The Mathematics of String Theory [Graduate Level]

[Music] welcome to the iceberg of string theory a technical addition 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 2 hours than you will watching say 20

hours of mitchi okaku documentaries or Neil degrass Tyson rants why because you'll be shown the actual math instead of handwavy metaphoric explanations that leave you slack jawed deracinated from the equations and even misinformed my name is Kurt jongle 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 paragr into the all encompassing nature of the universe we'll cover the abstruse math Of string theory black holes as well as other tow 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 if you're confused at any point by the exposition then don't worry this is a strenuous subject ask questions in the comments and I'll answer personally or obviously someone Else will respond all

right now let's explore the iceberg of string theory layer one types of string theory in string theory there are five so-called consistent formulations or flavors there's type one there's type 2A type 2B heterotic sl32 and heterotic E8 cross E8 type one string theory is characterized by open and closed strings with the gauge group so32 coming from something called the Chan Paton factors At the end points of the open strings type 2 a and type 2 B are both closed string theories with type 2 a being non- chyal and type 2 B actually being chyro the

heterotic string theories are based on hybrid of 26 dimensional bonic string theory and a 10-dimensional super String 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 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 end points 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 Regis slope these will come up over and over by the way the regi slope is related to the so-called string tension all of these will'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 2 a with type 2B and then there's s Duality linking type 1 with heterotic so32

the fact of these dualities is 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 theories in 10 Dimensions they actually mean 9 + 1 so nine spatial dimensions and one time Dimension and when they say it's 11 dimensional they mean 10 + 1 the reason is they're usually talking about space-time Dimensions as a whole

apparently the m in M Theory stands for a matrix or membrane or mystery or mother but I think it stands for an upside down W for Whitten 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 world Sheet and are described by the poov 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 the string each with their own quantum number like energy charge Mass spin winding number if you introduce lome coordinates which you'll see on screen as Sigma plus and minus again on The world sheet then you can separate that polyol action into left and right moving components leading to the forier 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++ and say T minus minus which generates something

called the verao 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 Tilda that there's already one too many har bars in physics there's another subtlety here of matching left and right mode numbers in order to preserve luren Co variance 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 2 tensor field perturbations H muu 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 2 would be the particle of GRA gravity thus they're equivalent you get this by linearizing the Einstein field equations around a Flat background metric giving the equation on screen this is the aformentioned 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 paperian 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 reggi slope this connects type 2 a and type 2B super string theories how so it Maps a type 2A theory on a circle of radius R to a type 2B 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 SL 2z invariance of type 2B String Theory which acts on a complex parameter combining the string coupling GS and certain raymen raymen Fields by the way I've heard this pronounced Raymond I've heard this pronounced rayon I'm just going to stick with Raymond this makes it only slightly more difficult than taking the inverse of GS this s Duality Links type 1 String Theory with heterotic sl32 Theory which then gives insights into nonperturbative

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 Orient toold 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 Fons 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 fironic modes from a 10 dimension ditional super String Theory this Abomination is gotten to by compactifying 16 additional dimensions in the left moving sector on an internal lce 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 super string 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 it's V8 of the E8 Lee algebra and the heterotic theories are defined by their lattices through this construction you'll notice here that 26 = 10 + 8 + 8 though you'll also notice that 26 does Not equal 10 + 32 the reason is

that you don't deal with the group so 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 reggi slope the Regis slope denoted by a prime is a fundamental parameter in string theory that relates the mass squared of a string to its angular momentum J through the linear Reggie trajectory written on Screen susin 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 reggi slope is inversely proportional to String tension t with a prime equaling something proportional to the inverse of t for those Quantum field theorists interested in scattering amplitudes the Reggie 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 In quantum mechanics by the way you can also formulate the Regis slope as the square of the String's length and if you'd like a breakdown of natural units I have a two-hour lecture series breaking down this topic link is in the description World sheet symmetry the world sheet of a string is the two-dimensional surface that a string sweeps out in SpaceTime shown on screen here the basic symmetries of the world sheet include reparameterization Invariants and then something

called vile symmetry careful not to call it while symmetry if you do string theorist will respect you exactly 3% Less reparameterization in variance means you can choose whichever coordinates you like on the world sheet and it won't affect the physical predictions vile 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 polyol action here this little guy is invariant under both reparameterization and vile Transformations conformal Symmetry and the polyol 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 poov action before the G here is the regular space-time metric that we know and love and the H is the world sheet metric this symmetry leads to a Vanishing of of the trace of the angular momentum tensor yielding the verosa sural 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 ghost strings and brst cohomology 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 world Sheet symmetry Transformations though it should be noted that these are specific to something called covariant quantization we talk more about ghost particles and even the detection of

them here on this podcast with Kiara marletto Layer Two too 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 noyman said the point of math isn't to understand it but rather

to get used to it you would think that Fields medalist Richard bards 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 this video itself took me 2 months to write and even here I'm barely scratching the surface if you're interested in advanced math or physics or philosophy just stick with it don't be concerned that certain Concepts go over your head what goes over your head today you'll be able to eat for breakfast in one year prioritize

General acclamation over minute comprehension 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 DB brains also known as derish lay brains serve as endpoints for open strings the action for a d brain is given by the dra born infeld action shown on screen here where T is the brain tension and Gamma is the induced metric and the caligraphic 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 Randle sundrome model exemplifies the brain World scenario with two three brains embedded in a 5D anti- Deiter SpaceTime where one of these brains represents our universe the RS metric is given on screen where K is the adus curvature scale R is the compactification radius and F 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 plank

scale is transformed into the te 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 pronounced the durac equation durac and not D Rock Equation because I just can't help but think about Dwayne Johnson writing a hyperbolic pte string cosmology and inflation string cosmology is a framework for investigating inflationary models compactification schemes such as calb Yao manifolds orbifolds and flux compactifications all of which we'll talk about later impact virtually every cosmological quantity how the modu field from

