_time": 2581.34,
"text": " If you want to know more about dark dimensions, which suggests that dark matter is associated with these extra dimensions, then watch this video by Sabine Hassenfelder, linked in the description. In fact, if you want to know about almost any physics topic, just Google Sabine and that physics term, it's always a useful, though polemical, starting point. Conifold Transitions. Conifolds may sound like a type of manifold or a variety, but they actually refer to the singularities on a variety."
},
{
"end_time": 2623.916,
"index": 115,
"start_time": 2611.169,
"text": " Conifolds help us understand topology changes in string compactifications as they involve transitions between distinct Kalibi-Yau manifolds. When a CY3 develops a conical singularity, then this transition commences."
},
{
"end_time": 2653.251,
"index": 116,
"start_time": 2624.224,
"text": " This can be resolved either through something called a small resolution or a deformation, both of which result in a new Calabi-Yau manifold. In a small resolution, the conifold point transitions into a projective cycle of finite size, smoothing it out, while in a deformation, the conifold point is replaced by a non-vanishing three-form flux. Governed by Picard-Lefchet's monodromy, the periods of holomorphic three-forms transform through this process. Conifold transitions can be described by the exchange of massless"
},
{
"end_time": 2682.91,
"index": 117,
"start_time": 2653.439,
"text": " Instant Tons are topologically non-trivial solutions to the anti-self-dual Yang-Mills equations, where F is the field strength tensor and the tilde F is the Hodge dual in four-dimensional Euclidean space, and we're talking about classical Yang-Mills equations here. Okay, so all of that is a mouthful, but you can think of them as what extremizes the action in certain Yang-Mills theories."
},
{
"end_time": 2711.288,
"index": 118,
"start_time": 2683.2,
"text": " or, to translate that a tad, what are physical solutions? These solutions are characterized by their topological charge, or instanton number, k. Donaldson invariants are topological invariants of smooth, compact, oriented 4-manifolds that were put forward by Simon Donaldson, a Fields medalist, in the 1980s. The construction of these invariants involves counting the number of instantons on a 4-manifold m, modulo gauge transformations, subject to certain constraints on their characteristic classes."
},
{
"end_time": 2741.271,
"index": 119,
"start_time": 2711.681,
"text": " Characteristic classes are invariants of vector bundles. There exists an extension of these Donaldson invariants, which are useful for physicists, called the Seeberg-Witten invariants, which are used to describe the low energy effective action of n equals 2 supersymmetric Yang-Mills theories. In string theory, tachyon condensation is a process involving tachyons, particles with imaginary mass, which can destabilize the vacuum state and trigger infinite transitions to a lower energy state."
},
{
"end_time": 2770.589,
"index": 120,
"start_time": 2741.681,
"text": " This is well studied in the context of open-string tachyons attached to d-brains, where the tachyon potential has the form on screen here. The tachyon condensation drives the system toward a stable configuration, effectively removing the d-brains from the spectrum and reducing the energy of the system. Sen's conjecture, which we'll talk about later, states that the endpoint of tachyon condensation corresponds to the annihilation of the d-brain, resulting in a closed-string vacuum."
},
{
"end_time": 2789.821,
"index": 121,
"start_time": 2770.913,
"text": " Supersymmetry."
},
{
"end_time": 2815.145,
"index": 122,
"start_time": 2790.606,
"text": " supersymmetry is a symmetry between balsonic and fermionic degrees of freedom governed by the supersymmetric algebra with the compatibility between the q's on screen here and they're the supercharge operators those alphas with the dots are spinner indices and the p represents the spacetime momentum operator it's not so intimidating this implies that for every balsonic particle there exists a fermionic superpartner and vice versa"
},
{
"end_time": 2845.333,
"index": 123,
"start_time": 2815.52,
"text": " Historically, the concept of supersymmetry was independently discovered by three groups in about the 1970s, so early 1970s by Galfan and Lichtman, Ramon and Neveu and Schwartz. In string theory, supersymmetry is a consequence of the cancellation of worldsheet anomalies we talked about earlier, and that also avoids tachyonic instabilities. Broken supersymmetry is said to be imperative in addressing the so-called hierarchy problem, controlling the Higgs boson mass and providing viable candidates for dark matter."
},
{
"end_time": 2875.299,
"index": 124,
"start_time": 2846.886,
"text": " Extended supersymmetry. Extended supersymmetry theories are super interesting. The ordinary supersymmetry that you hear about on pop-side channels is actually n equals 1 supersymmetry, but there are other extended versions with n greater than 1. This just means that it has more generators, and thus more superpartners, and thus more particles. For instance, the n equals 2 superconformal algebra given on screen here, where GR is the superconformal generator and LR the verisora generators,"
},
{
"end_time": 2892.125,
"index": 125,
"start_time": 2875.299,
"text": " Due to the additional constraints imposed by supersymmetric generators, the number of free parameters is actually reduced, increasing the predictive power, which is like minimizing the overfitting. It sounds like because we have many more particles being predicted, that it's much more of a broad theory in terms of its predictions."
},
{
"end_time": 2907.466,
"index": 126,
"start_time": 2892.449,
"text": " Extended SUSE leads to smaller massless particle content and further cancellation of anomalies. When you extend your supersymmetry past n equals 1, you get as a benefit more control over non-perturbative effects"
},
{
"end_time": 2932.671,
"index": 127,
"start_time": 2907.466,
"text": " and enhanced stability of the vacuum, essential for constructing consistent and stable string vacua, and phenomenologically viable models. For 10-dimensional superstring theories, we usually have either that n equals 1 or n equals 2, though you can also have differing amounts of supersymmetry on the left and the right modes, like we talked about in the heterotic case earlier. Now, you may be wondering about higher values of n"
},
{
"end_time": 2955.93,
"index": 128,
"start_time": 2932.671,
"text": " Low Energy Effective Gravity"
},
{
"end_time": 2982.534,
"index": 129,
"start_time": 2957.278,
"text": " In string theory, the low-energy effective action governs the dynamics of massless fields and connects familiar gravitational physics to the underlying string theoretical framework. Formally, the effective action is described as follows, where G denotes the spacetime metric and phi is the dilaton and H is the Neville-Schwarz three-form field strength. The dilaton field introduces the string coupling via the formula on screen, which modulates the strength of string interactions."
},
{
"end_time": 2995.998,
"index": 130,
"start_time": 2982.534,
"text": " N equals two quantum field theories."
},
{
"end_time": 3024.309,
"index": 131,
"start_time": 2996.34,
"text": " Extended supersymmetry, and supersymmetry in general, isn't just for string theory, but for quantum field theory. In the n equals 2 supersymmetric quantum field theory, topological invariants, like we mentioned before, there's Donaldson and Seeberg-Witten invariants, have massive roles to play. Donaldson invariants emerge from the moduli space of anti-self-dual connections in twisted supersymmetric Yang-Mills theory, while a certain twisting procedure aligns the Lorentz and R symmetry groups, resulting in a topological theory."
},
{
"end_time": 3049.036,
"index": 132,
"start_time": 3024.65,
"text": " In the context of n equals 2 supersymmetric quantum field theories, this twisting process refers to the modification of the supercharges such that it becomes a scalar under Lorentz transformations. The partition function of the twisted n equals 2 super Yang-Mills theory localizes on the moduli of anti-self-dual connections, and the observables are given by the correlation functions of operators corresponding to cohomology classes."
},
{
"end_time": 3075.606,
"index": 133,
"start_time": 3049.36,
"text": " Seiberg-Witten invariants, which generalize Donaldson invariants, come about from the low energy effective action of n equals 2 super Yang-Mills theory. This is governed by the Seiberg-Witten curve and the pre-potential, which encode the modulized space of vacua. These invariants are computationally more tractable and can be expressed as integrals over differential forms. Interestingly, there's a correspondence called the Seiberg-Witten-Donaldson correspondence, which relates these two types of invariants."
},
{
"end_time": 3089.787,
"index": 134,
"start_time": 3075.606,
"text": " These invariants have connections to string theory, notably in type IIa and heterotic string compactifications, where n equals 2 quantum field theories appear on d-brain world volumes. Multiverse of the string landscape"
},
{
"end_time": 3113.695,
"index": 135,
"start_time": 3090.35,
"text": " The string theory landscape refers to the vast array of 10 to the 500, sometimes as quoted, possible vacua resulting from string theory's extradimensional compactifications, such as on collibial manifolds and even through other techniques like flux compactifications. These vacua lead to a multitude of different gauge groups, particle content, and cosmological constants for the low-energy effective field theories."
},
{
"end_time": 3137.961,
"index": 136,
"start_time": 3113.695,
"text": " Historically, the term landscape was first used by Lee Smolin in the Life of the Cosmos book. Each vacuum represents a possible universe with its corresponding physical laws resulting in a multiverse concept. It's unknown if each of these universes exist. Are we just one of the 10 to the 500 universes? The fermionic string"
},
{
"end_time": 3160.145,
"index": 137,
"start_time": 3138.695,
"text": " The fermionic string action is given on screen here, where the gammas are the world-sheet gamma matrices, and the nabla denotes the world-sheet covariant derivative. This action is invariant under supersymmetry transformations, and thus we say it's supersymmetric. In the 1970s, this guy named Pierre Raymond, and then this other guy named John Schwartz, and then this other guy named André Neveux,"
},
{
"end_time": 3183.78,
"index": 138,
"start_time": 3160.435,
"text": " developed the Raymond-Nevus-Schwartz formulation, the RNS formulation. This implements the GSO projection, which is something that removes tachyonic and unphysical states from the spectrum. Operator product expansion. This allows the computation of correlation functions by expressing the product of two operators in proximity as a weighted sum of operators at a single point."
},
{
"end_time": 3210.623,
"index": 139,
"start_time": 3184.104,
"text": " Here, phi denotes the primary fields, c represents the OPE coefficients, h signifies the conformal weights, as usual, and z and w denote the world sheet coordinates. The OPE significantly simplifies the calculations of amplitudes for processes in string theory by utilizing the conformal structure of the world sheet. In string theory, vertex operators correspond to string mode creation or annihilation, and their OPEs encode information about string interactions."
},
{
"end_time": 3229.224,
"index": 140,
"start_time": 3210.623,
"text": " For instance, in balsonic string theory, the OPE of two tachyon vertex operators is given by this formula, which relates the interaction amplitude of two tachyons. Imagine each operator as a character in a story. When there are two characters, so operators interact, which means they come close on the world sheet,"
},
{
"end_time": 3243.66,
"index": 141,
"start_time": 3229.343,
"text": " Holographic Theories"
},
{
"end_time": 3272.449,
"index": 142,
"start_time": 3244.258,
"text": " One fateful year in 1997, Maldeseyna put forward a conjecture known as the ADS-CFT correspondence, which states that there's a duality between gravitational theories on an anti-de Sitter space and conformal field theories on the boundary of those spaces, generally expressed as the partition functions of each equaling one another. You can think of the partition function as a way of saying, hey, this function contains all the information about the system."
},
{
"end_time": 3297.125,
"index": 143,
"start_time": 3272.688,
"text": " This duality provides an extremely powerful tool for studying strongly coupled gauge theories. How? By mapping them to weakly coupled gravitational theories, and vice versa. Sometimes you get a formula connecting the ADS radius, r, with the strong coupling g, the number of colors, n, and the Regislope alpha prime is given by r to the fourth proportional to a product of all of them, with alpha being squared."
},
{
"end_time": 3323.012,
"index": 144,
"start_time": 3297.961,
"text": " This is why the ADS-CFT correspondence is said to be quote-unquote more accurate in the large n limit, where the classical gravity approximation is valid. It's also another reason why the theorists aren't terribly concerned about it being an ADS space and not a DS one, so a desider space. A desider space wouldn't have a boundary like this, at least not necessarily, but if we're taking n to infinity anyhow, then the radius goes there as well."
},
{
"end_time": 3352.517,
"index": 145,
"start_time": 3323.012,
"text": " The hope is that there will eventually be some translation or application to decider space, thus describing our universe. For clarity, the n here corresponds to the gauge group rank, so usually it's SU2, which is n equals 2, SU3 is n equals 3, and when someone says that they're considering the large n limit, what that means is to consider numbers of n, so integers, sorry, natural numbers of n, which are far larger than say 2 or 3, even all the way up to"
},
{
"end_time": 3380.333,
"index": 146,
"start_time": 3353.012,
"text": " Celestial Holography"
},
{
"end_time": 3398.609,
"index": 147,
"start_time": 3380.896,
"text": " Celestial holography is one of the most beautiful sounding terms in all of physics. It studies encoding scattering amplitudes in asymptotically flat spacetimes onto a celestial sphere at null infinity. In other words, it's another way of looking at holography in string theory that isn't just ADS-CFT."
},
{
"end_time": 3428.712,
"index": 148,
"start_time": 3398.916,
"text": " An integral component in this approach is the celestial spheres parameterization by conformal coordinates, omega and omega tilde, with the Mellon transformation associating bulk amplitudes with celestial correlators, given by this formula on screen here, where delta represents the conformal weights and H denotes the dimension of the local operators. The vertex operators V in the worldsheet CFT correlate to local operators on the celestial sphere within string theory, with their conformal weights defining the string states masses and spins."
},
{
"end_time": 3447.637,
"index": 149,
"start_time": 3429.241,
"text": " Celestial holography can be thought of as a Rosetta stone between scattering amplitudes, conformal symmetry, and string theory. Historically, celestial holography emerged as an outcome of investigating the symmetries of soft theorems, which is actually a hilarious term, meaning the study of particles with momentum approaching zero."
},
{
"end_time": 3460.128,
"index": 150,
"start_time": 3447.944,
"text": " The Mellon Transform applied to scattering amplitudes is the connection between bulk physics and conformal structures in a similar manner to how the Fourier Transform unearths connection between, say, time and frequency."
},
{
"end_time": 3483.609,
"index": 151,
"start_time": 3460.418,
"text": " Celestial holography generalizes the BMS symmetry, which was researched by Strominger, which itself goes back to the Bondi-Metzner-Sachs group in 1962. Strominger studied this symmetry in about 2013 or so, and celestial holography can be seen as an extension of this work. The celestial sphere refers to scry plus or minus, or the future and past null light cones."
},
{
"end_time": 3508.763,
"index": 152,
"start_time": 3485.043,
"text": " In a recent video by Sabrina Pasteurski, she contrasts celestial CFT with ADS4CFT3. The primary difference between BMS CFT and celestial holography is that the latter focuses on encoding scattering amplitudes and asymptotically flat spacetime onto a celestial sphere at null infinity, while BMS CFT is more concerned with the symmetries of soft theorems."
},
{
"end_time": 3523.814,
"index": 153,
"start_time": 3508.763,
"text": " Super Currents"
},
{
"end_time": 3552.585,
"index": 154,
"start_time": 3525.043,
"text": " In Type II string theories, supercurrents encode worldsheet supersymmetric transformations. The supercurrent, G plus or minus, is given by this formula here, where the psi's are the worldsheet fermions, and the x's represent the spacetime coordinates, and H is the Neville-Schwartz three-form field string. These supercurrents satisfy the superconformal algebra, including the verisora algebra for the energy-momentum tensor, and the U1 current, J, with an additional anti-commutation relation."
},
{
"end_time": 3577.005,
"index": 155,
"start_time": 3553.729,
"text": " Why is string theory so successful at producing results in other seemingly unrelated areas of math? Why is it so fruitful that entire new fields of mathematics are spawned? This is a puzzle because this usually happens with physical theories that have evidence associated with them, like quantum mechanics,"
},
{
"end_time": 3605.009,
"index": 156,
"start_time": 3577.005,
"text": " with his study of infinite-dimensional Hilbert spaces and quantum cohomology and general relativity and quantum field theory. Part of the answer is sociological, but we don't know how much of the relative pure mathematical success of string theory is because of historical reasons of, say, power and arrogance, such as those outlined by Eric Weinstein, Lee Smolin, and Peter White, or how much of this is because string theory is indeed striking at the heart of physical reality."
},
{
"end_time": 3612.671,
"index": 157,
"start_time": 3607.961,
"text": " Layer 4 Defining String Theory"
},
{
"end_time": 3638.712,
"index": 158,
"start_time": 3613.439,
"text": " So, what is a string theory exactly? This isn't something that's asked in most string theory courses. You learn motivations, starting with the Reggi slope and then how Feynman diagram singularities can be smoothed out because you've now moved from one dimension to two dimensions. And then you start to explore more and more mathematical consequences. But few people stop to ask like, hey, when I hand you a theory, how do you know if it's a string theory?"