these compactifications influenced the Dynamics of inflation in string Inspired scenarios like the large volume scenario with Kor moduli and the radial dilaton dilatant we'll talk about later and if you're interested in the low energy effective action it's on screen the inflationary potential is V cosmic strings string theory predicts cosmic strings F and dterm inflation and axom monodromy models contrary to what people say that string theory has no predictions these actually do yield Testable predictions on the tensor to scaler ratio r on the scalar spectral index n and on non- Galan aties 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 Hagadorn temperature usually

denoted by th if you see my tutorial master class on undergrad physics in 2 hours which is Linked in the description then you'll see why I'm fond of this Tia 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 quasistatic Hagadorn phase for fractal-like scale invariant Spectra of fluctuations the specific Dynamics and Interactions of the string gas during these

hagador phases are important verasa algebra Symmetry algebras and infinite generators the verasa algebra is a extension of the wit 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 C ofs the algebra's representations are characterized by the igen values of l0 known as the inform formal

weights Delta in string theory the central charge is related to the space-time 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 verasa 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 backer equation which shows up in Solon 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 back equations are seen in statistical mechanics Quantum groups and not Theory regarding statistical Mechanics it enables the construction of integrable lattice models such as the six vertex model via the algebraic beta onsets offering Exact Solutions for correlation functions and their thermodynamic properties

actually Edward Frankle talked about the beta ons Sals in this podcast on this channel theories of everything here among other topics like Consciousness and the failure of strength Theory the link is in the description in Quantum group Theory the Quantum Yang backer equation results in the discovery of quantum deformations of Lee algebras called drfi Jimbo Quantum groups it's denoted here by this U with usually a q is underneath and in Brackets is the Lee algebra G so not the group G but the Lee 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 invariants of knots and Links such as the Jones polinomial and and the homfly polinomial which generalizes the Jones polinomial by the way correlators are a physicist fancy way of saying greens functions and that's a mathematicians 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

world sheet and can be obtained by varying the poov action with respect to the world sheet metric H the requirement of conformal symmetry leads to the traceless condition which in turn gives rise to the vural 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 Zar conformal weights here H and H Tilda characterize the fields and 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 one and heterotic super string theories anomalies happen when you have classical symmetries like even gauge symmetries and dimorphism in variants 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 SpaceTime 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 calibran field and X8 is the eight form characteristic class of the gauge bundle which is what was introduced by Green Schwarz integrate all of that over the t-dimensional SpaceTime 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 super symmetry 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 one string theory specifically with that gauge group mentioned sl32 was subsequently extended to the chyo heterotic E 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 dimorphism 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 calb raymen 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 remon 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 Witten 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 if it's the latter then no Ed sorry the mid 1980s are indeed the first string Revolution Louisville integrability 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 matrics 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 vso amplitude the amplitude here based on Oilers beta function represents early steps toward String Theory historically it was discovered in 1968 how does this relate To the strong nuclear force well initially it was

applied to mezon scattering it employs mandal Stam variables so s and t for squared energy and momentum transfer respectively with Alpha Prime as reggis 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 SpaceTime 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 anti- symmetric two tensor field B is known as the Cal ramond 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 fi is a scalar Field and it sets the string coupling constant via this formula which is usually just the exponential of fi 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 Richie scaler and H is the derivative of the calman field and G is The metric tensor determinant a choice of these background Fields affects the

compactification schemes as well as de 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 Witten super potential W where H is

the ne Schwarz NE Schwarz three form fluxes so nsns three form and too is an axio dilaton 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 desitter vacua which is needed because we live in a desitter Space not an anti- Deiter one

layer three at this point congratulations you now know more than nine out of 10 people who say that they either like or dislike String Theory sher's anti-gravity Joel Sherk is one of the founders of string theory who unfortunately died unexpectedly in tragic circumstances only months after the supergravity workshop at Stonybrook in 1979 the workshop proceedings were dedicated to his memory with a statement that Sherk 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 Sherk published A Little

Gnome 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-g graviton the Anti-g graviton couples to a conserved current J associated with the quarks and lepton 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 anti-gravity clash with the equivalence principle it seems to But this Clash can be resolved if a scale field acquires a nonzero 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 Sher showed that this anti-gravity is a quality of any extended super symmetric gravitational model this paper has received little attention and no one that I personally know in the string Community other than say David Chester has mentioned it

actually there are conflicting stories about sh death a friend of his noted that Sherk suffered a breakdown his wife left with their children and he later committed suicide the swamp land the swamp land conjecture originates from vafa work in 2005 it pause its 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 swamp land now swamp land sounds like a negative word but actually the larger the swamp land the better because you'll be able to narrow down the space of possible Solutions in string theory two Central conjectures in the swamp land 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 that there exists a particle with charge q and mass m such that this formula on screen this inequality is satisfied where m is the plank Mass MPL 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 Fe 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 swamp land 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 trans Plank and censorship conjecture so rather than suggesting the absence of Stable de space this just puts constraints on the existence of meta stable de vacua in string theory 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 sensorship 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 planking censorship conjecture on the other hand is another censorship principle that has connections to the swamp land 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 plank length or at energies higher than the plank scale

should not be observable in other words it asserts that any observable structure in our universe should have originated from trans planking scales through causal local and unitary processes without requiring any transplan in physics moonshine and string theory moonshine refers to 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 bards using vertex operator algebras and their associ I 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 a Taurus give partition functions with modular invariant and characters encoding group representation data to translate that Attad formally speaking you'll see

a formula on screen and this is for any ABCD belonging going into SL2 Z umbrell moonshine on the other hand relates something called nir lates to Mato and other sporadic groups I don't know how to pronounce these names I'm a self- studier I've only read these by the way I've spoken to Richard borchard on this podcast here about string theory and moonshine Link in the description entropic Gravity 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 entropy is more

accurately defined as quantifying the number of micro States corresponding to a given macro States rather than a measure of quote unquote disorder but Kurt what are micro States great question man I love this audience micro states are different configurations that Each of your energy levels can take in our context it would be string excitations exotic dualities there are more dualities than just T and S Young had one 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 monton Olive Duality or Electric magnetic