},
{
"end_time": 3658.541,
"index": 159,
"start_time": 3638.712,
"text": " Is it the presence of a Nambu-Gado action or that Polyakov one? Is it somehow that the tension parameter shows up? Is it any theory with extended objects even if they're more than one dimensionally extended? Sometimes this question becomes so general that it will lead even the creators of string theory to call any quantum field theory a string theory."
},
{
"end_time": 3681.015,
"index": 160,
"start_time": 3658.848,
"text": " Actually, it would be far more accurate to say that a string theory is an example of a type of quantum field theory, where you either have strings or brains. By the way, it's unclear even what a quantum field theory is, and you can see the talks by Natty Seidberg, Dan Freed, and Nima Arkani Hamed. Those are in the description as well as they're on screen right now. The Second String Revolution"
},
{
"end_time": 3706.408,
"index": 161,
"start_time": 3682.722,
"text": " The second string revolution highlighted the non-perturbative aspects of string theory and led to major advancements. So what happened? In 1995, there was the proposal of the existence of another theory called M theory, an 11 dimensional framework which would encompass all five major string theories. Sometimes people say it will quote unquote unite them, but it's more accurate to say it encompasses them or relates them."
},
{
"end_time": 3718.336,
"index": 162,
"start_time": 3706.92,
"text": " M theory relates to type 2a string theory via compactification on a circle with radius r. So you follow what's on screen, this formula where L11 is the 11-dimensional Planck length and LS is the string length."
},
{
"end_time": 3742.91,
"index": 163,
"start_time": 3718.814,
"text": " M-theory, when compactified on a Z2 orbifold, also connects it to heterotic E8 x E8. The inception of M-theory can be traced back to Witten and Horová's attempts to understand the strong coupling limit of Type IIa string theory. This ignited a spark in both the physics and math community, from which our current flame is a descendant of. The pre-Big Bang scenario cosmological model"
},
{
"end_time": 3773.575,
"index": 164,
"start_time": 3743.797,
"text": " In string cosmology, the pre-Big Bang scenario suggests that time predates the conventional Big Bang, with a contracting phase followed by a diluting phase and then a bounce, leading to the observed expanding universe. There are several theories on cosmogony, something I may do an iceberg on, so that is theories of how the universe came to be and where it's going. Some suggest that time emerged from space, some suggest that both emerged from something non-spacetime-like, such as Hawking and Hartle, but today, here, we have something different."
},
{
"end_time": 3799.428,
"index": 165,
"start_time": 3773.916,
"text": " Here it's suggested that time existed even prior to the Big Bang. This framework emerges from the low energy effective action of string theory, so given by this formula on screen, where phi is the dilaton field, H is the antisymmetric tensor field strength, and V is the dilaton potential. A key feature is scale factor duality, and also given by the transformations on screen, with A being the scale factor and eta being the conformal time."
},
{
"end_time": 3819.565,
"index": 166,
"start_time": 3799.428,
"text": " The pre-Big Bang model describes a universe evolving from a weakly-coupled, highly-dilute state, so dilaton-driven inflation, to a strongly-coupled, hot, dense state before transitioning to the standard Big Bang epoch. Dilute, in this case, by the way, means what you think it means, namely sparse and cool matter rather than dense and hot matter."
},
{
"end_time": 3837.125,
"index": 167,
"start_time": 3819.855,
"text": " Actually, Gabriel Venziano, the founding father of string theory, was urged by the legendary Stephen Hawking himself to consider the cosmological implications of string theory during a 1986 visit to Boston University, laying the groundwork for future developments in string cosmology. Hagedorn's Universe"
},
{
"end_time": 3862.398,
"index": 168,
"start_time": 3838.183,
"text": " Is there a maximum temperature? This is an interesting question because we think that there's a minimum temperature, namely absolute zero. So is there some finite version of absolute infinite temperature? Well, in string theory, the Hagedorn temperature, TH, signifies just this. At this temperature, a phase transition occurs, characterized by the prolific production of huge strings, called long strings, actually."
},
{
"end_time": 3885.196,
"index": 169,
"start_time": 3863.046,
"text": " The Hagedorn temperature is given by TH equals the inverse of 2 pi times the square root of the Regi slope. You can see by adjusting the string tension, TH can be made lower than even the Planck temperature. The Hagedorn universe could be a candidate for the state of the universe before the Big Bang. In this scenario, the universe would be in a highly energetic string-dominated phase,"
},
{
"end_time": 3900.299,
"index": 170,
"start_time": 3885.486,
"text": " Interestingly, in the 1960s, Ralph Hagedorn proposed the concept of the Hagedorn temperature in the context of the statistical bootstrap model applied to hadrons before the development of string theory. Non-commutative geometry and string theory."
},
{
"end_time": 3921.186,
"index": 171,
"start_time": 3902.09,
"text": " Non-commutative geometry, coming from the Moyle product, so the Moyle star product, replaces the usual point-wise multiplication of functions with a non-commutative product defined as follows, where theta is an anti-symmetric tensor representing the non-commutativity, and those derivatives are derivatives with respect to x and x' respectively."
},
{
"end_time": 3948.916,
"index": 172,
"start_time": 3921.186,
"text": " In string theory, this geometry emerges due to a constant, Neveu-Schwarz B field. Historically, non-commutative geometry was developed in the 1940s by Joseph Moyle and later popularized by Elaine Konis. The Cyberg-Witten map relates the commutative and non-commutative fields by introducing correction terms, called theta corrections. The discovery of non-commutative geometry and its connection to string theory reveals unintuitive structures potentially underlying the universe."
},
{
"end_time": 3974.565,
"index": 173,
"start_time": 3949.701,
"text": " 2-0 super conformal field theory. Now this is super interesting, at least to me. If you know what a category is, then you know that you can generalize them to two categories and three categories all the way up to infinity categories. 2-0 theory is related to two categories, which involve objects, so states, morphisms, so think transformations, and two morphisms, which are higher level structures, arrows between arrows."
},
{
"end_time": 3982.722,
"index": 174,
"start_time": 3974.957,
"text": " 2-0 theory was proposed in the mid-90s, centered around the dynamics of strings within M5 brains and their interactions with M2 brains."
},
{
"end_time": 4007.09,
"index": 175,
"start_time": 3983.131,
"text": " Specifically, it deals with strings on the boundary of M5 brains that form the edges of M2 brains. 2-0 theory employs differential forms, which will be used for representing fields, and homological algebra, for studying properties like boundary and symmetry. Mathematically, you can use algebraic structures called L-infinity algebras, which generalize Lie algebras to accept any number of inputs, rather than the standard 2."
},
{
"end_time": 4022.995,
"index": 176,
"start_time": 4007.09,
"text": " 2-0 theory is considered to be a candidate for the most symmetric field theory, potentially exhibiting a form of maximal supersymmetry. For more, watch this brief lecture by Christian Siemen, who's a professor of mathematical physics at Harriet Watt University."
},
{
"end_time": 4050.947,
"index": 177,
"start_time": 4024.684,
"text": " F-theory is a 12-dimensional, non-perturbative framework, again, there's that ill-defined word, which generalizes type IIb string theory with varying string couplings and compactifying on something called elliptically-fibred Calabi-Yau fourfolds, not threefolds, this time it's Cy4. Vafa, the founder of this theory, also linked F-theory compactifications to M-theory compactifications on elliptically-fibred Calabi-Yau manifolds."
},
{
"end_time": 4065.93,
"index": 178,
"start_time": 4050.947,
"text": " this elliptic vibration is given by the weierstrass equation on screen here but this time f and g are sections of a line bundle given by k minus four and k minus six on the base manifold b and kb is the canonical bundle on the base manifold"
},
{
"end_time": 4086.561,
"index": 179,
"start_time": 4066.152,
"text": " Historically, the discovery of F-theory in the 1990s by Kamranwafa and collaborators was a pivotal step toward unifying different string theories and understanding their non-perturbative behavior. Apparently the F stands for father, but I think that the F stands for F.U. Witten, my theory is better. The quantum hall effect in string theory."
},
{
"end_time": 4099.735,
"index": 180,
"start_time": 4087.995,
"text": " The quantum Hall effect is a phenomenon observed in two-dimensional electron systems subjected to extreme magnetic fields, which leads to the quantization of the Hall conductance, where nu is the filling factor."
},
{
"end_time": 4124.155,
"index": 181,
"start_time": 4100.094,
"text": " This effect's connection to string theory comes through d-brains, specifically through d2-brains with an external magnetic field. Why? It's because you can understand the world volume action, which is a generalization of the world sheets action, for a d2-brain by including a Chern-Simons term. Integrating this term over the brain results in a Hall conductance expression similar to the quantum Hall effects formula."
},
{
"end_time": 4148.302,
"index": 182,
"start_time": 4124.155,
"text": " Type IIb string theory compactification on a six-dimensional manifold with a non-trivial three-flux form can give a four-dimensional low-energy effective theory resembling the quantum hall effect, where again the filling factor is related to the quantized three-form field flux. In other words, string theory gives another perspective on the quantum hall effect. Not invariance and Chern-Simon's theory."
},
{
"end_time": 4169.189,
"index": 183,
"start_time": 4149.48,
"text": " Knot invariants, including the Jones polynomial, serve as tools to classify and distinguish knots. It seems like knots are trivial, but that's, haha, not the case. The Jones polynomial is given by this formula on screen, where k denotes the knot, and this absolute value of k represents the number of crossings."
},
{
"end_time": 4183.78,
"index": 184,
"start_time": 4169.189,
"text": " While R is the R matrix stemming from the quantum group, so UQ of SL2C, and the UQ stands for the quantum deformation of the universal enveloping algebra. When Q equals 1, you get the regular universal enveloping algebra."
},
{
"end_time": 4206.732,
"index": 185,
"start_time": 4184.224,
"text": " The connection between knot invariance and the three-dimensional Chern-Simons theory, which is a topological quantum field theory with the action given on screen, is partially what's responsible for Ed Whitten getting the Fields Medal, a medal which has been analogized to the Nobel Prize of math. In this action, A denotes the gauge connection, and M represents a three-manifold, and K is the level."
},
{
"end_time": 4232.073,
"index": 186,
"start_time": 4206.971,
"text": " Through the Wilson loop operator given on screen here, which takes into account a loop in M and a gauge group representation R, we secure a link between not invariance and Chern-Simons theory. But the question is how? The vacuum expectation value of the Wilson loop operator remains invariant under ambient isotopy and can be determined via the path integral formulation of Chern-Simons theory. Now, this is remarkable."
},
{
"end_time": 4260.896,
"index": 187,
"start_time": 4232.5,
"text": " Who would have thought that a quantization procedure, like the path integral, would have anything to do with knots? The Wilson loop can be used to detect knots and links through the lens of quantum field theory, analogously to how electric charges and magnetic monopoles interact in electromagnetism. You may even say string theorists tie themselves into knots over this. I'm here all evening, people. String Field Theory"
},
{
"end_time": 4283.882,
"index": 188,
"start_time": 4261.954,
"text": " String field theory is a background-independent approach to studying string dynamics, with its action formulated as following. Keep in mind, one of the largest criticisms of string theory from Lee Smolin in the early days is that string theory lacks a background-independent formulation. In general relativity, spacetime is the background and it has dynamics to it in a two-way street with matter slash energy."
},
{
"end_time": 4303.916,
"index": 189,
"start_time": 4284.309,
"text": " That is, matter affects space-time and vice versa, as you've heard. When someone says that string theory is background dependent, they mean that most often what you do in string theory is you specify a background rather than you calculate one, and even when you provide one, it doesn't have the same sort of dynamics to it that one would think a quantum theory of gravity should have."
},
{
"end_time": 4326.561,
"index": 190,
"start_time": 4304.428,
"text": " Loop quantum gravity instantiates such dynamics from the get-go, but it lacks quantum field theory, whereas string theory instantiates quantum field theory from the get-go, but lacks such dynamics. The string field, on the other hand, represents an infinite component field that encodes all string excitations, while the BRST charge operator Q makes a reappearance."
},
{
"end_time": 4355.35,
"index": 191,
"start_time": 4326.8,
"text": " After quantization, an interaction term, here, is introduced, resulting in the string field theory potential. This potential allows us to analyze non-perturbative effects, such as the aforementioned tachyon condensation. The homotopy algebraic structure, known as the A-infinity algebra, encodes the associative star product, as well as higher homotopy products. The problem with string theory, compared to, say, regular string theory, is that it's primarily developed for bosonic strings and not superstrings,"
},
{
"end_time": 4368.882,
"index": 192,
"start_time": 4355.623,
"text": " This is partially why it's not been so popular. By the way, do you know who one of the pioneers of string field theory is? Michio Kaku! Yes, the dubious Mr. Quantum Supremacy."
},
{
"end_time": 4386.305,
"index": 193,
"start_time": 4370.469,
"text": " BPS states allow us to understand non-perturbative aspects of string theory. These are states that preserve a portion of supersymmetry, commonly half, and have a mass formula with some bound, where m is the mass, q denotes charges, and gs represents the string coupling constant."
},
{
"end_time": 4415.145,
"index": 194,
"start_time": 4386.305,
"text": " BPS states provide three major benefits. So number one, they're resistant to quantum corrections since their supersymmetry preservation constrains loop corrections. This ensures the stability of properties like masses and interactions despite both perturbative and non-perturbative effects. Number two, there are dual relations mapping BPS states in one theory to those in another. So Prasad and Sommerfield's original work connected BPS states to soliton solutions in around 1975."
},
{
"end_time": 4432.944,
"index": 195,
"start_time": 4415.299,
"text": " 3. They count microscopic states, allowing us to calculate the black hole entropy through the Strominger-Waffel formula given on screen, where S is the Bekenstein-Hawking entropy as we know and love and C is the central charge of the CFT and Q1 and Q5 and N are black hole associated charges."
},
{
"end_time": 4461.425,
"index": 196,
"start_time": 4433.336,
"text": " In other words, BPS states serve as stepping stones for understanding the topography of string theory, not topology. That's a common malapropism. As you study string theory and quantum field theory at the more theoretical level, BPS states are everywhere. Sen's conjecture Sen's conjecture, proposed by Ashok Sen, concerns the behavior of these BPS states, particularly in type 2 string theories, as the string coupling gets varied."
},
{
"end_time": 4483.456,
"index": 197,
"start_time": 4461.92,
"text": " This conjecture posits that these bps states with non-zero mass remain stable even as you vary the coupling constant. The technicality here is that you have to assume that there are no other bps states with the same quantum numbers passing through masslessness during the process. This condition is expressed mathematically as follows, with m denoting the mass of the bps states."
},
{
"end_time": 4502.637,
"index": 198,
"start_time": 4483.763,
"text": " Twister String Theory and Cosmological Models"
},
{
"end_time": 4527.278,
"index": 199,
"start_time": 4503.746,
"text": " Penrose's twister theory can be used to compute scattering amplitudes in gauge and gravity theories, with a twister denoted a z with a superscript alpha, which conforms to the incidence relation given on screen here. Specifically, the twister z alpha encodes information about light rays in spacetime and this incident relation relates these twisters to specific points in flat spacetime, so Minkowski spacetime."
},
{
"end_time": 4538.677,
"index": 200,
"start_time": 4527.756,
"text": " In twister string theory, the action is given on screen here, where the twisters connection to space time comes from something called the Penrose transform, which maps twisters to space time fields."
},
{
"end_time": 4558.865,
"index": 201,
"start_time": 4539.002,
"text": " This transform is defined as something that takes holomorphic sheaf cohomology classes on twister space and maps them to solutions of massless field equations on Michalski spacetime, often expressed as follows, where the calligraphic O represents holomorphic functions on twister space and the latter part represents massless fields in spacetime."
},
{
"end_time": 4579.872,
"index": 202,
"start_time": 4559.087,
"text": " You can liken the Penrose transform to a change of basis in linear algebra, except here we're transforming from one representation of a system, so twisters, to another spacetime fields, under a certain set of rules. Constructing De Sitter vacua in type IIb string theory, using flux compactifications, stabilizes moduli and yields a positive cosmological constant."
},
{
"end_time": 4608.968,
"index": 203,
"start_time": 4579.872,
"text": " The use of twister theory in constructing de Sitter vacua comes from the formulation and solution of supersymmetric constraints within these cosmological models, particularly in relation to the complex geometry of the compactified dimensions, often represented as transformations in the complex structure of collibial manifolds. By the way, if you can think of other applications of twisters to string theory, then let me know. Maybe there's one for non-geometric flux compactifications. Maybe we'll write a paper together."