Duality there's K3 vibration Duality there's open and closed string Duality there's F3 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 Cali ya 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 Kor structure or the simplec structure and it has applications in enumerative Geometry which we'll talk about again later monton and all of Duality is an S duality in super symmetric 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 K3 vibrations Duality is about the relationship between K3 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 brains and closed strings in the Presence of raymen raymen fluxes f the/ heterotic Duality connects F Theory a 12-dimensional framework extending type 2B String Theory two heterotic strings via compactification on elliptically fibered calow manifolds mirror symmetry mirror symmetry this is a deep topic mirror symmetry is a duality relating two calbo manifolds M and W interchanging their complex and Kor structure the mirror map on

screen here is bidirectional and Relates the complex moduli say F of M to the simplec moduli say s of w where this F here is the pre potential and ta are the simplec 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 Kor and complex structure respectively the a model computes the grandma of Whitten inv variant and these little guys capture information about Holomorphic Curves in M while B computes what are called periods of the holomorphic 30

form on W the gafa Kamar vafa invariants on the other hand are their younger snpp your sister which reformulate the grma of Whitten 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 models techniques turns out there are Like 10 to the 10 examples of distinct data points of cy3 manifolds so that's kalio 3 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 cy3 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 subass of cab Yao manifold with exceptional holonomy so G2 smooth they're compact they're ronian they're seven-dimensional and they have a non- degenerate

three form fee 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 EX 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 ukian case to a manowski space much like Peter white does in his ukan twister unification which is explored here on this podcast if you want to know

more about dark Dimensions which suggest that dark matter is associated with these extra dimensions then watch this video by Sabine hossenfelder 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 pical 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 cab Yao 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 Cabo 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 nonvanishing three form flux governed by peard Leets monodromy the periods of holomorphic three forms transform through this process conifold transitions can be described by the exchange of massless closed string States like the wrapped D3 brain with zero tension around the vanishing

projective cycle instantons and Donaldson invariants instantons are topologically non-trivial solutions to the anti-self-dual Yang Mills equations where f is the field strength tensor and the Tilda f is the hod duel in four-dimensional ukian in 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 extremize 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 four 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 four manifold M Modular 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 seberg Whitten and variants which are used to describe the low energy effective action of n equals 2 super symmetric Yang Mills theories tachon condensation in string theory tachon Condensation is is a process involving

tacons 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 tach on attached to D brains where the tachon potential has the form on screen here the tachon condensation drives the system toward a stable configuration effectively removing the de brains from the Spectrum and reducing the energy of The system sense conjecture which we'll talk about later states that the n point of tachon condensation corresponds to the annihilation of the D brain resulting in a closed string vacuum

this concept is also used in the construction of non-bps brain configurations which we'll explore later by the way it's false to say the tachon fields imply faster than light travel it's only if you interpret the field as a particle and there are other interpretations such As being in a metastable state super symmetry super symmetry is a symmetry between bonic and fonic degrees of freedom governed by the super symmetric algebra with the compatibility between the q's on screen here and they the supercharge operators those Alphas with the dots are spinner indices and the P represents the

space-time momentum operator it's not so intimidating this implies that for every bosonic particle there exists a fironic super partner and Vice versa historically the concept of super symmetry was independently discovered by three groups in about the 1970s so early 1970 by galon and liman Ron and neevu and Schwarz in string theory super symmetry is a consequence of the cancellation of world cheet anomalies we talked about earlier and that also avoids tonic instabilities broken super symmetry is said to be imperative in addressing the so-called hierarchy problem controlling the higs Boson mass and providing viable candidates for

Dark Matter extended super symmetry extended super symmetry theories are super interesting the ordinary super symmetry that you hear about on popside channels is actually n equals 1 super symmetry but there are other extended versions with n greater than one this just means that it has more generators and thus more Super partners and thus more particles for instance the Nal 2 super Conformal algebra given on screen here where gr is the super conformal generator and LR the verasa generators due to the additional constraints imposed by super symmetric 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 But actually it constrains the theory extended Susi lead to smaller massless particle content and further cancellation of anomalies when you extend your super symmetry past n equals 1 you get as a benefit more control over non-perturbative effects and enhan stability of the vacuum essential for constructing consistent and stable ring vacua and phenomenologically viable models for 10-dimensional super string theories we usually have either

that n Equals 1 or n equals 2 though you can also have differing amounts of super symmetry 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 and the issue is that higher values lead to negative Dimensions so you may ask hey Kurt why is it that extended super symmetries with n equals larger than then 2 is talked about this is because when you compactify or you go to the low energy Limit quote unquote you get what appears to be

extra super symmetry 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 fi is the dilaton and H is the nevu Schwarz three form field Strength the dilaton field introduces the string coupling via the formula on screen which modulates the strength of string interactions in the low energy limits string excitations become negligible and the effective action approaches the Einstein Hilbert action rendering general

relativity accurate within this domain n equals two Quantum field theories extended super Symmetry and super symmetry in general isn't just for String theory but for Quantum field Theory in the N equals 2 super symmetric Quantum field Theory topological invariant like we mentioned before there's Donaldson and seberg Whitten invariance have massive roles to play Donaldson and variance emerged from the moduli space of anti-self-dual Connections in Twisted super symmetric Yang Mills Theory while a certain twisting procedure aligns the lorence and R symmetry groups resulting in a Topological theory in the context of n equal 2 super symmetric

Quantum field theories this twisting process refers to the modification of the supercharges such that it becomes a scaler under Loren Transformations the partition function of the Twisted n equals 2 super yangang 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 Cyberg Wht invariants which generalized Donaldson invariance come about from the low energy effective action of n equals 2 super yangang Mills Theory this is governed by the seberg Whitten 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 seberg Whitten Donaldson Correspondence which relates these two types of invariants these invariants have connections to string theory notably in type 2A and heterotic string compactifications where n equals 2 Quantum field theories appear on dbrain world volumes Multiverse of the string landscape the string theory landscape refers to the vast array of 10 to the 500 sometimes is quoted possible vacua resulting from string theories Extradimensional compactifications such as on Kow manifes 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 smolen 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 gamas are the world sheet gamma matrices and the nabla denotes

the world sheet covariant derivative this action is invariant under super symmetry Transformations and thus we say it super symmetric in the 1970s this guy named Pierre raymon and Then this other guy named John Schwarz and then this other guy named Andre nevu developed the Raymond neevu Schwarz formulation the RNs formulation this implements the gso projection which is something that removes tach onic 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 single point here

F Denotes the primary Fields C represents the op coefficients H signifies the conformal weights as usual and z and W denote the world sheet coordinates the op significantly simplifies the calculations of amplitudes for processes and 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 bonic string theory the op of two tachon vertex operators is given by this formula which relates the interaction amplitude of two tachon imagine each operator is a character

in a story when there are two characters so operators interact which means they come close on the world sheet then their combined effect can be represented by a new character a single operator in the expansion who carries certain traits let's say conformal weights and Coefficients influenced by the original characters holographic graic theories one fateful year in 1997 Mala put forward a conjecture known as the ads CFT correspondence which states that there's a duality between gravitational theories on an anti- Deiter 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 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 reggi slope Alpha Prime is given by R to 4th 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 in the large end 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 desitter space a desitter 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 De space thus describing our universe for clarity the N

here corresponds to the gauge group rank so usually it's su2 which is Nal 2 su3 is Nal 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 two or three even all the way up to Infinity now you may think that this is Unphysical and it is but increasing n simplifies the perturbation series such that only planer diagrams dominate planer diagrams are those diagrams which can be drawn on flat surfaces without Crossing

Lines this reduction in complexity was first demonstrated by Gerard tuft as far as I know and for more on this you can watch this lecture here by Witten Celestial holography Celestial holography is one of the most beautiful Sounding terms in all of physics it studies in coding scattering amplitudes in ASM totically flat space times 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 Tilda

with the melon 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 world sheet 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 melon transform applied to scattering amplitudes is the connection between bulk physics and conformal structures in a similar manner to how the 4A transform on Earth's connection between say time and frequency Celestial holography Generalizes the BMS symmetry which was researched by Stringer which itself goes back to the Bondi meter sax group in 1962 Stringer 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 are the future pass

null light cones in a recent video by Sabrina pki she contrasts Celestial CFT with ads4 CFT 3 the primary difference between BMS CFT and Celestial holography is that the latter focuses on encoding scattering amplitudes and ASM totically flat SpaceTime onto a celestial sphere at null Infinity while BMS CFT is more concerned with the symmetries of soft theorems fun fact a few years ago this formula is one that I would write and I would rewrite effectively as a doodle merely because I love the way that the math looked even though I had no idea what it

meant and I forgot about this Until I was writing this script right now supercurrents in type two string theories supercurrents encode World sheet super symmetric Transformations the super current G plus or minus is given by this formula here where the size are the world sheet firion and the X's represent the space-time coordinates and H is the neevu Schwarz three form field strength these supercurrents satisfy the super conformal algebra Including the verasa algebra for the energy momentum tensor and the U1 current J with an additional anti-commutation relation the question of mathematical applicability 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 smolen and Peter white or how much of this is because string Theory is indeed striking at the heart of physical reality layer four 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 reggis slope and then how fan 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 is 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 nambo action or that poov one is it somehow that the tension parameter shows up is it any Theory with extended objects even if they're more than on 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 seberg Dan Fred and Nema arani 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 plank length and LS is the string length M Theory when compact Define on a Z2 orbifold all also connects it to heterotic E8 cross E8 the Inception of M Theory can be traced back to Witten and horava attempts to understand the strong coupling limit of type 2A 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 dilute boting phase and then a bounce leading to The observed expanding Universe there are several theories on cosmogeny 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-space time like such

as Hawking and hardle but today here we have something different here it 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 fi is the dilaton field H is the anti-symmetric 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 Ada 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 venzano the founding father of string theory was urged by the legendary Steven 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 Hagadorn 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 hagador temperature is given by th equals the inverse of 2 pi * the root of the reggi slope you can see by adjusting the string tension th can be made lower than even the plank temperature the Hagadorn 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 Hagadorn proposed the concept of the Hagadorn 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 mo product so the mo 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 Prime respectively in string theory this geometry emerges due to a constant Neu Schwarz B field historically Non-commutative geometry was developed in the 1940s by Joseph Moy and later popularized by Elaine kis the Cyber Witten map relates the communative 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 20 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 two zero 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 two theory was proposed in the mid90s 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 twoo Theory employs differential forms which will be used for representing field and homological algebra for studying properties like boundary and

symmetry mathematically you can use algebraic structures called L Infinity algebras which generalize Lee algebras to accept any number of inputs rather Than the standard two 20 theory is considered to be a candidate for the most symmetric field Theory potentially exhibiting a form of maximal super symmetry for more watch this brief lecture by Christian Seaman who's a professor of mathematical physics at Harriet watt University F Theory F theory is a 12-dimensional non-perturbative framework again there's that IL defined word which generalizes type 2B String Theory with varying String couplings and compactifying on something called elliptically fibered calibby

Yao 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 fibered kyab Yao manifolds this elliptic vibration is given by the yr equation on screen here but this time f and g are sections of a line bundle given by K minus 4 and K minus 6 on the base Manifold B and KB is the canonical bundle on the Bas manifold historically the discovery of f theory in the 1990s by kran bafa 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 you 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 new 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 ation of

the world sheets action for a D2 brain by including a churn Simon term integrating This term over the brain results in a hall conductance expression similar to the quantum Hall effects formula type 2B 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 churn Simon's Theory not invariants including the Jones polinomial

serve as tools to classify and distinguish knots it seems like knots are trivial but that's haha not the case the Jones polinomial is given by this formula on screen where K denotes the not 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 SL2 C and the Ukq stands for the quantum deformation of the universal enveloping algebra when Q equals 1 you get the regular universal enveloping algebra the connection between not invariance and the three-dimensional CH Simon Theory which is a