},
{
"end_time": 4636.152,
"index": 204,
"start_time": 4610.06,
"text": " Yangians. The Yangian Algebra, a class of Hopf Algebras, generalizes the concept of Lie Algebras and has connections to so-called integrable systems. Formally, the Yangian Algebra is associated with a Lie Algebra and its defining relations are given on screen where j are the generators and f denotes the structure constants of the underlying Lie Algebra g and kappa stands for the level of the associated affine Kakmudi Algebra."
},
{
"end_time": 4654.428,
"index": 205,
"start_time": 4636.152,
"text": " Think of Yangians as a natural way to connect algebraic structures with integrable systems. Within string theory, Yangians come up in the ever-popular ADS-CFT correspondence, this time connecting Type IIb string theory on a five-dimensional sphere background to N equals 4 super Yang-Mills theory"
},
{
"end_time": 4671.869,
"index": 206,
"start_time": 4654.428,
"text": " The Yangian symmetry of integrable spin chain models emerges in the planar limit of n equals 4 theory."
},
{
"end_time": 4689.821,
"index": 207,
"start_time": 4672.244,
"text": " Spin chains are the one-dimensional version of the Ising model, and planar limits mean considering only the leading order term 1 over n in the expansion where n is the rank of the gauge group Sun. Note, one of these n's is calligraphic, and the other is not because, confusingly, they refer to different facets."
},
{
"end_time": 4710.401,
"index": 208,
"start_time": 4689.821,
"text": " Don't shoot the messenger. Physics is abound with such pedagogical bewilderments. Anyhow, Yangian symmetry contributes to constructing scattering amplitudes, especially within the Grassmannian formulation. Here, amplitudes emerge as integrals over the Grassmannian manifold, and the Yangian symmetry imposes constraints on the integrand, streamlining the computations."
},
{
"end_time": 4736.681,
"index": 209,
"start_time": 4710.401,
"text": " An example of a Grassmannian integral in the context of scattering amplitudes is given on screen where calligraphic A is the scattering amplitude for N particles and the Grassmannian is GR and we also have an integration measure D and a volume VOL. The PFs are faffions of certain matrices associated with the geometry of the problem. Don't worry if this looks like gibberish, we'll talk more about this when we get to the amplitude hedron. BMN matrix model."
},
{
"end_time": 4747.056,
"index": 210,
"start_time": 4738.2,
"text": " Now we get to one of the big boys, the BMN matrix model. This guy's a proposal for a non-perturbative formulation of Big Daddy M Theory itself."
},
{
"end_time": 4771.578,
"index": 211,
"start_time": 4747.193,
"text": " with action given on screen. Here, Xi are the Hermitian matrices representing transverse coordinates of D0 brains and the Xi are fermionic matrices related to the supersymmetric extension of the model. The action characterizes D0 brain dynamics in a specific plane wave background connected to the Penrose limits of ADS4 x S7 and ADS7 x S4 geometries."
},
{
"end_time": 4785.094,
"index": 212,
"start_time": 4771.92,
"text": " Inspired by Berenstein, Maldacena, and Nostas' Seminal 2002 paper, which within the large end limit, the BMN model has a phase structure with different phases corresponding to different M-theory geometries."
},
{
"end_time": 4802.056,
"index": 213,
"start_time": 4785.316,
"text": " Rigorously, this means that the BMN matrix model in the large N limit can be used to study M theory by analyzing its phase structure described by the eigenvalue distribution of matrices. These are related to the distribution of D0 brains though in transverse space."
},
{
"end_time": 4829.428,
"index": 214,
"start_time": 4802.056,
"text": " The action can be derived from the M-theory supermembrane action via something called truncation, which I won't get into here. Put simply though, truncation means what it sounds like. It's the process of simplifying something larger to something smaller. In this case, truncation means to take a full string theory and consider only a subset of its models or degrees of freedom, making calculations more manageable. The BMN model provides a strong weak coupling duality, which you remember means S-duality."
},
{
"end_time": 4841.869,
"index": 215,
"start_time": 4829.428,
"text": " This time it's between the gauge theory on the D0 brains and M theory in the plane wave background. This generalizes the ADS-CFT correspondence. BFSS matrix model."
},
{
"end_time": 4860.964,
"index": 216,
"start_time": 4842.892,
"text": " This is the older, scrawnier cousin of the BMN. The BFSS provides a non-perturbative definition of M-theory by considering quantum mechanical system of N-coincident D0 brains, whose actions are described by the action on screen here, where the XIs again represent 9 Hermitian matrices,"
},
{
"end_time": 4887.756,
"index": 217,
"start_time": 4860.964,
"text": " associated with the transverse coordinates of the D0 brain, and theta is a 16-component Majorana vial spinner introducing fermionic degrees of freedom. And g is the coupling constant, of course. I say scrawnier because this model has difficulties with the definition of the ground state. It's also formulated in flat spacetime. The action displays UN gauge symmetries and local symmetries, with the positive definite bosonic potential coming from the commutator and the fermionic term representing supersymmetry."
},
{
"end_time": 4916.51,
"index": 218,
"start_time": 4887.756,
"text": " In the large N limit, the BFS matrix model is conjectured to encompass M theory in something called the infinite momentum frame, ascribing the eigenvalues of the X matrices to the D zero brain positions in transverse space. Actually, if you like cone gauge quantized the M2 brain and add a turn Simon's mass term to the BFS, then you almost get to the previous BMN model. We talk more about both the BMN and the BFS models here in this podcast with string theorist Stefan Alexander."
},
{
"end_time": 4919.548,
"index": 219,
"start_time": 4917.517,
"text": " IKKT Matrix Model"
},
{
"end_time": 4949.77,
"index": 220,
"start_time": 4920.452,
"text": " This is the creepy, awkward, super-genius neighbor of both of the previous two. This model is a non-perturbative definition of, again, type IIb string theory, where spacetime emerges from the dynamics of n by n Hermitian matrices. The action is given on screen, and the gamma term here are the 10-dimensional version of the gamma matrices that you learn in undergrad for quantum mechanics. Bet you didn't know you could generalize them to virtually any dimension! The model's action has Lorentz symmetry and type IIb supersymmetry."
},
{
"end_time": 4978.609,
"index": 221,
"start_time": 4949.77,
"text": " Sometimes people say about physics theories that it has manifest Lorentz supersymmetry. I don't like the term manifest because it's an equivocal term much like saying the quote-unquote following exercise is trivial. The IKKT model was developed when its creators were investigating de-instanton contributions to type 2b superstring theory. The non-commutative geometry of spacetime is represented by commutators here which imply that the spacetime points no longer are sharply defined. Does this mean spacetime is doomed?"
},
{
"end_time": 5007.073,
"index": 222,
"start_time": 4979.002,
"text": " Well, that's something we talk about with Peter Voigt here in this podcast about string theory and spacetime. The large and limit of IKKT matrix model is conjectured to capture the full non perturbative dynamics of type two B superstring theory in 10 dimensions. So far, this hasn't been proven. And if anyone knows how, let me know again, we'll write a paper together. The low energy effective action of the model reproduces the 10 dimensional type two supergravity action supporting its connection to type two B superstring theory."
},
{
"end_time": 5018.473,
"index": 223,
"start_time": 5007.073,
"text": " Kovanov Homology"
},
{
"end_time": 5043.507,
"index": 224,
"start_time": 5019.599,
"text": " Chern-Simon action is a tool, a powerful tool, in the study of topological invariance and gauge theories. Unfortunately, it's restricted to exactly three dimensions by construction. Kovanoff's extension seeks to generalize the theory to four dimensions using a connection B on a gerb with the action as follows on a four-dimensional manifold N. Kovanoff homology is a categorification of the Jones polynomial."
},
{
"end_time": 5071.954,
"index": 225,
"start_time": 5043.916,
"text": " Historically, Chern-Simon's theory emerged in the late 1970s through the works of Xingxuan Chern and James Harris Simon, whose collaboration gave rise to new insights in various branches of mathematical physics. The key challenge is now to explore whether the structures of three-dimensional Chern-Simon's theory, such as Naughton Variance and Wilson Loops, can be successfully captured in a four-dimensional extension. We talk about Chern-Simon and Kovenov Homology here in this podcast with Dror Barnettin."
},
{
"end_time": 5102.534,
"index": 226,
"start_time": 5073.183,
"text": " Black holes and higher compositional laws. In the Stu model of string theory, so the STU model, which is a specific model of compactification involving certain fields called STU, we explore the connection between extremal black holes, so those of maximum charge, and by Garva's higher compositional laws, which generalize the classical compositional laws of quadratic forms to higher degrees. U-duality orbits of these black holes are characterized by their charge vectors in tensor Z2, tensor Z2."
},
{
"end_time": 5125.418,
"index": 227,
"start_time": 5102.807,
"text": " By the way, that's orbits in the group theoretic sense rather than in the planetary sense. It's hypothesized that these group orbits correspond to equivalence classes of something called Bargava's cubes, which are numerical representations of algebraic structures containing triples of balanced-oriented ideals, which is a specific way of structuring certain subsets of rings, in rings of discriminant D."
},
{
"end_time": 5150.862,
"index": 228,
"start_time": 5125.981,
"text": " It may be that black hole microstates correspond to narrow class group classes. The narrow class group is a concept from algebraic number theory. It's a generalization of the ideal class group, which is a fundamental group associated with a ring that measures the failure of unique factorization of that ring. The Bekenstein-Hawking formula written on screen here with delta mirroring d escapes a full explanation, so who knows?"
},
{
"end_time": 5170.162,
"index": 229,
"start_time": 5151.254,
"text": " Fun fact, Bargava is a genius mathematician who won the field medal and grew up less than an hour away from me. Shout out to my fellow Torontonians. P.S. the STU black hole is a supersymmetric extremal solution in n equals 2, d equals 4 supergravity characterized by three independent complex scalars STU."
},
{
"end_time": 5197.056,
"index": 230,
"start_time": 5170.418,
"text": " Its charge configuration is given by the integer matrix on screen here, where the electric and magnetic charges are denoted Q and P respectively, and the charge cube obtained via the tensor product Q tensor Q tensor Q yields a 6 by 6 by 6 integer tensor analogous to Bargava's cube of integers, which governs higher compositional laws describing the action of the arithmetic group SL2Z on integral and re-quadratic forms."
},
{
"end_time": 5204.326,
"index": 231,
"start_time": 5199.599,
"text": " Higher Dimensional Non-Geometric Backgrounds What does non-geometric mean?"
},
{
"end_time": 5234.548,
"index": 232,
"start_time": 5204.821,
"text": " How can you have a non-geometric space? Recall that a geometric space is one where you can have a globally well-defined notion of a metric, and that your space obeys the usual differential geometric rules, such as compatibility of coordinate patches and a defined notion of parallel transport. There are some spaces that don't have these qualities, yet are spaces in their own right. Maybe they have a metric, for instance, but it's only local, not global. Maybe they have non-commutative or non-associative spatial aspects."
},
{
"end_time": 5253.78,
"index": 233,
"start_time": 5234.548,
"text": " In string theory, we explore R-spaces and their R-fluxes and T-folds coming from non-trivial H-fluxes. The R-spaces and their associated R-fluxes are related to the geometric fluxes that come about in the process of compactification. They can be understood as generalizations of torsion in the underlying geometry,"
},
{
"end_time": 5266.049,
"index": 234,
"start_time": 5253.78,
"text": " and they play a role in determining the effective low-energy field theory. On the other hand, T-folds are a class of non-geometric backgrounds that come from compactifying string theory on manifolds with non-trivial H fluxes."
},
{
"end_time": 5284.957,
"index": 235,
"start_time": 5266.442,
"text": " These H fluxes are associated with the three-form field strength H of the Nevo-Schwarz-Nevo-Schwarz two-form potential B. The non-trivial H fluxes can lead to interesting topological features, for example, non-geometric monodromies and non-commutative geometry, each of whom have implications for the structure of the compactified theory."
},
{
"end_time": 5308.848,
"index": 236,
"start_time": 5284.957,
"text": " The non-geometric Q flux is characterized by the formula on screen. So what role does Q flux play? Well, it bestows our spaces their non-commutative and non-associative nature. Now the tricky part is how do you patch in a compatible fashion these local coordinate charts? How do we go about addressing this with something called double field theory that embeds doubled spacetime? It merges geometric"
},
{
"end_time": 5329.753,
"index": 237,
"start_time": 5309.189,
"text": " Algebraic K-theory"
},
{
"end_time": 5349.411,
"index": 238,
"start_time": 5330.316,
"text": " K-theory is a way to study topological invariance of vector bundles classified by growth in D groups K. The connections between K-theory and string theory come about when you classify D-brains. The RRs of the Ramon-Ramon field strength F exhibit quantization of the form, being members of the equivalence class, also the cohomology class,"
},
{
"end_time": 5376.425,
"index": 239,
"start_time": 5349.411,
"text": " where X denotes the space-time manifold commonly related to k-theory classes. There's a great introduction to k-theory and co-bordism theory in this lecture here. This is an advanced topic. Even defining what k-theory is, what a growth-indique group is, is going to take several minutes. Type IIb string theory classifies d-brains by k0 through even-rank k-theory classes, while type IIa does so by k1 via odd-rank k-theory classes."
},
{
"end_time": 5403.166,
"index": 240,
"start_time": 5376.817,
"text": " In simpler terms, K-theory allows us to discern the difference between objects that can be continuously deformed into one another, much like various D-brain configurations. These ideas, as applied to physics, came about through the work of Atiya, Witten, and Harava, among several others. For the string theoretic context we're interested in, specific supersymmetric theories, like the Witten index, a topological invariant counting BPS states,"
},
{
"end_time": 5431.732,
"index": 241,
"start_time": 5403.166,
"text": " corresponds to the K-theory Euler class, where the dimension here represents the rank of the K-theory group. The Atiyah-Herzabrotch spectral sequence establishes the link between K-theory and ordinary cohomology, and is an active research to this day, trying to understand how D-brains behave. I'm thinking of doing an iceberg on algebraic geometry. If you would like to collaborate on it, please let me know in the comments. W-strings. W-algebras extend the verisaur algebra,"
},
{
"end_time": 5459.087,
"index": 242,
"start_time": 5431.92,
"text": " By incorporating higher spin currents. To account for these currents, the BRST charge has to be modified, where T is the stress energy tensor and CN are the ghost fields. UN are the higher spin currents and the contour integration is over some curve C. As usual, physical states are required to satisfy the BRST condition. W strings are a different type of string, coming from W algebras, but they have issues. Some of them being that they have negative norm states,"
},
{
"end_time": 5486.954,
"index": 243,
"start_time": 5459.087,
"text": " and problems with unitarity. The exact connection between alternatives of string theory, including tachyonic, baryonic, non-geometric backgrounds, and fractional strings in W-algebras isn't currently clear. The stress energy tensor in the context of W-algebras provides geometric information about the world sheet. This is at the frontier of research, even though its inception dates back to the 1980s. The failure of string theory."
},
{
"end_time": 5513.319,
"index": 244,
"start_time": 5488.575,
"text": " The failure of string theory is something that's been talked about for decades before it was cool to dunk on string theory. Now it's all in vogue. Hey, I hate string theory. There were just four initial primary critics, Eric Weinstein, Lee Smolin, Peter Wojt, and Sabine Hassenfelder. Though you can lump several other physics and math professors there as well, they just weren't so vocal. Okay, so what is meant when people say that string theory has failed?"
},
{
"end_time": 5535.862,
"index": 245,
"start_time": 5513.319,
"text": " Okay, number one, there's a lack of experimental evidence. String theory has not provided any testable predictions that could be verified or falsified through experiments, which is a fundamental requirement for scientific theory. This is technically false. It's provided predictions that are just far out of the range of what we can currently test. Number two, a lack of connection to the standard model."
},
{
"end_time": 5548.985,
"index": 246,
"start_time": 5535.862,
"text": " Throughout this whole iceberg so far, I was careful to say the word quantum field theory rather than the standard model. And that's because despite the hype, string theory is far from being a unification of the standard model with gravity."
},
{
"end_time": 5571.101,
"index": 247,
"start_time": 5549.326,
"text": " Rather, it makes compatible gravity with quantum field theory. This is decidedly different. Quantum field theory is a general framework and it suffers from its own issues of inequivalent representations and not being rigorous, at least enough for the mathematicians. And the standard model itself is a far cry from being the unique quantum field theory. Number three, non-unique solutions."