topological Quantum field Theory with the action given on screen is partially what's responsible for Ed Witten getting the fields metal a metal which has been analogized to the Nobel Prize of math in this action a denot Notes 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 invariant and churn 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 CH Sinus 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 monopol 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 Mullen 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 SL energy that is matter affects SpaceTime and vice versa as you've heard When someone says that string theory is background dependent they mean that most 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 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 tachon 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 super strings 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 miio Kaku yes the dubious Mr Quantum Supremacy BPS States BPS states allow us to understand non-perturbative aspects of string

theory these are states that preserve a portion of super symmetry commonly half and have a mass formula with some bound where m is the 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 super symmetry 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 prad and Summerfield's original work connect BPS states

to soliton Solutions in around 1975 number three they count microscopic states allowing us to calculate the black hole entropy through the stamman jerv vafa formula given on screen where S is the beckenstein 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 s conjecture s conjecture proposed by asok Sen concerns the behavior of these BPS States particularly in type two 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 an interesting historical fact is that it was derived from

studying F Theory and also to date s conjecture has been supported by various calculations and examples but a rigorous proof of the conjecture remains an open problem much like ads CFT twister string theory and cosmological models pen roll's twister Theory can be used to compute scattering amplitudes Eng gaug and gravity theories with a twister denoted a z with a superscript alpha which conforms to the incident 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

meowski SpaceTime in twister String Theory the action is given on screen here where the Twister's connection to SpaceTime comes From something called the pen rolls transform which Maps Twisters to SpaceTime Fields this transform is defined as something that takes holomorphic sheath cohomology classes on twister space and Maps them to Solutions of massless field equations on melsky SpaceTime often expressed as follows where the caligraphic O represents holomorphic functions on twister space and the latter part represents massless fields in SpaceTime you can liken the Pen rolls 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 deit vacua in type 2B String Theory using flux compactifications stabilizes moduli and yields a positive cosmological constant the use of Twister theory in constructing de vacua comes from the formulation and solution of super Symmetric 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 Kia 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 Hop algebras generalizes the concept of Lee algebras and has connections to so-called integrable systems forly the yang an algebra is associated with a Lee algebra and its defining relations are given on screen where J are the generators and F denotes the structure constants of the underlying Lee algebra G and Kappa stands for the level of the associated aine KAC Moody 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 2B string theory on a five-dimensional sphere background to n equal 4 super Yang Mills theory in four dimensions historically ludvig fidv yes the ghost guy and Leon Tagan first introduced yangians in the context of the quantum inverse scattering problem the yangian symmetry of integrable spin chain models Emerges in the planer limit of nals 4 Theory spin chains are the one-dimensional version of the icing model and planer limits mean considering only the leading order term 1 / n in

the expansion where n is the rank of the gauge group Su note one of these NS 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 yangan Symmetry contributes to constructing scattering amplitudes especially within the grass Manan formulation here amplitudes emerge as integrals over the grass Manan manifold and the yangi and symmetry imposes constraints on the integran streamlining the computations an example of a grass Manan integral in the context of scattering amplitudes is given on screen where caligraphic a is

the scattering amplitude for n particles and the grass Manion is gr and we also Have an integration measure d and a volume v the PFS are fafan 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 amplitud hedron BMN matrix model now we get to one of the big boys the BMN Matrix matx model this guy is a proposal for non-perturbative formulation of Big Daddy M Theory itself with the action given on screen here XI Are the herian matrices

representing transverse coordinates of d0o brains and the S are fironic matrices related to the super symmetric extension of the model the action characterizes dzo brain Dynamics in a specific plane wave background connected to the pen rows limit of ads4 cross S7 and ads7 cross S4 geometries inspired by barenstein Mala agnostic 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 igen value distribution of matrices these are related to the distribution of d0o brains though in transverse space the action can be derived from the M Theory super membrane 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 gate H theory on the d0o brains and M Theory in the plain wave Background this generalizes the ads CFT correspondence bfss matrix model this is the older scraw 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 nine herian matrices associated with the transverse coordinates of the d0o brain and Theta is a 16 component myON of vial

Spinner introducing fironic 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 fironic term representing super symmetry in the large n limit the BFS matrix model model is Conjectured to Encompass M Theory in something called the infinite momentum frame ascribing the igen values of the X matrices to the d0 brain positions in transverse space

actually if you like cone gauge quantize the M2 brain and add a turn Simon's Mass term to the bfss 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 Stefon Alexander ik kkt matrix model this is is The creepy awkward Super Genius neighbor of both of the previous two this model is a non-perturbative definition of again type 2 b string theory where SpaceTime emerges from the Dynamics of n byn Herm Mission matrices the action is given on screen and

the gamerm 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 Lorence Symmetry and type 2B super symmetry sometimes people say about physics theories that it has manifest Lawrence super symmetry I don't like the term manifest because it's an equivocal term much like saying the quot quote following exercise is Trivial the ik KT model was developed when its creators were investigating D instanton contributions to type 2B super String 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 white here in this podcast about string theory and SpaceTime the large n limit of ik kkt matrix model is conjectured to capture the full non-perturbative dynamics of type 2B super String 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 2

supergravity action supporting its connection to type 2B super String Theory additionally the discrete ly cone quantization formulation has a direct link to the L cone gauge fixed type 2B green Schwarz action we've seen before kov homology the turn 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 kav'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 kavanov homology is a categorification of the

Jones polinomial historically CH simus Theory emerged in the late 1970s through the works of Shing Shen churn 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 churn Simon Theory such as not in variants and Wilson Loops can be successfully captured in a four-dimensional extension we talk about churn Simon and cenov homology here in this podcast with drawer bar neton black holes and higher compositional Laws in the stew model of string theory so the St model which

is a specific model of compactification involving certain Fields called St we explore the connection between extremal black holes so those of maximum charge and by gara's higher compositional laws which generalize the classical compositional laws of quadratic forms to higher degrees uality orbits of these black holes are characterized by their charge vectors in 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 barava 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 micro States 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 beckenstein Hawking formula written on screen here with Delta mirroring D escapes a full explanation so who knows fun fact barava is a genius mathematician who won

the fields medal and grew up less than an hour away from Me shout out to my fellow torontonians yes the St black hole is a super symmetric extremal solution in Nal 2 D = 4 super gravity characterized by three independent complex scalers St 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 6X 6X 6 integer tensor analogous to bava's Cube of integers which governs higher compositional

law describing the action of the arithmetic group SL2 Z on integral iny 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 defin 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 tfold 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 Neu shortz n shortz two form Potential B the non-trivial H fluxes can lead to interesting topological features for example non-geometric monodromies and non-commutative GE geometry each of whom have implications for the structure of the compactify theory the non-geometric Q flux is characterized by the formula on screen so what role does qlux 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 and non-geometric fluxes in this massively intertwined structure the strong constraints shown on screen here ensure generalized theomorphism and variance it's interesting that dual structures emerge over and over in physics not just in string theory what does this mean about Reality algebraic K Theory K theory is a way to study topological invariance of vector bundles classifi by grow in De groups K the connections between K Theory and String Theory come about when you classify 5D brains the RR so the Ramone rone field strength

F exhibit quantization of the form being members of the equivalence class Al all 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 cobordism 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 2B String Theory classifies D brains by k0 through even rank K Theory classes while type 2A 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 dbrain configurations these ideas as applied to physics came about through the work of Atia Witten and Hara among several others for the string theoretic context we're interested in specific super symmetric theories like the Witten index a topological invariant counting BPS States corresponds to the K Theory Oiler class or the dimension here represents the rank of the K Theory group the AA herab spectral sequence establishes the link between K Theory And ordinary cohomology and is an active research to

this day trying to understand how Deb 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 algebra's the veras 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 u n are the higher spin currents and the Contour integration is over some curv 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 unit it the exact connection between Alternatives of string theory including tonic bionic 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 Mullen Peter white and Sabine hossenfelder 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 predic that could be verified or falsified Through experiments which is a fundamental requirement for a scientific

theory this is technically false it's provided predictions they're 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 number four academic pressure the academ 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 for 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 wht's publication record on Google Scholar you can notice a decrease in string related articles with time number seven even Ed witton's Collaborator Edward Frankle discusses the failure of the original promise of string theory to provide

a unique theory of everything going unacknowledged by string theories creators over and over again the podcast with Edward Frankle is shown on screen here and in the description layer five non-bps brains in M Theory compactifications on Kow threefolds non-bps brains are intriguing objects they're extremal Solutions meaning that They saturate the so-called bomal bound and that means basically that they have a minimum amount of energy given a fixed set of quantum numberers 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 super symmetry non-bps brains preserve Even less resulting in non-vanishing central charges and non-trivial scaler potentials this quote unquote attractor mechanism is given by the effective potential formula on screen here and governs the behavior of moduli fields Z at the Horizon of extremal black holes Z denotes the central charge and capital di signifies the Kor 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 super symmetry 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 you may think that this is obvious but it's not because in other unified theories such as string theories there are Non-geometric phases there are non-perturbative effects that can lead to situations where the gravitational force is not the weakest Force 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 Nal 8 and Nal 2 supergravity correspondences to find a connection between Quantum States and space-time geometry the key idea Here is to consider extremal black holes as cits so a higher dimensional generalization of cubits or cits and we do this through the lens of hop vibrations and Jordan algebras to be precise Rios demonstrated that rank one elements in Jordan algebras of degree 2 and three can be associated with cubits and C trits respectively

in particular cubits are formalized by H2O while cits are formalized by h3o and the O is do tonians what's cool is that when you Take into account U Duality groups these Transformations can be understood as Quantum information theories s 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 fee which is a scaler field responsible for the so-called genus expansion of string amplitudes this genus expansion is like expanding in finement 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 constant and G like we mentioned before without a subscript is the genus and then it equals e to ^ 2 Gus 2 and all of that times V the importance of the dilaton field resembles the significance of the

higs field in the standard model geometric quantization now geometric Quan ization is a method for constructing quantum theories from classical systems so you start by identifying the classical Systems phase space with a complex line bundle dubed the quote unquote prequantum line bundle associating this bundle entails a covariant derivative though you have to make use of the Kor potential and the resulting curvature aligns with the phase space's simplec form Quantum states are viewed as sections of this line bundle satisfying Q SI where SII represents a section of the prequantum 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 Fineman path integral or canonical quantization as quantization the langin program The langland program is a broad set of conjectures in number Theory and representation Theory that's at the Forefront of research in 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 langland's 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 field to representations of its dual Group G Che which is actually another reductive algebraic group now the langland program is so broad that it has various subprograms like the geometric langin correspondence which is a version of the langin 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 chyal algebra on the space of conformal blocks this connects the representations of the loop group Gat to the space of D modules on the modulized stack ghat check so the Dual group local systems which provides a deeper insight into the action of the chyal algebra on the space of conformal blocks within the scope of Electric magnetic Duality the four-dimensional Nal 4 super yangang

Mills Theory provides an example of s Duality which has a specific connection to the langin correspondence through the identification of the electric and magnetic gauge groups with the langin dual groups I talk more about both string theory and the langland program here in this podcast with Edward Frankle there's also this lecture by Ed Witten on gauge Theory geometric languin and all that link to all resources are in The description modular forms and string partition functions modular forms originated from the work of gaus reman and kleene their complex analytic functions with specific transformation properties under discrete

subgroups of SL2 R usually SL to Z in string theory the world sheet conformal theories partition function Z must be invariant under modular Transformations for consistency this restricts allowed compactification luses 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 4A series coefficients in these modular forms connecting the mathematical structure and the physical properties of black holes in String Theory actually Andrew Stringer and cin vafa established this connection in about 1996 interview with both of those so Stringer 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 turn Simon term you are correct they both originate from the same structure a three form constructed from the gauge field the turn Simon 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 that the wzw term is relevant for string Sigma models and conformal field theories while the CH s term plays a role in the topological field theories and is Associated again with the invariance and linking numbers and Jones polinomial 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 Le 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 some called The paukov W consistency condition black hole string transitions in a paper recently published by malesa 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 Pinsky made decades earlier their analysis