},
{
"end_time": 5589.189,
"index": 248,
"start_time": 5571.101,
"text": " Recall, there are various possible vacua that outnumber the particles in the observable universe. The academic and research environment has often favored string theory, leading to pressure on physicists to conform to this framework, at the expense of exploring other ideas."
},
{
"end_time": 5612.722,
"index": 249,
"start_time": 5589.189,
"text": " Number five. In terms of data showing string theories decline as a litmus test, you can see the failure by the lack of Wikipedia entries in the history section of string theory, where every decade prior warranted its own section before the last two decades. Despite thousands and thousands of more people working on string theory than ever before, it only has a single entry. Furthermore, it's the smallest entry."
},
{
"end_time": 5635.299,
"index": 250,
"start_time": 5612.722,
"text": " Number six, another data point you can use as a rough heuristic is that browsing Ed Whitten's publication record on Google Scholar, you can notice a decrease in string-related articles with time. Number seven, even Ed Whitten's collaborator Edward Frankel discusses the failure of the original promise of string theory to provide a unique theory of everything, going unacknowledged"
},
{
"end_time": 5659.582,
"index": 251,
"start_time": 5635.299,
"text": " Think Verizon, the best 5G network is expensive? Think again. Bring in your AT&T or T-Mobile bill to a Verizon store today and we'll give you a better deal. Now what to do with your unwanted bills? Ever seen an origami version of the Miami Bull? Jokes aside, Verizon has the most ways to save on phones and plans where you can get a"
},
{
"end_time": 5681.118,
"index": 252,
"start_time": 5660.93,
"text": " Layer 5 Non-BPS Brains"
},
{
"end_time": 5701.203,
"index": 253,
"start_time": 5681.834,
"text": " In m-theory compactifications on Kali-Biao threefolds, non-BPS brains are intriguing objects. They're extremal solutions, meaning that they saturate the so-called Bogomolnyi bound, and that means basically that they have a minimum amount of energy given a fixed set of quantum numbers like charge."
},
{
"end_time": 5720.52,
"index": 254,
"start_time": 5701.374,
"text": " Some of these non-BPS brains are called non-BPS attractors and have a connection to something called the weak gravity conjecture, which I'll explain shortly, though we did talk about it earlier in the Swampland program. Unlike BPS brains or BPS states, which preserve half the supersymmetry, non-BPS brains preserve even less."
},
{
"end_time": 5749.036,
"index": 255,
"start_time": 5720.52,
"text": " resulting in non-vanishing central charges and non-trivial scalar potentials. This quote-unquote attractor mechanism is given by the effective potential formula on screen here and governs the behavior of moduli fields ZI at the horizon of extremal black holes. Z denotes the central charge and capital DI signifies the Caylor covariant derivative and capital GI is the gauge kinetic function. An intuitive way that at least I understand non-BPS attractors"
},
{
"end_time": 5768.985,
"index": 256,
"start_time": 5749.036,
"text": " is that they can be thought of as objects that attract scalar fields to specific values in the moduli space near the horizon, stabilizing them and breaking any remaining supersymmetry. By the way, the weak gravity conjecture is the statement that gravity will always be the weakest force in any consistent quantum theory of gravity."
},
{
"end_time": 5786.169,
"index": 257,
"start_time": 5768.985,
"text": " Black Hole-Cudit Correspondence"
},
{
"end_time": 5803.541,
"index": 258,
"start_time": 5787.432,
"text": " In recent work by Rios, extremal black holes in 5D and 6D are investigated within the framework of string theory, making use of n equals 8 and n equals 2 supergravity correspondences to find a connection between quantum states and spacetime geometry."
},
{
"end_time": 5826.647,
"index": 259,
"start_time": 5803.848,
"text": " The key idea here is to consider extremal black holes as qudits, so a higher dimensional generalization of qubits or qutrits. And we do this through the lens of Hopf vibrations and Jordan Algebras. To be precise, Rios demonstrated that rank 1 elements in Jordan Algebras of degree 2 and 3 can be associated with qubits and qutrits respectively."
},
{
"end_time": 5850.145,
"index": 260,
"start_time": 5826.647,
"text": " In particular, Q bits are formalized by H2O, while Q trits are formalized by H3O, and the O is the octonions. What's cool is that when you take into account U-duality groups, these transformations can be understood as quantum information theory's SLOOCC operations, acting on charge vectors Q by GQ, where G denotes the U-duality group."
},
{
"end_time": 5867.875,
"index": 261,
"start_time": 5850.145,
"text": " So what does this mean? It means that extremal black holes can be viewed as a cosmic quantum circuit and their entropic and dynamic properties may potentially be emulated by quantum algorithms. See the paper here for more information. Dilaton and genus expansion."
},
{
"end_time": 5883.37,
"index": 262,
"start_time": 5868.268,
"text": " The dilaton field is one of the most important fields in string theory, denoted by this phi, which is a scalar field responsible for the so-called genus expansion of string amplitudes. This genus expansion is like expanding in Feynman diagrams, except for strings."
},
{
"end_time": 5900.299,
"index": 263,
"start_time": 5883.37,
"text": " It also determines the strength of string interactions with the coupling constant given as the exponential. The dilaton field equation expressed on screen here shows the relationship between the dilaton field and the curvature R, the Regis slope as usual, and the three-form as usual H. In the genus expansion,"
},
{
"end_time": 5923.507,
"index": 264,
"start_time": 5900.299,
"text": " string amplitudes are classified according to the topology of their world sheet surfaces with the genus G representing the number of handles or even equivalently holes in the world sheet. This can be interpreted as a perturbative series in the string coupling where each term is proportional to G to the 2G minus 2 and the first G there it's difficult to say is the string coupling constants in G"
},
{
"end_time": 5938.319,
"index": 265,
"start_time": 5923.507,
"text": " Geometric Quantization"
},
{
"end_time": 5958.012,
"index": 266,
"start_time": 5938.865,
"text": " Now geometric quantization is a method for constructing quantum theories from classical systems. So you start by identifying the classical system's phase space with a complex line bundle, dubbed the quote-unquote pre-quantum line bundle. Associating this bundle entails a covariant derivative, though you have to make use of the K-layer potential."
},
{
"end_time": 5975.964,
"index": 267,
"start_time": 5958.012,
"text": " and the resulting curvature aligns with the phase space's symplectic form. Quantum states are viewed as sections of this line bundle, satisfying Q psi, where psi represents a section of the pre-quantum line bundle, and Q is a quantum operator derived from the classical Hamiltonian."
},
{
"end_time": 6000.094,
"index": 268,
"start_time": 5975.964,
"text": " Specifically though, the quantum states are better described as sections of a quotient bundle obtained by dividing the pre-quantum line bundle by the chosen polarization. This is a large topic that I'll be exploring more on an upcoming podcast with Eva Miranda, so feel free to subscribe to see it as many students are only taught the Feynman path integral or canonical quantization as quantization. The Langlands Program."
},
{
"end_time": 6018.729,
"index": 269,
"start_time": 6000.947,
"text": " The Langlands program is a broad set of conjectures in number theory and representation theory that's at the forefront of research and math, some of the most pure of pure math. That's why it's surprising that it has connections to physics, particularly in two-dimensional quantum field theories and 4D gauge theories."
},
{
"end_time": 6042.756,
"index": 270,
"start_time": 6018.729,
"text": " See, there's something called the Langlands Correspondence. This relates automorphic representations of a reductive algebraic group G, which you can think of as the group of symmetries of a number field, to representations of its dual group G-check, which is actually another reductive algebraic group. Now, the Langlands program is so broad that it has various sub-programs like the Geometric Langlands Correspondence,"
},
{
"end_time": 6059.599,
"index": 271,
"start_time": 6042.756,
"text": " which is a version of the Langlands correspondence for curves over algebraically closed fields. This has physics quote-unquote applications, and I say that lightly because it's not clear if you can call them even applications, and it shows connections between the action of the chiral algebra on the space of conformal blocks."
},
{
"end_time": 6085.691,
"index": 272,
"start_time": 6059.599,
"text": " This connects the representations of the loop group G hat to the space of D modules on the moduli stack G hat check, so the dual group local systems, which provides a deeper insight into the action of the chiral algebra on the space of conformal blocks. Within the scope of electric magnetic duality, the four dimensional n equals four super Yang-Mills theory provides an example of s duality, which has a specific connection to the Langlands correspondence"
},
{
"end_time": 6108.404,
"index": 273,
"start_time": 6085.691,
"text": " through the identification of the electric and magnetic gauge groups with the Langlands dual groups. I talk more about both string theory and the Langlands program here in this podcast with Edward Frankel. There's also this lecture by Ed Witten on gauge theory, geometric Langlands and all that. Link to all resources are in the description. Modular forms and string partition functions."
},
{
"end_time": 6128.302,
"index": 274,
"start_time": 6108.66,
"text": " Modular forms originated from the work of Gauss, Riemann, and Klein. They're complex analytic functions with specific transformation properties under discrete subgroups of SL to R, usually SL to Z. In string theory, the world-cheap conformal theory's partition function, Z, must be invariant under modular transformations for consistency."
},
{
"end_time": 6150.162,
"index": 275,
"start_time": 6128.592,
"text": " This restricts allowed compactification lattices and conformal field theories. Modular forms like the elliptic genus ZEG represent BPS states contributing to black hole microscopic degeneracy. Black hole entropy can be derived from Fourier series coefficients in these modular forms, connecting the mathematical structure and the physical properties of black holes in string theory."
},
{
"end_time": 6178.046,
"index": 276,
"start_time": 6150.162,
"text": " Actually, Andrew Strominger and Kamran Vafa established this connection in about 1996. Interview with both of those, so Strominger and Vafa will be coming up on toe on this topic as well as on the topic of modular bootstraps and CFTs. String Sigma Models with West Zumino Witten Terms. So the WZW term is represented by the integral here, where K is the level and A is the gauge field. It acts as a topological invariant and quantizes H flux."
},
{
"end_time": 6203.746,
"index": 277,
"start_time": 6178.422,
"text": " If you think this looks like the Chern-Simons term, you are correct. They both originate from the same structure, a three-form constructed from the gauge field. The Chern-Simons term is typically found in 3D topological field theories and is represented by the integral of a three-form, just like the WZW term. Both involve the gauge field, their exterior derivatives, and then the wedges. However, there are differences in their coefficients and the overall context in which they appear."
},
{
"end_time": 6230.64,
"index": 278,
"start_time": 6203.746,
"text": " The WZW term is relevant for string sigma models and conformal field theories while the Chern-Simons term plays a role in the topological field theories and is associated again with the invariance and linking numbers in Jones polynomial. The WZW term can be thought of as a curvature term necessary to maintain the consistency of string theory with the central charge formula being corrected to what's on screen here where H check is the dual coxeter number of the Lie group G."
},
{
"end_time": 6245.981,
"index": 279,
"start_time": 6230.64,
"text": " This combined action here retains conformal invariance as long as the background fields comply with the equations of motion and the WZ terms satisfy something called the Polyakov-W consistency condition. Black hole string transitions."
},
{
"end_time": 6272.346,
"index": 280,
"start_time": 6246.34,
"text": " In a paper recently published by Maldacena and Witten in 2023, they look into the connection between black holes and highly excited strings. Actually, this revisits the self-gravitating string solutions by Horowitz and Polczynski made decades earlier. Their analysis of the linear sigma models for the heterotic strings demonstrates a smooth transition from the Horowitz-Polczynski solutions to black holes,"
},
{
"end_time": 6300.128,
"index": 281,
"start_time": 6272.346,
"text": " a connection hindered in type 2 superstring theories by differing supersymmetric indices. The entropy S of charged black hole solutions derived from these string solutions through generating techniques adheres to the relation given on screen here, where Q and P represent the charges and S0 is the entropy of the neutral solution. This is super exciting because it shows a new avenue for connecting black hole entropy, quantum states, and string theory."
},
{
"end_time": 6319.428,
"index": 282,
"start_time": 6302.005,
"text": " JT Gravity and Black Holes We've talked about the ADS-CFT duality before, denoted here, at least up to a Legendre transformation. Something we haven't mentioned before is that there's actually a simplified two-dimensional dilaton gravity model called JT Gravity."
},
{
"end_time": 6348.285,
"index": 283,
"start_time": 6319.889,
"text": " If I could pronounce the author's names, I would, but I can't, so I'll show it on screen. This time, we have a correspondence of ADSC2 and CFT1 that is a two-dimensional dilaton gravity model defined by the action given here where H is the induced metric on the boundary and K is the extrinsic curvature. So why is JT gravity important? Because its equations of motion are trivial in the bulk, but they are non-trivial at the boundary, meaning that we have an elementary context"
},
{
"end_time": 6373.677,
"index": 284,
"start_time": 6348.285,
"text": " for exploring holography. Its solutions encompass ADS2 black holes whose entropy can be associated with the dilaton field. Now the asymptotic behavior, again this means far away from some region of interest, of the dilaton field provides a practical regularization to quantify thermodynamic properties. Machine learning. This is a new field that has exploded in interest in the past decade."
},
{
"end_time": 6396.937,
"index": 285,
"start_time": 6373.831,
"text": " Recall the vast landscape of string theory. Researchers are seeing how the heck can neural networks tackle the parameter space, so the 10 to the 500 proposed vacua. A notable application involves the exploration of these CY manifolds, like we mentioned before, where a machine learning algorithm predicts Hodge numbers from the input adjacency matrix of the quiver diagram of the toric diagram."
},
{
"end_time": 6421.237,
"index": 286,
"start_time": 6396.937,
"text": " This leads to a regression problem formulated as follows, with A being the adjacency matrix and F is the learned function. Yang He, along with several others, were pioneers in applying machine learning to this field in 2017. As for F-theory compactifications, machine learning deduces the gauge group and matter content from the singularity structure of an elliptically-fibrid kalibi-yau 4-fold given as input."
},
{
"end_time": 6441.084,
"index": 287,
"start_time": 6421.237,
"text": " Chiral Factorization Algebras"
},
{
"end_time": 6469.991,
"index": 288,
"start_time": 6441.647,
"text": " Chiral factorization algebras, which are spearheaded now by Emily Cliff, are a rigorous method for investigating quantum field theories. You've heard that quantum field theory suffers from the problem of being rigorously defined. This is only partially true. There are actually several rigorous formulations. It's just that none of them capture the full breadth of quantum field theory. A chiral algebra is a vertex operator algebra, V, that fulfills the operator product expansion relation on screen here."
},
{
"end_time": 6475.06,
"index": 289,
"start_time": 6469.991,
"text": " for both v of w and v of z that are members of this vertex operator algebra."
},
{
"end_time": 6501.203,
"index": 290,
"start_time": 6476.323,
"text": " This association has singularities with simple poles at most, embodying locality in chiral conformal field theories. So, how do factorization algebras fit into this? Well, firstly, they're a generalization of vertex operator algebras, and secondly, they provide a systematic, ground-up methodology to develop conformal field theories. For a VOA-V, the related factorization algebra F of V"
},
{
"end_time": 6529.189,
"index": 291,
"start_time": 6501.203,
"text": " assigns a state space to each interval i in R. Factorization maps are such that we have this relation here, which almost looks like an exponential property for disjoint unions i1 and i2 within i, and these preserve the OPEs. By exploiting the world-cheap conformal symmetry, holomorphic and anti-holomorphic factorization of correlators is achievable, thus reducing calculations to one-dimensional conformal field theories. Geometric Unity"
},
{
"end_time": 6549.514,
"index": 292,
"start_time": 6530.145,
"text": " When thinking about string theory, it's useful to think about alternatives. Usually, loop quantum gravity is proposed as the primary contender, but that's only a contender in the quantum gravity stage, not on the toe unification stage. That is to say, it's not clear how loop quantum gravity is a unification of general relativity and the standard model."
},
{
"end_time": 6565.094,
"index": 293,
"start_time": 6549.514,
"text": " That whole Toll Unification stage is a decidedly different stage than the Quantum Gravity one, and there are not many combatants on it. Wolfram is one such combatant, Peter Wojt is another, Garrett Lisi is another, Eric Weinstein is another, with his Geometric Unity approach."
},
{
"end_time": 6590.282,
"index": 294,
"start_time": 6565.094,
"text": " Usually Eric explains it as a theory where the four-dimensional space-time that we know and love is not fundamental but rather emergent, but I think that's doing geometric unity a disservice. One of the reasons that I like geometric unity is because it takes seriously as a primitive a four-dimensional manifold which is then used to construct other unfamiliar structures and familiar ones. Geometric unity is quite intricate and can well have its own iceberg."
},
{
"end_time": 6613.746,
"index": 295,
"start_time": 6590.282,
"text": " But what other structures? Well, the Observer's, for instance, which is characterized by a triple X4, Y14, and embedding into a higher dimensional Riemannian monofold. These embeddings are local Riemannian and induce a metric on X4, thus generating a normal bundle. At some point, you choose a signature, which then gives the so-called chimeric space Y7,7."
},
{
"end_time": 6631.834,
"index": 296,
"start_time": 6613.746,
"text": " The main principle bundle in GU is as follows, where the first guy is the double cover of the frame bundle of the chimeric bundle, H is the unitary group of 64,64, and this row, this variation on row, is the representation of the spin group on complex Dirac spinners."
},
{
"end_time": 6658.166,
"index": 297,
"start_time": 6631.834,
"text": " From this, you get what looks like space-time spinners and internal quantum numbers. There are other arguments for recovering bosonic particles as well. Often in the discussion of toes is the discussion of Grand Unified Theories, or GUTs, but just so you know, GUTs aren't toes. However, there's one GUT called the SU10 model, or the Georgi-Glashow model. There's also the SPIN10 Georgi model, and there's a SPIN4 x SPIN6 Patissala model."
},
{
"end_time": 6681.988,
"index": 298,
"start_time": 6658.166,
"text": " These all have significance in GU, with the number 10 here being related to the 10 degrees of freedom in the four-dimensional Romanian metric. Geometric unity is quite intricate and can well have its own iceberg. Non-critical strings Non-critical strings deviate from the critical dimension, which is 10 as we know for superstrings and then 26 for bosonic strings."
},
{
"end_time": 6702.073,
"index": 299,
"start_time": 6682.193,
"text": " They're related to the cancellation of conformal anomalies, which is a different type of anomaly we haven't discussed. To study non-critical strings, random matrices are usually introduced. Consider the random matrix ensemble on screen here, where M is an N by N Hermitian matrix, V of M is a potential function, and lambda is a coupling constant."
},
{
"end_time": 6723.131,
"index": 300,
"start_time": 6702.466,
"text": " This ensemble is a discretized world-sheet action for non-critical strings with M representing discretized world-sheet fields and V of M encapsulating string interactions. In the 1980s, the study of non-critical strings using random matrices led to the discovery of the double scaling limit by Bresen, Edzicson, Parisi,"
},
{
"end_time": 6739.377,
"index": 301,
"start_time": 6723.131,
"text": " The scaling behavior of the matrix model near critical points exposes properties of non-critical strings such as string susceptibility, which is determined by the specific heat exponent alpha via the relation that gamma equals 2 minus alpha."
},
{
"end_time": 6764.053,
"index": 302,
"start_time": 6739.377,
"text": " To maintain conformal invariance in non-critical strings, Louisville theory is used, and we do so by introducing a Louisville field, phi that couples to the world sheet curvature, effectively compensating for the deviation from the critical dimension. By the way, Louisville theory is something you use to maintain conformal invariance when working with dimensions different from the critical dimension. Type 0A and 0B."
},
{
"end_time": 6786.015,
"index": 303,
"start_time": 6764.548,
"text": " Tachyonic states are characterized by an imaginary mass and faster than light propagation, though this only happens if you interpret as a particle and if you interpret the coupling constant as being mass. In bosonic string theory, for instance, the mass squared of a string state is given as follows, where n is the excitation level and a is the normal ordering constant."
},
{
"end_time": 6807.108,
"index": 304,
"start_time": 6786.015,
"text": " It turns out that in addition to the five flavors of string theory that you know and love, there are several more, two of them being these type 0a and type 0b, but these are characterized by these flagecious tachyons, as well as because they describe only bosons, thus you hear little about them. Fractional Strings and Non-Integer Conformal Weights"
},
{
"end_time": 6828.114,
"index": 305,
"start_time": 6807.978,
"text": " Fractional strings are strings characterized by non-integer mode numbers. This means that the strings' vibrational modes don't conform to simple harmonic patterns. Conventional conformal weights result from the normal ordering of the verisora generators L0 and then L bar 0, with integer conformal weights tied to the quantization oscillators."
},
{
"end_time": 6848.473,
"index": 306,
"start_time": 6828.114,
"text": " However, for fractional strings, we have non-integer conformal weights which defies typical quantization. We have to re-examine string spectra and world sheet symmetries because of these, if we're to take them seriously. The modified conformal weights can be discerned through the formula here for H, where K squared signifies the spacetime momentum,"
},
{
"end_time": 6871.937,
"index": 307,
"start_time": 6848.473,
"text": " and M stands for the fractional mode number, and alpha prime is the, again, the Reggi slope, the traditional verisora constraints are impacted, culminating in the updated conditions, which are on screen here. Now here's the question, what the heck could a fraction of a harmonic mean? It's not clear to me how to visualize them. Fractional strings aren't studied anywhere near as much as regular strings, which is, again, why you haven't heard of them."
},
{
"end_time": 6889.087,
"index": 308,
"start_time": 6872.261,
"text": " When people talk about the five flavors, always keep in mind, we're talking about vanilla, chocolate, strawberry, mint, and cookie dough. But those aren't the only flavors. There's also, hey, there's peanut butter cup, something that the toe logo looks like, by the way. Unconventional twisted heterotic string theory."
},
{
"end_time": 6905.64,
"index": 309,
"start_time": 6889.94,
"text": " Unconventional twisted heterotic string theory is a different approach than usual to heterotic string theory has been proposed by introducing twisted boundary conditions using a twist operator omega as an automorphism of the world sheet satisfying that omega squared equals one."
},
{
"end_time": 6932.09,
"index": 310,
"start_time": 6905.64,
"text": " which acts on the left moving sector by modifying the oscillators alpha mu n to omega for all n and mu. The twisted action is given by simply a sum of both the right and the left one, though the left one now has a twistedness in it. So the twisted left moving action is derived by replacing the conventional oscillators with their twisted counterparts. By choosing specific twist operators, different massless spectra and gauge groups can be obtained."
},
{
"end_time": 6960.486,
"index": 311,
"start_time": 6932.09,
"text": " The Schurck-Schwarz mechanism is a historical example of a twisted string theory applied to the breaking of supersymmetry, and this requires compactifying an extra dimension with a twist. To ensure world-cheap conformal symmetry and consistency, the choice of this twisting must commute with the BRST charge, allowing quantization of the twisted heterotic strings via the familiar BRST cohomology. Monstrous M-theory in 26 plus 1 dimensions"
},
{
"end_time": 6975.384,
"index": 312,
"start_time": 6962.278,
"text": " In monstrous M-theory, a recent extension of the standard M-theory to 26 plus 1 dimensions by Chester, Rios and Morani, the massless spectrum of M-theory is shown to have connections to the so-called monster group."
},
{
"end_time": 6998.848,
"index": 313,
"start_time": 6975.725,
"text": " This is what we discussed earlier in the Monstrous Moonshine Conjecture. The deep origins or motivation for the decomposition of the Greece algebra into 98280, direct summed with 98304 and then we have 1, was unknown to Conway. Mirani realized that one of these middle factors, 98304, is a would-be quote-unquote gravitational, so a spin one and a half field."
},
{
"end_time": 7023.814,
"index": 314,
"start_time": 6998.848,
"text": " which is typically found in supergravity. The 98280 was understood to be half the leech lattice, possibly like a Z2 orbifold or the identification of positive roots. The 1 is, of course, the dilaton. The new approach suggests an n equals 1 spectrum in these 27 dimensions or an n equals 2 spectrum in 26 dimensions, so 25 space dimensions."
},
{
"end_time": 7039.343,
"index": 315,
"start_time": 7023.814,
"text": " Analyzing M-theory for n equals 1, so minimal supergravity, in this spatially odd dimensional setting, isn't simple. The moduli space geometry is linked up with the monster group's complexified elements, analogous to the vertex operator algebra representations."
},
{
"end_time": 7066.715,
"index": 316,
"start_time": 7039.565,
"text": " This fusion of the largest sporadic groups representation theory with high energy physics potentially reveals new symmetries in spacetime and I'm excited to see where this research goes, especially as I personally don't know of many applications of the Greece algebra to physics. By the way, if you're wondering about how did they get around Nomm's theorem, they found a way with nested brain worlds. So this no-go theorem applies only when you reduce down to three plus one dimensions."
},
{
"end_time": 7089.633,
"index": 317,
"start_time": 7069.292,
"text": " Double field theory is a t-duality approach to string theory, where you augment spacetime by doubling its dimension, combining the winding and momentum modes of strings into double coordinates, where x tilde and x represent signifying winding and momentum modes, respectively. The DFT framework involves a double metric,"
},
{
"end_time": 7106.152,
"index": 318,
"start_time": 7089.872,
"text": " that sees the conventional metric and the B field as equal. The DFT action is the following, where phi represents the dilaton as usual and r is the Ricci scalar in the double dimension geometry and 2D is the doubled space-time coordinate count, where D is the initial count."
},
{
"end_time": 7134.94,
"index": 319,
"start_time": 7106.152,
"text": " Although the DFT action respects generalized diffeomorphisms, incorporating transformations that blend both Xi and tilde Xi, a stringent constraint must still be instituted for consistency and to retrieve the standard string theory by curtailing these degrees of freedom to the initial dimension count, though this is still being debated today. DFT serves as a geometric method to grasp T-duality, its so-called unifies diverse string theories under a shared framework."
},
{
"end_time": 7146.886,
"index": 320,
"start_time": 7136.323,
"text": " Loop Quantum Gravity LQG is a non-perturbative, background-independent quantum gravity framework reconciling quantum mechanics and general relativity."
},
{
"end_time": 7170.708,
"index": 321,
"start_time": 7147.449,
"text": " It employs Ashtakar variables representing the gravitational field via an SU2 connection A and its conjugate E. The last one is called a densitized triad. Spin networks are graphs with vertices labeled with an intertwiner I and edges by irreducible representations J of SU2 form the foundation of loop quantum gravity."
},
{
"end_time": 7197.261,
"index": 322,
"start_time": 7170.708,
"text": " Just like the motivation for string theory, loop is also quite simple mathematically speaking, and also, like string theory, its humble beginnings, belie its subsequent tortuous flowering. Loop quantum gravity creates the Hilbert space basis of gravitational field quantum states, each representing a quantized three geometry for the three spatial dimensions. Discrete spectra come about for area A and volume V operators."
},
{
"end_time": 7222.108,
"index": 323,
"start_time": 7197.568,
"text": " Transition amplitudes between spin networks originate from spin-foam evaluation, modifying the famous quantum field theoretic path integral technique. Loop quantum gravity was developed or discovered, depending on your philosophical framework, in the 80s by Ashtakar, Rovelli, and Smolin,"
},
{
"end_time": 7247.329,
"index": 324,
"start_time": 7222.108,
"text": " Plenty of work was also done in the 90s by John Byas as well. To this day, it's seen as an antagonist to string theory, but Lee Smolin told me in a recent podcast just last week that string theory and loop quantum gravity are two sides of the same coin. Layer six. Quantum entanglement."
},
{
"end_time": 7257.961,
"index": 325,
"start_time": 7248.49,
"text": " One of the most astounding subjects in modern PopSci is quantum entanglement, with its ostensible faster than light signaling. Let's explore this by starting with entropy."
},
{
"end_time": 7285.282,
"index": 326,
"start_time": 7258.353,
"text": " If we take the von Neumann entropy, where rho is the reduced density matrix, then the holographic entropy, which includes the Ryu-Taki-Yanagi formula as a special case, connects the entanglement entropy with the area of a minimal surface gamma. This relationship gave rise to the so-called ER equals EPR conjecture or heuristic, whatever you want to call it. But what does this mean? It suggests that entangled pairs of particles are equivalent to wormholes."
},
{
"end_time": 7304.974,
"index": 327,
"start_time": 7285.282,
"text": " Now, if that wasn't remarkable enough, it has the further implication that space-time geometry itself emerges from the entanglement structure of underlying quantum states. But what about that firewall argument? That one that suggests a breakdown of the equivalence principle at the black hole event horizon due to maximal entanglement?"
},
{
"end_time": 7316.647,
"index": 328,
"start_time": 7305.435,
"text": " The firewall argument, well, it was proposed by four researchers named Almherry, Mirov, Polchinsky, and Sully, abbreviated as Amps, raised concerns about the validity of ER equals EPR."
},
{
"end_time": 7340.845,
"index": 329,
"start_time": 7316.886,
"text": " According to Amps, a black hole that's maximally entangled with another system, for instance Hawking radiation, can't also be entangled with its own interior, as that would violate the so-called monogamy of entanglement principle. Consequently, the smooth spacetime structure near the horizon, as predicted by general relativity, would break down, and the observer would experience a firewall instead."
},
{
"end_time": 7366.391,
"index": 330,
"start_time": 7340.845,
"text": " This argument has led to this huge debate among physicists, with some proposing possible resolutions such as the soft hair proposal by Hawking, Perry and Strominger, or the idea of state dependence, which states that the experience of an observer falling into a black hole depends on the specific quantum states of the system. This is all fascinating and highly speculative. Let me know if you'd like me to do an iceberg on black holes."
},
{
"end_time": 7398.422,
"index": 331,
"start_time": 7368.439,
"text": " A rigorous analysis of string field theory in the context of non-commutative geometry necessitates the introduction of the Moyel star product into the action given on screen here. The Moyel star product is defined by the following where A is the algebra of the functions on the phase space. Here this theta is a constant anti-symmetric matrix that characterizes the non-commutativity of spacetime coordinates and f and g are again the functions on phase space."
},
{
"end_time": 7422.824,
"index": 332,
"start_time": 7398.712,
"text": " The Moilstar product is an associative but non-commutative product that generalizes the usual pointwise product of functions on phase space in the context of non-commutative geometry. What the Moilstar product does effectively is to deform the commutation relations of the spacetime coordinates and the corresponding fields. This leads to a modification of the usual commutation relations, propagators, and interaction vertices."
},
{
"end_time": 7441.783,
"index": 333,
"start_time": 7423.183,
"text": " In non-commutative spacetime, coordinates satisfy the following algebra. This is supposed to capture some of the fuzziness of spacetime at the string scale. Non-commutative geometry, though, has its roots with mathematicians like Elaine Konis, John von Neumann, and Marie Gerstenharber, all of whom explored it in different contexts before it found its application in string theory."
},
{
"end_time": 7458.916,
"index": 334,
"start_time": 7441.783,
"text": " The appearance of the star product in the scalar field's kinetic term, mass term, and interaction term actually comes from the Seeberg-Witten map that we talked about before, which in turn comes from the open string low energy effective action of the noncommutative scalar field. Quantum Groups and String Theory"
},
{
"end_time": 7476.647,
"index": 335,
"start_time": 7459.855,
"text": " Quantum groups are denoted as follows with this U and the subscript Q of a Lie algebra G and what they are non-commutative deformations of the universal enveloping algebra of the Lie algebra with a deformation parameter Q. Now the defining relation for certain generators is AB equals QBA."
},
{
"end_time": 7492.671,
"index": 336,
"start_time": 7476.647,
"text": " The R matrix, which satisfies the Yang-Baxter equation, encodes the non-commutativity with the defining relation on screen here. Importantly, quantum groups retain the structure of Hopf algebras, allowing a description of both algebra and co-algebra actions."
},
{
"end_time": 7509.275,
"index": 337,
"start_time": 7493.097,
"text": " In the limit when q goes to 1, quantum groups reduce to their classical counterparts, both in terms of Lie algebras and Lie groups. By the way, Hopf algebras are algebraic structures that simultaneously generalize groups, associative algebras, and Lie algebras. How so?"
},
{
"end_time": 7536.015,
"index": 338,
"start_time": 7509.684,
"text": " They have two algebra maps, so a co-product here, which encodes the algebraic structure, and a co-unit, which encodes the identity element of the group-like structure. Hop algebras also possess something called an antipode map, which provides something like the inverse of the group-like elements, and they satisfy this relation on screen here. Their relevance to string theory originates from integrable systems and conformal field theories through the underlying world sheet CFT and its quantum group symmetry."
},
{
"end_time": 7556.254,
"index": 339,
"start_time": 7536.015,
"text": " The connection to braid groups comes from the R matrix, which describes the braid properties of tensor categories associated with those quantum groups. For rational CFTs, the fusion rules, which are given by the Verlinde formula, can be derived using quantum group representations relating conformal weights of primary fields to representation labels."