of the linear Sigma models for the heterotic strings demonstrates a smooth transition from The hor its Pinsky solutions to black holes a connection hindered in type two super string theories by differing super symmetric 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&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 Leandra 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 ads C2 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 ASM totic 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 propose 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 toic 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 fibered C fourfold given as input where s represents the singularity structure capital G is the gauge group M highlights the matter content and the lowercase G is the function learned by the model potentially I can do a podcast on just machine learning so let me know if you'd like to see that in the comment section Below chyal factorization algebras chyal factorization algebras which are spearheaded now by Emily Cliff are a rigorous method for investigating Quantum field Theory 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 chyal 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 op Association has singularities with simple poles at most most embodying locality in chyro conformal field theories so how do factorization algebras fit into this well firstly they're are generalization of vertex operator algebras and secondly they Provide

a systematic groundup methodology to develop conformal field theories for a vaav 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 cheet 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 toe 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 white is another Garrett Ley is another Eric Weinstein is another with his geometric Unity approach usually Eric explains it as a theory where the four-dimensional SpaceTime that

we know 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 Observers for instance which is characterized by a triple X4 y14 and embedding into a Higher dimension ranian monif these embeddings are local ranian and induc symmetric on X4 thus

generating a normal bundle at some point you choose a signature which then gives the so-called chimeric space y 7 comma 7 the main principal bundle ngu 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 64a 64 and this row this variation on row is the representation of the spin group on Complex direct 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 Georgie Glam model there's also the spin 10 Georgie model and there's a spin 4 cross spin 6 Pati Salam 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 super strings 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 byn hermion 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 matricies led to the discovery of the double scaling limmit by breesen zixen pery and Zuber 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 louville theory is Used and we do so by introducing a louville field fi that couples to the world sheet curvature effectively compensating for the deviation from the critical Dimension by the way louville theory is something you use

to maintain conformal invariance when working with Dimensions different from the critical Dimension type 0 a and 0 B tonic 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 squar 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 type0 a and type 0 B but these are characterized by these fici tachon 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 verasa generators l0 and then L Bar Z 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 reexamine string Spectra and World sheet symmetries because of these if we were to take them seriously the modified conformal weights can be discerned through the formula here for H where k^ s signifies the SpaceTime momentum and m stands for the fractional mode number and Alpha Prime is the again the Reggie slope the traditional verasa 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 such 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 Squ equal 1 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 Sher Schwarz mechanism is a historical example of

a twisted String Theory applied to the breaking of super Symmetry and this requires compactifying an extra Dimension with a Twist to ensure World sheet 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 20 6 +1 Dimensions in monstrous M Theory a recent extension of the standard M Theory to 26 plus1 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 and then we have one was Unknown to Conway morani realized that one of these middle factors 98304 is a wouldbe quote unquote gravitino so a spin one and A2 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 one is of course the dilaton the new approach suggests an nal1 Spectrum in

these 27 Dimensions or or and Nal 2 Spectrum in 26 Dimensions So 25 space dimensions analyzing M Theory for Nal 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 Noms theorem they found a way with nested brain worlds so this nogo theorem applies only when you reduce down to 3 + 1 Dimensions double field Theory 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 Tilda 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 F represents the dilaton as usual and R is the richy scaler in the Double Dimension geometry and 2D is the doubled SpaceTime Coordinate count where D is the initial count although the DFT action respects generalized dimorphisms incorporating Transformations that blend both XI and Tilda 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 it's 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 ashar 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 found 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 Bell 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 Given on screen here here where gamma is the Barbaro EMZ parameter and L is of course the plank length 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 ashar relli and smolan plenty of work was also

done in the 9s by John Bas as Well to this day it's seen as an antagonist to string theory but Lee smolen 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 popsy is quantum entanglement with its ostensible faster than light signaling let's explore this By starting with entropy if we take the vanan entropy where row is the reduced density Matrix then the holographic entropy which includes the rayu takian Nagi 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 equal epr conjecture or heris 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 almur moral Pinsky and Sully abbreviated as amps raises concerns about the validity of er equals epr according to amps a black coal 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 strainger or the idea of State dependence which states that the experience of an observer falling into a black hole depends on the specific Quantum state 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 Mo stars and non-commutative geometry a rigorous analysis of Stringfield theory in the context of non-commutative geometry necessitates the introduction of the mo Star product into the action given on screen here the mo 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 face space the mo Star product is an associative but non-commutative product that generalizes the usual point-wise product of functions on face space in the context of non-commutative geometry what the mo Star product does effectively is to deform the commutation relations of the space-time 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 Ela kis John Von noyman and Marie gersen Harbor all of whom explored it in different context before it found its application in string theory the Appearance of the Star product in the scalar Fields kinetic term Mass term and interaction term actually comes from the seberg Whitten 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 Lee algebra G and what they are non-commutative deformations of the Universal enveloping algebra of the Lee algebra with a deoration parameter Q now the defining relation for certain generators is Abal QB 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 hop algebras allowing a description of both algebra and coalgebra actions in the limit when

Q goes to one Quantum groups reduce to Their classical counterparts both in terms of Lee algebras and Le groups by the way hop algebras are algebraic structures that simultaneously generalize groups associative algebras and Lee algebras how so they have two algebra Maps so a co-product here which includes the algebraic structure and a co-unit which includes 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 they're relevant Vin 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 verin formula can be derived using Quantum group representations relating conformal weights of primary fields to representation labels the pair with the lovely names dfield and Jimbo independently introduced Quantum groups in the 1980s primarily to investigate integrable systems dfield was also mentioned in the book with Edward Frankle

and again the Edward Frankle 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 Le groups and their corresponding geometry which are 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 multi-dimensional flexibility is apparent the exceptional Le 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 EF equations of motion amplitud hedron the amplitud hedron 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 Yosh shabach one with John Veri another with Bernardo castrop and Susan Schneider and yet another one with Philip Goff the topics usually center around Consciousness

though here we'll talk about NE arani Hamed's amplitud hedron what this is is A specific type of convex polytope within Ark that encodes scattering amplitudes in Nal 4 super symmetric Yang Mills Theory this is realized by a relationship with the Positive grass Manan this means it's a space of K byn matrices with positive minors mathematically the amplitud hedron is induced from a mapping of the positive grass Manion under a specific positive map given on screen here where the map is defined by taking the positive grass Manion to the amplitud hedron Via a linear map as