},
{
"end_time": 7575.759,
"index": 340,
"start_time": 7556.254,
"text": " The pair with the lovely names Drinfield and Jimbo independently introduced quantum groups in the 1980s primarily to investigate integrable systems. Drinfield was also mentioned in the book with Edward Frankel and again the Edward Frankel podcast is on screen here. Love and Math is the book. Exceptional Field Theory"
},
{
"end_time": 7603.78,
"index": 341,
"start_time": 7577.056,
"text": " This is a geometrical scaffold housing varied representations of string theory and 11-dimensional supergravity, employing the terminology of exceptional Lie groups and their corresponding geometry, which are the exceptional in the name exceptional field theory. Its action is on screen here, where G is the EFT metric, D is the dilaton, and H is a measure of the three-form field string. With capital D adopting different values in this exceptional field theory, multidimensional flexibility is apparent."
},
{
"end_time": 7622.108,
"index": 342,
"start_time": 7604.019,
"text": " The exceptional Lie groups transform into global symmetry groups, resulting in exceptional geometries. Now, while EFTs don't unite all string theories, it explores them as specific sectors corresponding to unique solutions of the EFT equations of motion. Amplitude Hedron"
},
{
"end_time": 7643.814,
"index": 343,
"start_time": 7623.029,
"text": " The Amplituhedron is something that Donald Hoffman readily brings up, so it's useful to have an explanation here. Donald has been interviewed several times on this channel before, once solo, with the technical exploration of his theories, another with Joscha Bach, one with John Vervecky, another with Bernardo Kastrup and Susan Schneider, and yet another one with Philip Goff."
},
{
"end_time": 7669.002,
"index": 344,
"start_time": 7643.814,
"text": " The topics usually center around consciousness, though here we'll talk about Nima Arkhani Hamed's amplitude hydron. What this is, is a specific type of convex polytope within RK that encodes scattering amplitudes in N equals 4 supersymmetric Yang-Mills theory. This is realized by a relationship with the positive Grassmannian. This means it's a space of K by N matrices with positive minors."
},
{
"end_time": 7689.667,
"index": 345,
"start_time": 7669.224,
"text": " Mathematically, the amplituhedron is induced from a mapping of the positive Grassmannians under a specific positive map given on screen here, where the map is defined by taking the positive Grassmannian to the amplituhedron via a linear map as follows, with the constraint that all k plus 1 minors of C are non-negative."
},
{
"end_time": 7715.026,
"index": 346,
"start_time": 7689.667,
"text": " Scattering amplitudes can then be computed via integration over the canonical form of the amplitude hedron, providing a way that avoids some complexities of some Feynman diagrams. The amplitude hedron is connected to string theory through that good old celebrated AdS-CFT correspondence relating N equals 4 super Yang-Mills theories to type 2b super string theories in an AdS5 cross S5 background."
},
{
"end_time": 7734.275,
"index": 347,
"start_time": 7715.196,
"text": " It should be specified that it's the scattering amplitudes rather than the amplituhedron itself that connects to this correspondence, with the convex polytope being this calculational tool like a middleman. In 2013, the amplituhedron was introduced by Nima Arkani-Hamed and his collaborator Jora Slav,"
},
{
"end_time": 7750.964,
"index": 348,
"start_time": 7734.275,
"text": " Partially inspired by the study of ancient math objects called a sociahedra. These date back to the 1960s and appear in various branches of mathematics including algebraic topology and combinatorics. The amplitude hedron is appealing because it suggests that we might not need fields."
},
{
"end_time": 7777.108,
"index": 349,
"start_time": 7750.964,
"text": " Double copy theory."
},
{
"end_time": 7808.08,
"index": 350,
"start_time": 7778.166,
"text": " The double copy theory establishes a remarkable correspondence between gauge and gravity theories through something called KLT relations, where gravity amplitudes can be expressed as the square of Yang-Mills gauge theory amplitudes. I like this phrase poetically, but for me it should be expressed a bit more rigorously because, at least for myself, when I hear that the Dirac equation is the square root of the Klein-Gordon equation, or that spinors are the square root of some other structure, personally just confuses me more until I see the math."
},
{
"end_time": 7833.712,
"index": 351,
"start_time": 7808.712,
"text": " When we say that gravity amplitudes are the quote-unquote square of the Yang-Mills amplitudes, we mean that the gravity scattering amplitude can be obtained as a product of two Yang-Mills scattering amplitudes with a modified kinematic substitution given on screen here. This corresponds to the closed string amplitude being constructed from the open string amplitude in the KLT relations. This, by the way, links closed and open string amplitudes."
},
{
"end_time": 7861.476,
"index": 352,
"start_time": 7833.712,
"text": " The color kinematics duality requires that the kinematic numerators satisfy the same Jacobi identities as the color factors. Following this duality, if we have CA equals CB plus CC, then NA is defined as NB plus NC. This allows us to express the graviton scattering amplitude as the square of gluon scattering via something called the BCJ double copy construction. This encompasses the KLT relations which were discovered in the late 1980s."
},
{
"end_time": 7884.787,
"index": 353,
"start_time": 7861.596,
"text": " I forgot to mention that we also have to enforce momentum conservation given on screen here. Now this entire double copy theory is interesting to me because it produces a significant reduction in computational complexity for scattering amplitudes and gravity theories while still drawing connections between gauge and gravity theories similar in spirit to what the amplitude hedron did. You plane integral."
},
{
"end_time": 7908.473,
"index": 354,
"start_time": 7885.93,
"text": " The low-energy effective action for the Type IIb string theory on K3 surfaces relies heavily on the evaluation of U-plane integrals. Recall that K3 surfaces are smooth, compact, complex two-dimensional manifolds with a trivial canonical bundle and a holonomy group SU2, and they're important because of their role in supersymmetry, mirror symmetry, and Calabi-Yau manifolds in compactification."
},
{
"end_time": 7932.927,
"index": 355,
"start_time": 7908.473,
"text": " In this context, the U-plane is the moduli space parameterized by the complex coupling constant, where this theta represents the Raymond-Raymond scalar field and gs denotes the string coupling constant. The BPS states describe the spectrum of stable configurations in the theory. By the way, I've heard other names for the U-plane like the S-duality orbit or the Coulomb branch or the Seiberg-Whitten moduli space and the moduli of vacua."
},
{
"end_time": 7955.64,
"index": 356,
"start_time": 7932.927,
"text": " The integrand takes the form of an exponential multiplied by d and f, where the integer n and the degeneracies d of n specify the bps spectrum, and the modular forms f of k capture the automorphic properties. To evaluate U-plane integrals, you have to use something called the Rademacher expansion. Now, I probably butchered that, and at first I thought that was the same Rademeister as in the moves, but it's something different."
},
{
"end_time": 7982.858,
"index": 357,
"start_time": 7955.947,
"text": " This Rademacher expansion expresses the modular forms as a sum of Poincare series, which allows us to isolate pertinent information from the integrand as follows, where S represents the Clustermann sum and S is a modular parameter. The U-plane integrals are connected with the Mach modular forms, a class of non-holomorphic modular forms, generalizing the classical Eisenstein series, not Einstein, but Eisenstein, and that links number theory and geometry to string theory."
},
{
"end_time": 8009.735,
"index": 358,
"start_time": 7984.497,
"text": " M Theories and Multiple Dimensions of Time. We usually talk about 10 plus 1 dimensions of space-time or 3 plus 1, etc. There's always this plus 1 at the end. This means it's one-dimensional. However, there is work by bars that has two dimensions of time. But what does this mean mathematically? So mathematically, the concept of multiple time dimensions are captured by extending the metric tensor to include extra temporal components"
},
{
"end_time": 8029.087,
"index": 359,
"start_time": 8009.735,
"text": " Or you may see it as x squared plus y squared plus z squared minus t squared. It just has extra minuses after it. In Barr's work, he introduces a second spacetime coordinate, t prime, described by a d plus two dimensional spacetime. You can take this even further to discuss 3D time in the same way that we discuss 3D space."
},
{
"end_time": 8041.8,
"index": 360,
"start_time": 8029.087,
"text": " How? In the context of extending super Yang-Mills theories through exceptional periodicities, this recent work by Rias, by Chester, by Morani, they consider the super algebra in D equals 27 plus 3 dimensions."
},
{
"end_time": 8069.889,
"index": 361,
"start_time": 8042.363,
"text": " The descending dimensional sequence from a super algebra in D equals 27 plus 3 to 26 plus 1 reduces the dimensions directly along an 11 dimensional brain world volume yielding an N equals 1 super algebra in D equals 11 plus 3 which upon successive dimensional truncation aligns with the N equals 1 super algebra in D equals 10 plus 1 and D equals 11 plus 1 as well as type 2A to B strings."
},
{
"end_time": 8083.712,
"index": 362,
"start_time": 8070.162,
"text": " This suggests an 11-dimensional brain world volume origin for string dualities in both M and F theory, though this time with signature 11-3. Wilson surfaces and loop space connections"
},
{
"end_time": 8113.626,
"index": 363,
"start_time": 8085.503,
"text": " These guides provide insights into the underlying symmetries and structures of M-theory. In particular, Wilson surfaces generalize the concept of Wilson loops, expressed as follows, which represent the parallel transport of particles in gauge fields. In M-theory, Wilson surfaces describe higher dimensional extensions and interactions with M-brains, such as M2-brains coupled to the 3-form potential C3, and M5-brains coupled to the 6-form potential C6."
},
{
"end_time": 8136.22,
"index": 364,
"start_time": 8114.309,
"text": " Wilson surfaces are expressed similarly as before, where Cn are the n-form potentials and sigma is a p-dimensional sub-manifold. The loop space connections, denoted by alpha, are introduced as calligraphic alpha equals a plus b2 plus c3 plus so and so on, where Roman a is the usual gauge connection and b2 and Cn are higher-form connections."
},
{
"end_time": 8154.224,
"index": 365,
"start_time": 8136.22,
"text": " Loop space connections build on Wilson loops. How? They extend parallel transport to deal with extended objects looping through space. This helps us understand the non-perturbative features of M theory as well as it's meant to reveal more dualities. Arithmetic Geometry"
},
{
"end_time": 8181.288,
"index": 366,
"start_time": 8155.623,
"text": " In arithmetic geometry, one studies algebraic varieties over number fields and zeta functions, like the Haase-Wei zeta function. These zeta functions contain information about the distribution of rational points and other geometric invariants, such as those talked about by one of the Millennium Prize problems, the Birch-Swinerton-Dyer conjecture. Though, this conjecture refers specifically to the rank of an elliptic curve group and the order of vanishing of its associated L function."
},
{
"end_time": 8204.787,
"index": 367,
"start_time": 8183.08,
"text": " in String Theory Compactifications and the Arithmetic Properties of Zeta Functions. This is heavily related to the discoveries in mirror symmetry in 1991 by physicists and mathematicians Candelas, Assa, Green, and Parks. See this talk here about the Langlands and arithmetic quantum field theory, though this is not about string theory. Categorical Symmetries"
},
{
"end_time": 8231.323,
"index": 368,
"start_time": 8206.084,
"text": " In higher category theory, categorical symmetries come from an abstraction of traditional symmetries represented by group actions. Let's focus on two groups. So mathematically, a two group is viewed as a strict monoidal category, with all objects and morphisms being invertible. In symbols, a two group is a collection of objects and morphisms and multiplications and inversions and identities, where there are only two objects, so G0 and G1."
},
{
"end_time": 8255.111,
"index": 369,
"start_time": 8231.323,
"text": " Categorical symmetries come up in string theory through higher gauge theories, which describe extended objects like D-brains and M-brains, and these D-brains can be associated with gerbs, they can be associated with two categorical generalizations of line bundles, and twisted versions of ordinary bundles. Their transition functions are described as elements of the automorphism 2 group of the principal U1 bundle, not just BU1."
},
{
"end_time": 8274.957,
"index": 370,
"start_time": 8255.503,
"text": " Recall that a BU1 is defined as the classifying space of U1 bundles, so in other words BU1 is the same as U1, except you mod out by contractible spaces on which U1 acts freely. Historically, categorical symmetries originated from John Biases and Jane Dolan's study of higher-dimensional algebras in the 1990s."
},
{
"end_time": 8303.2,
"index": 371,
"start_time": 8274.957,
"text": " M5 brains have categorical symmetries. Why? Their self-dual three-form is governed by a three-categorical structure, specifically through the two connection components as follows. The three-form field strength H is induced by a two connection on a gerb, with A being a one-form connection and B is a two-form connection, such that the field strength can be expressed as follows, with F equals dA. The role of three algebras is also useful in describing world-sheet dynamics."
},
{
"end_time": 8306.084,
"index": 372,
"start_time": 8304.565,
"text": " Higher spin gravity."
},
{
"end_time": 8336.254,
"index": 373,
"start_time": 8307.705,
"text": " If you listen to this podcast, you'll hear me say often that it's not so clear gravity is merely the curvature of spacetime. Yes, you heard that right. You can formulate the exact predictions of general relativity, but with a model of zero curvature with torsion, non-zero torsion, that's Einstein Cartan. You can also assume that there's no curvature and there's no torsion, but there is something called non-matricity. That's something called symmetric teleparallel gravity. Something else I like to explore are higher spin gravitons."
},
{
"end_time": 8363.558,
"index": 374,
"start_time": 8336.613,
"text": " Higher spin gravity theories are characterized by massless fields with spin greater than 2, such as Vasilyev's higher spin gravity in ADS-4. The action for these theories has a form similar to the above, where H and phi represent the higher spin fields. These theories possess infinite dimensional gauge symmetries, but so does general relativity, given you ordinarily consider the diffeomorphism group. So how is this different than usual?"
},
{
"end_time": 8387.978,
"index": 375,
"start_time": 8363.831,
"text": " The difference lies in the types of gauge transformations and the structure of the gauge fields. In higher spin gravity, gauge transformations are associated with tensor fields of higher rank, so S-1, while general relativity involves vector fields. Therefore, it exhibits that conjecture duality with certain large n CFTs with higher spin symmetries, such as the O-n vector model CFT. Historically,"
},
{
"end_time": 8410.64,
"index": 376,
"start_time": 8387.978,
"text": " Franz Dahl's work during the late 1970s and early 1980s laid the foundation for higher spin gravity, notably with his equation for massless fields of arbitrary spin. In some ways, you can think of this as allowing for more ways to quote-unquote wiggle in space-time rather than the regular two degrees of freedom of ordinary gravity theories. The Atiya-Singer Index Theorem"
},
{
"end_time": 8434.514,
"index": 377,
"start_time": 8412.295,
"text": " This index theorem is a landmark result in differential geometry and topology. What it does is compute something called the analytical index of elliptic differential operators, and by doing so, shows the connection between the topology of a manifold and the solutions of partial differential equations on it. An analytical index is the difference between the dimensions of the kernel and the co-kernel of an elliptic operator."
},
{
"end_time": 8452.039,
"index": 378,
"start_time": 8434.514,
"text": " and elliptic differential operators are linear partial differential operators that satisfy a certain condition called the ellipticity condition which guarantees the existence of solutions and good estimates for their behavior expressed as follows for large psi where p is called the principal symbol of the operator"
},
{
"end_time": 8480.094,
"index": 379,
"start_time": 8452.039,
"text": " meaning the highest order homogeneous part of the operator in local coordinates, and psi is a point in the cotangent bundle. Because of this, elliptic operators have favorable properties such as the existence of smooth solutions and well-posedness. In string theory, the theorem has found application in establishing anomaly cancellation conditions when applied to the elliptic Dirac operator on the worldsheet. The index is associated with the topological invariance of the worldsheet like the Euler characteristic"
},
{
"end_time": 8506.459,
"index": 380,
"start_time": 8480.094,
"text": " and the Herzabrutsch signature, through the following expression, where A is the A-roof and L is the L-genus of the manifold X, and CH is the churn character of the relevant bundle. The A-roof is defined as the Pfaffian of the curvature form, divided by the Pfaffian of the tangent bundle, so this expression on screen here, and the Pfaffian is a polynomial function associated with a skew symmetric matrix, such that the square of the Pfaffian equals the determinant of the matrix."
},
{
"end_time": 8532.193,
"index": 381,
"start_time": 8506.459,
"text": " Modulize Stabilization"
},
{
"end_time": 8561.101,
"index": 382,
"start_time": 8533.746,
"text": " Recently, a paper was published by Bassiori, which provides non-perturbative terms in the superpotential and the combined effects of logarithmic loop corrections and two non-perturbative superpotential, Caylor moduli-dependent terms. How so? They, the authors, derive the following effective potential, which takes into account both the perturbative and non-perturbative contributions, where a, b, c, psi, and eta are coefficients that depend on various parameters of the theory."
},
{
"end_time": 8582.534,
"index": 383,
"start_time": 8561.101,
"text": " So what does this mean, Kurt?"
},
{
"end_time": 8594.206,
"index": 384,
"start_time": 8583.183,
"text": " Well, my friend, the result shows that fluxes exist for large and even moderate volume compactifications, which defines a decider space and stabilizes moduli fields."
},
{
"end_time": 8618.968,
"index": 385,
"start_time": 8594.565,
"text": " So, why is this important, Kurt? Well, this is an important finding because it demonstrates the existence of stable, decider vacua in type 2b string theory, which was previously known to be extremely challenging. The obtained effective potential appears to be promising for cosmological applications, such as cosmological inflation models, understanding dark energy, and the universe's expansion."
},
{
"end_time": 8633.097,
"index": 386,
"start_time": 8618.968,
"text": " Dark energy."
},
{
"end_time": 8657.227,
"index": 387,
"start_time": 8634.224,
"text": " Dark energy is about the expansion of the universe. Some think it's as simple as, well, it's just the cosmological constant, and others think it has to do with more mysterious modifications of the laws. The study of string cosmology is about examining string theory's implications on the universe's evolution, including dark energy and accelerated expansion. Let's consider the low energy effect of action,"
},
{
"end_time": 8672.722,
"index": 388,
"start_time": 8657.227,
"text": " By now you should be familiar with these symbols, but for those who skipped around and want a refresher, that capital G is the metric, the dilaton field is phi, the NSNS3 form strength is H, and F is the RRP form field strength, and the lone G is the determinant of the metric."
},
{
"end_time": 8690.452,
"index": 389,
"start_time": 8672.722,
"text": " By compactifying extra dimensions to four-dimensional spacetime, you get a 4D action and a scalar potential which is affected by these fields. This leads to something called a quintessence-like dark energy scenario. Quintessence is a scalar field with a potential responsible for the accelerated expansion of the universe"
},
{
"end_time": 8716.169,
"index": 390,
"start_time": 8690.452,
"text": " Is string theory the flashlight we need to illuminate the dark corners of the universe? Ambatwister String Theory"
},
{
"end_time": 8742.978,
"index": 391,
"start_time": 8717.466,
"text": " You've heard of twisters, but have you heard of ambitwisters? What are they? Well, they generalize twisters by considering the complexified phase space of null geodesics instead of Minkowski spacetime. The ambitwister space is a huge space that contains twister space as a subspace. Ambitwister string theory is a framework that uses both twister and ambitwister spaces to describe scattering amplitudes of massless particles."
},
{
"end_time": 8756.715,
"index": 392,
"start_time": 8743.2,
"text": " The worldsheet action is expressed as follows, with A and P being auxiliary fields related to the twister variables. Conformal symmetry, which of course is present in conventional string theory, is also there in ambitwister strings."
},
{
"end_time": 8774.548,
"index": 393,
"start_time": 8756.903,
"text": " What's the difference? Their target space comprises the space of complex null geodesics rather than just regular space-time. The CHY formula gives a compact representation for tree-level amplitudes of massless particles expressed as integrals over the moduli space of punctured Riemann spheres."
},
{
"end_time": 8794.138,
"index": 394,
"start_time": 8774.821,
"text": " This can be understood as a considerably efficient method of representing many particle interaction outcomes. Sir Roger Penrose's pioneering work on twister theory in the 1960s laid the groundwork for ambitwister strings to emerge decades later. Although ambitwister string theory simplifies scattering amplitudes, encoding soft limit"
},
{
"end_time": 8809.514,
"index": 395,
"start_time": 8794.138,
"text": " Non Archimedean Geometry"
},
{
"end_time": 8832.227,
"index": 396,
"start_time": 8811.476,
"text": " There's another field called the P-adic numbers. So, P-adic numbers are defined as equivalence classes of Cauchy sequences of rational numbers converging with respect to something called the P-adic norm. Now, just as there's non-Euclidean geometry, there's also something called non-Archimedean geometry. The P-adic numbers, denoted as Q with the subscript P,"
},
{
"end_time": 8861.766,
"index": 397,
"start_time": 8832.227,
"text": " form the completion of the rational numbers Q with respect to the Pyatik valuation, augmenting Q by incorporating something like digits and infinite amount of digits to the left, rather than to the right, as we're conventionally used to. Pyatik string theory was originated by Volovich in the 1980s, and it embeds the string worldsheet into Pyatik spacetime using the adapted Polykov action. Notice the Pyatik norm here. This allows invariance under Pyatik reparameterizations and Weyl transformations."
},
{
"end_time": 8872.108,
"index": 398,
"start_time": 8861.988,
"text": " Piatek string amplitudes have factorization properties similar to their Archimedean counterparts, allowing for Piatek analogues of Veneziano and Verasoro-Chapiro amplitudes."
},
{
"end_time": 8899.855,
"index": 399,
"start_time": 8872.398,
"text": " Tachyonic condensation occurs in the peatic setting, giving a non-perturbative description of D-brains. The Adelic product formula, associating products of amplitudes with certain topological invariants, hints at connections between peatic and Archimedean string theories, although this remains wonderfully speculative. Another physical theory that involves the peatic numbers is the so-called invariant set theory by Tim Palmer, which suggests that the universe evolves on a fractal attractor."
},
{
"end_time": 8909.974,
"index": 400,
"start_time": 8900.213,
"text": " Inumerative Geometry"
},
{
"end_time": 8938.404,
"index": 401,
"start_time": 8911.527,
"text": " Topological string theory has applications in enumerative geometry, particularly through the use of Gromov-Witten invariance. Now, those are those correlation functions that count the number of holomorphic curves within a Kolob-Yau manifold weighted by their genus G and homology class C that we talked about approximately an hour ago. These invariants are computed in the A model, so the symplectic one, and the B model, so the complex one, for topological string theories."
},
{
"end_time": 8964.138,
"index": 402,
"start_time": 8938.404,
"text": " They give information on the modular space of Kali-B.Yau manifolds, Yukawa couplings, and string theory compactifications, and what's important for this topic, the intersection number for counting problems in enumerative geometry. In other words, rational curves on a quintic threefold. Gromov-Witten invariance generalized classical intersection theory. Symplectic modular symmetry in heterotic string vacua."
},
{
"end_time": 8995.077,
"index": 403,
"start_time": 8965.759,
"text": " Ishiguro, Kabayashi, and Otsuka recently examined the unification of flavor, CP, and U1 symmetries coming from symplectic modular symmetry in the context of heterotic string theory on Kolibiao 3-folds. They found that these symmetries can be unified into the symplectic group's modular symmetries of Kolibiao 3-folds with H being the number of moduli fields. Together with the Z2-CP symmetry, they're enhanced to this group here, which is the generalized symplectic modular symmetry."
},
{
"end_time": 9010.094,
"index": 404,
"start_time": 8995.077,
"text": " They have S3, S4, T' and S9 non-abelian flavor symmetries on explicit toroidal orbifold with and without resolutions on Z2 and S4 flavor symmetries on three parameter examples of collibial threefolds."
},
{
"end_time": 9037.961,
"index": 405,
"start_time": 9010.094,
"text": " Layer 7."
},
{
"end_time": 9065.742,
"index": 406,
"start_time": 9038.899,
"text": " Congratulations on making it this far. Now we're in the deepest layer in one of the most thorny subjects, not only in physics, not only in math, but in all fields imaginable. It's useful to understand the math of string theory, even if string theory ends up missing the mark, because the problems being addressed here are problems at the heart of the physical universe. However, of course, you shouldn't mistake in the physical universe as being synonymous with reality. This is a point that Hilary Putnam makes."
},
{
"end_time": 9093.848,
"index": 407,
"start_time": 9066.032,
"text": " Despite this, understanding string theory gives you a bedrock at the fount of reality, the reality that can be established mathematically and logically. Let's get on with the iceberg. This is a grueling problem in physics. We often assume that there is such a correspondence, which is just yet to be found rigorously, but even defining it rigorously is formidable."
},
{
"end_time": 9109.77,
"index": 408,
"start_time": 9093.848,
"text": " Further, there are nine major problems. Number one, mapping between gravitational and field theory configurations. The issue is to find an exact dictionary that conjoins gravitational states with the states of the boundary conformal field theory."
},
{
"end_time": 9139.07,
"index": 409,
"start_time": 9110.043,
"text": " When you have configurations with less symmetry, it's not clear how to do this. Number two, ADS space as a regulator for flat space physics. The use of ADS space as a regulator to extrapolate to flat space physics involves taking the limit where the ADS radius of curvature R goes to infinity. This process, while keeping the local physics unchanged, isn't fully developed, especially in understanding how the ADS boundary conditions translate to flat space observables."
},
{
"end_time": 9165.845,
"index": 410,
"start_time": 9139.07,
"text": " Number three, holography in light-like boundaries. Understanding holography for light-like boundaries, as in the case of Minkowski spacetime, differs significantly from time-like boundaries typical of ADS CFT. The existence and nature of large end limits for theories that aren't gauge theories and for theories with less or no supersymmetry isn't anywhere as developed either. Number five, sub-ADS locality."
},
{
"end_time": 9184.838,
"index": 411,
"start_time": 9166.647,
"text": " How do you understand the emergence of bulk physics at scales smaller than the ADS radius? The solvable models we have currently of holography don't capture the locality expected from gravity in the bulk, which should be evident at scales much smaller than this radius. Number six, time evolution."
},
{
"end_time": 9202.824,
"index": 412,
"start_time": 9185.896,
"text": " The bulk reconstruction techniques developed so far primarily address static or equilibrium situations. The dynamical evolution of non-trivial states, particularly those involving black hole formation and thermalization, aren't well understood."
},
{
"end_time": 9233.951,
"index": 413,
"start_time": 9204.087,
"text": " Bulk operators must be dressed gravitationally to be gauge invariant, but the precise nature of this dressing in context with significant back reaction isn't fully understood as well. Dressing in this context, by the way, means incorporating the influence of gravitational fields generated by the operator itself on its definition, ensuring to feel morphism and variance. This is particularly relevant for operators that couple strongly to gravity. Number eight, entanglement wedge reconstruction."
},
{
"end_time": 9258.575,
"index": 414,
"start_time": 9234.838,
"text": " So the conjecture that the boundary subregion R is dual to the entanglement wedge W, rather than to the causal wedge CR, raises the question about the reconstruction of these bulk operators. The entanglement wedge can extend beyond the causal wedge, potentially including regions behind horizons, which complicates the understanding of bulk locality and the encoding of bulk information in the boundary theory."
},
{
"end_time": 9286.032,
"index": 415,
"start_time": 9258.575,
"text": " By the way, the entanglement wedge refers to the region of spacetime in the bulk that can be reconstructed from boundary subregion entanglement, while the causal wedge is the bulk region causally connected to that boundary subregion. And lastly, number nine, black hole interior. The description of the black hole interior is still an open problem in ADS-CFT. What is the existence of firewalls? What is the fate of an infalling observer? We don't know."
},
{
"end_time": 9315.572,
"index": 416,
"start_time": 9287.363,
"text": " fuzzballs and the microstructure of black holes. The fuzzball proposal in string theory suggests that black holes possess a microstructure composed of stringy excitations or fuzzballs which replace the classical event horizon as well as the singularity. This stems from the correspondence between black holes and d-brain bound states. It's an attempt to describe the near horizon geometry using the dual conformal field theory."
},
{
"end_time": 9343.353,
"index": 417,
"start_time": 9315.862,
"text": " To translate that a tad, the Fuzzball conjecture replaces the mysterious core as well as the edge of black holes with information storing strings. The Bekenstein-Hawking formula agrees with the degeneracy of these Fuzzball states, accounting for the microstates that generate the black hole entropy. The Fuzzball conjecture was first proposed by string theorist Mathur and his collaborators in 2002. You'll hear this term plenty, microstructures and microstates."
},
{
"end_time": 9363.951,
"index": 418,
"start_time": 9343.353,
"text": " To be specific, the microstructure generally refers to the arrangement of string excitations that comprise the black hole's interior, while microstates are these distinct field configurations that these excitations can take, each one corresponding to a unique quantum state. Essentially, they represent the different ways that strings can"
},
{
"end_time": 9390.589,
"index": 419,
"start_time": 9363.951,
"text": " vibrate or be bound together within the fuzzball giving rise to the black holes entropy now how do you generalize these fuzzballs not only to non-extremal black holes but to other broader classes of black holes this is an open problem also what's the exact mechanism for retrieving information from these fuzzballs we don't know but the answer to these can help resolve the black hole information paradox so good luck"
},
{
"end_time": 9418.336,
"index": 420,
"start_time": 9391.903,
"text": " Achieving background independence in string theory remains a large unsolved problem, but it's not as unsolved as it was a decade ago. There's more and more progress about background independence results in certain scenarios. For instance, this recent lecture a few months ago by Ed Whitten. But why is this such a confounding conundrum? Well, it's because string theory's perturbative roots demand a predefined background."
},
{
"end_time": 9442.312,
"index": 421,
"start_time": 9418.336,
"text": " However, Kurt, what if you incorporate the Poisson tensor derived from the Polyakov action? Does that not allow for curved backgrounds? Not exactly. Accommodating dynamic backgrounds is different than merely curved backgrounds. It requires a non-perturbative foundation for string theory. But Kurt, what about matrix models or the generalizations like tensor models or higher spin holography? Great point!"
},
{
"end_time": 9467.585,
"index": 422,
"start_time": 9442.312,
"text": " You are on it today. The issue is extending those results to a more general setting. And just so you know, a universally accepted non-perturbative definition remains unfound. This was one of the major critiques of one of the earlier Lee Smolin books of string theory. By the way, a podcast with Lee Smolin was just released about a week ago. Check the description or click subscribe to get notified. Pure Spinner Formalism"
},
{
"end_time": 9491.664,
"index": 423,
"start_time": 9468.507,
"text": " In superstring theory, an alternative to traditional Raymond Neville Schwartz and Green-Schwartz formalisms exist and they're called pure spinner formalism. So, what makes this PSF different? The formalism employs what are called pure spinners, which are a special class of spinners being self-dual and annihilated by a maximal isotropic subset of gamma matrices N."
},
{
"end_time": 9515.06,
"index": 424,
"start_time": 9491.664,
"text": " The formalism also simplifies calculations, especially for higher loop amplitude, using the simpler BRST charge given on screen. Now this baby girl is less complicated than her counterpart in RNS formalism. The pure spinner space can be constructed as a quotient of the common spinner space that you know and love, by the maximal isotropic subspace, represented mathematically here."
},
{
"end_time": 9539.104,
"index": 425,
"start_time": 9515.06,
"text": " where D signifies a Dirac spinner and N is the null subspace. These spinners enable a covariant quantization of the super string, eliminating the oddities of the picture-changing operators as well as ghost fields. Nathan Berkovitz birthed the pure spinner formalism in his pursuit for more symmetric solutions to super string theory constraints. Waterfall fields and hybrid inflation."
},
{
"end_time": 9560.674,
"index": 426,
"start_time": 9540.128,
"text": " In new work published in just 2022, which by the way is only a blink of the eye in this field, Antonietas, Lacombe, and Leon Torres presented a cosmological inflation scenario within the framework of type 2b flux compactifications. What makes their work different? They used three magnetized D7 brain stack."
},
{
"end_time": 9579.787,
"index": 427,
"start_time": 9561.135,
"text": " The inflation is associated with a metastable decider vacuum and the inflation is identified with the volume modulus. The authors propose that the inflation ends due to a waterfall field, which drive the evolution of the universe from a nearby saddle point toward a global minimum with tunable vacuum energy."
},
{
"end_time": 9603.131,
"index": 428,
"start_time": 9579.787,
"text": " This tunable vacuum energy could potentially describe the current state of our universe. The authors detail their model, including the implementation of what's called hybrid inflation, also the analysis of open string spectrums, and the dynamics of the waterfalls on this decider vacuum and inflation. The authors conclude that their model successfully implements the main principles of hybrid inflation."
},
{
"end_time": 9619.872,
"index": 429,
"start_time": 9603.131,
"text": " The introduction of these waterfall fields in this model is a pioneering mechanism for driving the universe's evolution from a metastable decider vacua to a global minimum, potentially even explaining dark energy. String Net Condensation and Emergent Spacetime"
},
{
"end_time": 9640.247,
"index": 430,
"start_time": 9620.742,
"text": " This is a mechanism in topological quantum field theory. String nets suggest that space-time isn't fundamental but comes from something pre-geometric in condensed matter systems. Elementary excitations in a lattice such as spins and qubits form string-like structures that, when condensed, lead to phase transitions."
},
{
"end_time": 9669.36,
"index": 431,
"start_time": 9640.247,
"text": " The ground state of a topologically ordered system is described by a superposition of string net configurations with the string net wave function given here where L denotes the string label on edge E and delta is the branching rule at vertex V. The emergent spacetime geometry is a result of collective string net behavior. So you may ask, where does the metric come into play? The metric emerges from interactions between string nets and their corresponding tensor networks."
},
{
"end_time": 9693.797,
"index": 432,
"start_time": 9669.77,
"text": " The low-energy excitations resemble particles in a 3-plus-1-dimensional spacetime as the emergent gauge fields in gravity are realized via fusion and braiding of anionic excitations in the system. Anions are exotic quasi-particles in two-dimensional systems. The emergent gauge group structure relies on anion fusion rules, while emergent gravity stems from topological entanglement entropy."
},
{
"end_time": 9719.394,
"index": 433,
"start_time": 9694.206,
"text": " Whether this is how the world works or not, this gives new tools for those studying the building blocks of space time. Eclectic flavor groups. This is a brand new area of research. The best resource I found was this 2020 open access article on screen here. Eclectic flavor groups combine traditional discrete flavor symmetries with modular flavor symmetries."
},
{
"end_time": 9746.476,
"index": 434,
"start_time": 9719.394,
"text": " They analyze a model based on the Delta 54 traditional flavor group and the finite modular group Sigma Prime 3, resulting in the eclectic flavor group given on screen here. Keep in mind that it's called eclectic and not electric. I made this mistake at least 10 times when writing the script because of pesky muscle memory. This scheme is highly predictive, constraining the representations and modular weights of matter fields and hence the structure of the scalar potential and super potential."
},
{
"end_time": 9772.449,
"index": 435,
"start_time": 9746.903,
"text": " the superpotential and scalar potential transform under the eclectic flavor group such that they combine to an invariant action. Discrete R-symmetries emerge intrinsically from the eclectic flavor groups and this model's predictive power is showcased by the severe restrictions on the possible group representations and modular weights for matter fields, which in turn control the superpotential and scalar potential structures."
},
{
"end_time": 9796.647,
"index": 436,
"start_time": 9772.449,
"text": " The Caylor potential is Hermitian and modular invariant with leading contributions given by the standard form and additional terms suppressed by the volume of the orbifold sector. Because of the connection between R symmetries and modular transformations within these eclectic flavor groups, this research may provide insight into discrete symmetries in string compactifications. O Minimal Structures"
},
{
"end_time": 9821.783,
"index": 437,
"start_time": 9797.585,
"text": " Originally introduced by Louvain d'Andries in the 80s, these O-minimal structures are a way of simplifying the topology of semi-algebraic sets. The key idea is to break down any definable set in an O-minimal structure into a finite number of cells, so basic building blocks like intervals and their higher dimensional analogs. You can do so by following the cell decomposition theorem."
},
{
"end_time": 9850.486,
"index": 438,
"start_time": 9822.278,
"text": " In string theory, considering the moduli space of Kali-Biao manifolds, more explicitly on screen here, where CYN represents the set of all Kali-Biao n-folds in O-minimal structures calligraphic O and M sub-calligraphic O denote the corresponding moduli space. This is brand new research and the best paper I found is by Grimm on taming the landscape of effective theories, that is, using O-minimal structures to explicate the Swampland."
},
{
"end_time": 9875.879,
"index": 439,
"start_time": 9851.937,
"text": " String Universality String Universality is the conjecture that every consistent quantum gravity theory corresponds to the vacuum of some string theory or string theory compactification. It's based on the fairly braggadocious belief that string theory encompasses all possible quantum theories of gravity, at least within certain conditions like a fixed number of dimensions and certain amounts of supersymmetry."
},
{
"end_time": 9897.892,
"index": 440,
"start_time": 9875.879,
"text": " We can symbolically represent this conjecture as follows, where QG is the space of all consistent quantum gravity theories and the calligraphic STV is the space of all string theory vacua and this is a surjective map. This conjecture is part of a broader set of ideas known as the Swampland program that we talked about earlier. In fact, string universality is seen as the endpoint"
},
{
"end_time": 9925.282,
"index": 441,
"start_time": 9897.892,
"text": " of the Swampland program, where string theory is the ultimate quantum theory of gravity. But, you may ask, what about loop quantum gravity? Recall, loop quantum gravity is a non-perturbative and background-independent approach, which attempts to quantize gravity directly by focusing on the geometric and topological aspects of spacetime. Importantly, it does not rely on supersymmetry, which is a key ingredient in many of the string theoretic constructions."
},
{
"end_time": 9944.445,
"index": 442,
"start_time": 9925.282,
"text": " Now, advocates of string universality would just argue that, hey, loop quantum gravity is not a complete nor consistent quantum theory of gravity, or some may say it will eventually be subsumed by string theory anyhow. This is a point that Ed Witten made in a recent book called Conversations on Quantum Gravity. But what does it mean to have a consistent quantum theory of gravity?"
},
{
"end_time": 9970.862,
"index": 443,
"start_time": 9944.445,
"text": " I find it helpful to replace the word consistent with non-pathological, because to me consistency has a particular mathematical logic meaning, and quantum field theorists don't use the word consistency in this sense. The pathologies that I refer to could be violating any one of the following, so unitarity, which you can think of as conserving probability, causality is another one which you can think of as no faster than light propagation of information or communication."
},
{
"end_time": 9984.565,
"index": 444,
"start_time": 9970.862,
"text": " String Theory and the Search for Aliens"
},
{
"end_time": 10014.855,
"index": 445,
"start_time": 9985.503,
"text": " String theory's extra-compactified dimensions raise questions, such as whether unconventional biochemistry, including potentially higher-dimensional life, may exist. Let's clarify that this connection is extremely speculative, far from what can be tested currently scientifically. At least, we think so. Now, there is the case to be made, as Lee Smolin does, that we may already possess data to answer such questions, and it's staring us right in the face we just lack the theoretic understanding to interpret the data."
},
{
"end_time": 10040.589,
"index": 446,
"start_time": 10014.855,
"text": " Brains can be seen as generalizations of strings, as you well know, given that you're now at Layer 7, serving not only as the boundaries where these strings terminate, but also as fundamental, multi-dimensional structures in their own right. Could other advanced civilizations be making use of these spaces either for faster-than-light travel, or constructing wormholes for slower-than-light travel but vast-distance travel?"
},
{
"end_time": 10058.456,
"index": 447,
"start_time": 10040.589,
"text": " Or even as places for their own existence. Can you manipulate local vacuum states to create pocket universes? Interestingly, Alexander Westfall, a string theorist, gave a talk 10 years ago to SETI, the academic organization behind the search for extraterrestrial life."
},
{
"end_time": 10087.09,
"index": 448,
"start_time": 10058.456,
"text": " It was about the string theory landscape that we talked about near the beginning of this iceberg. Each quote-unquote bubble universe in this multiverse may have different fundamental properties leading to a proliferation of possibilities for the emergence of life. There may even be avenues for communication. String Consciousness You've heard of Penrose and Hameroff's idea that the same mechanism responsible for quantum gravity"
},
{
"end_time": 10114.258,
"index": 449,
"start_time": 10087.432,
"text": " is twinly responsible for consciousness. It's known as orchestrated objective reduction and we've covered it here on this podcast with Hameroff himself. If this is the case, and if it's also the case that we have string universality, which connects all quantum gravities to string theory, then it's not so far-fetched to conjoin string theory and consciousness. Questions of consciousness such as the hard problem and the so-called problem of other"
},
{
"end_time": 10143.37,
"index": 450,
"start_time": 10114.514,
"text": " I'm skeptical because it's going to become part of physics. Yet, of course, whatever you think about consciousness, it's an important part of us. I don't know how we perceive anything including physics."
},
{
"end_time": 10174.701,
"index": 451,
"start_time": 10145.06,
"text": " And that has to do, I think, with the mysteries that bother a lot of people about quantum mechanics and its applications to the universe. So quantum mechanics kind of has an all-embracing property that, to completely make sense, it has to be applied to everything in sight, including ultimately the observer. But trying to apply quantum mechanics to ourselves makes us extremely uncomfortable, especially because of our consciousness, which seems to clash with that idea."
},
{
"end_time": 10190.384,
"index": 452,
"start_time": 10176.015,
"text": " Consider Carl Jung. In one sense, what Carl was doing was psychology but in another sense, what he was doing was attempting a rudimentary form of the physics of the mind. That is, what are the natural laws that govern the psyche?"
},
{
"end_time": 10220.111,
"index": 453,
"start_time": 10190.947,
"text": " You may say, hey, well, they're not mathematical, and so they don't count as the same sort of laws, and that's exactly right. What's also true is that before Newton and before Kepler, before anyone who placed mathematics at the fount of the world, there were hundreds of years of philosophizing with imprecise language and models of the times about nature, such as Thales, a pre-Socratic Greek philosopher, who suggested that water is the origin of all things and the lodestone has a soul."
},
{
"end_time": 10239.155,
"index": 454,
"start_time": 10220.111,
"text": " The Search for a Final Theory"
},
{
"end_time": 10268.575,
"index": 455,
"start_time": 10240.503,
"text": " Is the unification of general relativity with the standard model the last stumbling block in the reductive search for regularities at the sustentation of the world? Do we have to solve every major physics problem such as the matter-antimatter asymmetry? Do we live in a privileged place in the universe? Should the final theory, if it's meant to be a theory of indeed everything in the literal sense, explain consciousness or purpose?"
},
{
"end_time": 10290.589,
"index": 456,
"start_time": 10269.019,
"text": " Should a final theory be able to explain even itself? What does it even mean to explain? How essential is mathematical beauty or simplicity in guiding us? Where does the direction of time fit in? Not to mention initial conditions and boundary values. Would a final theory also tell us which interpretation of quantum mechanics is correct?"
},
{
"end_time": 10318.643,
"index": 457,
"start_time": 10291.084,
"text": " Is the notion of causality to be redefined, even abandoned? Is it the case that the true theory of everything is by definition unfalsifiable, and thus the final theory is one that lies outside the purview of Popperian science? What about what lies outside in principle observation, like singularities? What about observation itself? Where do you fit in?"
},
{
"end_time": 10340.555,
"index": 458,
"start_time": 10319.206,
"text": " These questions are ones that date back decades, even millennia. We simply don't know. I certainly don't know. But on this channel, Theories of Everything, each of these are explored in extreme detail, as rigorously as we can. The universe is just waiting for someone like you to take a crack at it."
},
{
"end_time": 10369.241,
"index": 459,
"start_time": 10344.138,
"text": " Alright, congratulations. That was a strenuous exercise. I'm sure at least it was for myself. String theory is a fascinating and deep rabbit hole. Personally, I loved learning about string theory. The past few months that I've spent working on this video has invigorated me, even if I'm not sold when people say that string theory has elicited new math and that's some justification or testament to it being on a more correct"
},
{
"end_time": 10391.63,
"index": 460,
"start_time": 10369.241,
"text": " I don't buy that, but I have found it incredibly fun, absolutely loved it. It's wonderfully engrossing in the same way that some people find listening to Beethoven is engrossing. Now I'd say I'm a neophyte in this all and if someone wants to collaborate with me then please comment the word collab, c-o-l-l-a-b. This way I can control f"
},
{
"end_time": 10417.637,
"index": 461,
"start_time": 10391.63,
"text": " and find others who want to work on icebergs I have several ideas for instance the extraterrestrial iceberg explained or the free will iceberg or the iceberg on theories of time or the consciousness iceberg or the iceberg of entropy or the iceberg of causality several several ideas your comments below will help me prioritize because these take months to make literal months I would like to thank at this point"
},
{
"end_time": 10444.189,
"index": 462,
"start_time": 10417.79,
"text": " All the editors, there were four of them. So that's Prajwal, Colin, Akshay, and most of all, Zach. Thank you, thank you so much. A combined hundreds of hours, 400, I believe, by the time this is done. And that's not including the hours that I put in myself in the editing and the writing and the rewriting and the voiceovers and then changing. And then, hey, I know it may seem that looking this good is just effortless for me. And it is, it is. I'll be honest."
},
{
"end_time": 10467.278,
"index": 463,
"start_time": 10444.974,
"text": " There will be a correction section in the description because there are bound to be several notational mistakes, even verbal ones simply the omission"
},
{
"end_time": 10494.002,
"index": 464,
"start_time": 10467.278,
"text": " of a word or the addition of an extra syllable that shouldn't be there. Anyone who's edited a video for months knows that it can all just look like white noise at a certain point, like static. If you're confused, make sure to ask a question in the comments and I will respond or someone else will respond. There are other topics I wanted to cover here like Wolfram's theory. I ran out of time. I also wanted to do asymptotic safety and what it means to have negative dimensions of space."
},
{
"end_time": 10512.756,
"index": 465,
"start_time": 10494.002,
"text": " I also wanted to cover string quantum field theory, which isn't exactly string theory, but for more on this, see the work of Lucas Cardoso. But just so you know, there is a whole podcast with Wilfrum on his theory of everything. It's on screen here. If you're interested, there are two, as Wilfrums appeared at least twice, actually three times on this channel."
},
{
"end_time": 10538.626,
"index": 466,
"start_time": 10512.756,
"text": " There are four ways of supporting me. If you choose to, you should know that I do this out of pocket. There's no major funder. There's no connections that I have. Unfortunately, I get bitter about it because sometimes I look at other podcasters or other video creators who have friends who are in high places who connect them with other guests and connect them with other connections. And I'm just here lonely in Toronto like an umbratic weasel."
},
{
"end_time": 10563.677,
"index": 467,
"start_time": 10539.241,
"text": " But if you would like to support theories of everything to make more content like this, then there are four ways. So there's PayPal for direct payments, like one time payments. There's crypto for the same reason. There's Patreon, which is monthly. And then now there's also you can join here on YouTube monthly. Thank you so much for staying with me for two hours, maybe two and a half. I'm unsure how long this will end up being, but it's been a blast. Take care."
},
{
"end_time": 10589.019,
"index": 468,
"start_time": 10567.927,
"text": " Okay, now on to some brief channel updates. Stick around for the next minute as they may concern you. Firstly, thank you for watching, thank you for listening. There's now a website, curtjymongle.org, and that has a mailing list. The reason being that large platforms like YouTube, like Patreon, they can disable you for whatever reason, whenever they like."
},
{
"end_time": 10615.367,
"index": 469,
"start_time": 10589.019,
"text": " That's just part of the terms of service. Now, a direct mailing list ensures that I have an untrammeled communication with you. Plus, soon I'll be releasing a one-page PDF of my top 10 toes. It's not as Quentin Tarantino as it sounds like. Secondly, if you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like helps YouTube push this content to more people like yourself"
},
{
"end_time": 10636.288,
"index": 470,
"start_time": 10615.367,
"text": " Plus, it helps out Kurt directly, aka me. I also found out last year that external links count plenty toward the algorithm, which means that whenever you share on Twitter, say on Facebook or even on Reddit, etc., it shows YouTube, hey, people are talking about this content outside of YouTube, which in turn greatly aids the distribution on YouTube."
},
{
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"start_time": 10636.288,
"text": " Thirdly, there's a remarkably active discord and subreddit for theories of everything where people explicate toes, they disagree respectfully about theories and build as a community our own toe. Links to both are in the description. Fourthly, you should know this podcast is on iTunes, it's on Spotify, it's on all of the audio platforms. All you have to do is type in theories of everything and you'll find it. Personally, I gained from rewatching lectures and podcasts"
},
{
"end_time": 10685.026,
"index": 472,
"start_time": 10662.09,
"text": " I also read in"
},
{
"end_time": 10712.108,
"index": 473,
"start_time": 10685.026,
"text": " There's also crypto. There's also just joining on YouTube. Again, keep in mind it's support from the sponsors and you that allow me to work on toe full time. You also get early access to ad free episodes, whether it's audio or video. It's audio in the case of Patreon video in the case of YouTube. For instance, this episode that you're listening to right now was released a few days earlier. Every dollar helps far more than you think. Either way, your viewership is generosity enough. Thank you so much."
}
]
}No transcript available.