follows with the constraint that all K +1 minors of c are non- negative scattering amplitudes can then be computed via integration over the canonical form of the amplitud hedron providing a way that avoids some complexities of some findan diagrams the amplitud hedron is connected to string theory through that good old celebrated ads CFT correspondence relating and equals 4 super yangang 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 amplitud hedron itself that connects to this correspondence with the convex

polytope being this calculational tool like a middleman in 2013 the amplitude hedron was introduced by Nima arani Hamed and his collaborator Jorah slav partially inspired by the study of ancient math objects called called associahedra these Date back to the 1960s and appear in various branches of mathematics including algebraic topology and combinatorics the amplitud hedron is appealing because it suggests that we might not need Fields fields are often considered as these accounting tools when demanding hamiltonian time Evolution and ignoring Advanced causation the amplitud hedron provides a way to consider causality with Elementary particles traveling backward In

time possibly through wormholes while still maintaining local minkowski SpaceTime along with Wormhole boundaries however this connection is still extremely speculative 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 Mill's 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 dur equation is the square root of the Klein Gordon equation or that Spinners 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 Mill 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 jacobe

identities as the color factors following this Duality if we have caal CB plus CC then na is defined as NB plus n c 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 he drawned you plane integral the low energy effective action for the type 2B string theory on K3 surfaces relies heavily on the evaluation of uplane integrals recall that K3 surfaces are smooth compact complex two-dimensional manifolds with a Trivial canonical bundle and a homy group su2 and they're important because of the role in super symmetry mirror symmetry and cabal manifolds in compactification in this context the up plane is the modui space parameterized by the complex coupling constant where this Theta represents

the Raymond Raymond scaler field and G s 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 up plane like the S Duality orbit or the kolon branch or the cyberg Wht moduli space and the moduli of vacua the integrant takes the form of an exponential multipli 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 up plane integrals you

have to use something called the ramacher expansion now I probably butchered that And at first I thought that was the same Ram Meister as in the moves but it's something different this ramacher expansion expresses the modular forms as a sum of Po Car Series which allows us to isolate pertinent information from the integrant as follows where s represents the cluster sum and S is a modular parameter the uplane integrals are connected with the mock 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 SpaceTime or 3 + 1 Etc there's always this plus one at the end this means it's onedimensional 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 tens to include extra temporal components or you may see it as x^2 + y^2 + z^ 2 minus t^2 it just has extra minuses after it in Bar's

work he introduces a second SpaceTime coordinate T Prime described by a D+ 2dimensional 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 yangang Mills theories through exceptional Periodicities this recent work by Rios by Chester by morani they consider the super algebra in D equal 27 + 3 Dimensions the descending dimensional sequence from a super algebra in D = 27 + 3 to 26 + 1 reduces the dimensions directly along an 11-dimensional brain World volume yielding an Nal 1

super algebra in D = 11 + 3 which upon successive dimensional truncation aligns with the n = 1 super algebra in D = 10 + 1 and D = 11 + one as well as type 2 A2B Strings this suggests an 11-dimensional brain World volume origin for string dualities in both M and F Theory though this time with signature 113 Wilson surfaces and loop space connections these guid provide insights into the underlying symmetries and structures of M Theory in particular Wilson's surfaces generalized the concept of Wilson Loops expressed as follows which represent the parallel trans

transport of particles and gauge Fields in M Theory Wilson surfaces describe higher dimensional extensions and interactions with M brains such as M2 brains coupled to the three form potential C3 and M5 brains coupled to the six 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 caligraphic Alpha = a plus B2 plus C3 Plus so on 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 H way zeta function these Zeta functions contain information about the distribution of rational points and other geometric invariance such as those talked about by one of the Millennium prize problems the Birch Swinton dire conjecture though this conjecture refers specifically to

the rank of an elliptic curve group and the Order of Vanishing of its Associated L function you can connect BPS States 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 candelis ASA green and Parks see this talk here about the langland 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 buu1 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 contractable spaces on Which you one act freely historically categorical symmetries originated from John bases and Jame Dolan's study of higher dimensional algebras in the 1990s M5 brains have categorical symmetries why their selfdual 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 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 fals 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 nonzero Torsion that's Einstein carton you can

also assume that there's no curvature and there's no torsion but there is something called non-metricity that's something called symmetric Tel parallel gravity something else I like to explore are higher spin gravitons higher spin spin gravity theories are characterized by massless Fields with spin greater than two such as vasil's higher spin gravity in ads4 the action for these theories has a form similar to the above Where h and f represent the higher spin Fields these theories possess infinite dimensional gauge symmetries but so does general relativity given you ordinarily consider the dimorphism 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 minus1 while general relativity involves Vector Fields therefore it exhibits that conjectured Duality with certain large n cfts with higher spin symmetries such as the o n Vector model CFT historically fron doll's work during the late 1970s and early 1980s laid the foundation for higher spin gravity notably with his equation for massless fields of arbit spin in some ways you can think of

this as allowing for more ways to quote unquote wiggle in SpaceTime rather than the regular two degrees of freedom of Ordinary gravity theories the AA singer index theorem this index theorem is a landmark result in differential geometry and topology what it does is comput 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 sigh where p is called the principal symbol of the operator meaning the highest order homogeneous part of the operator in Local coordinates and size of the point in the cotangent bundle because of this elliptic operators have favorable properties such as the existence of smooth Solutions and wellposedness in string theory the theorem has found application in establishing anomaly cancellation conditions when applied to the elliptic Dura

operator on the world sheet the index is associated with the topological invariance of the world sheet like the oiler characteristic and The hers BR 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 fafan of the curvature form divided by the fafan of the tangent bundle so this expression on screen here and the fafan is a polinomial function associated with a skew symmetric Matrix such that the square of

the fafan equals the determinant of the Matrix you can Think of the AA singer index theorem as a generalization of the gaus Bon theorem that is a method to associate purely topological invariance to the curvature of a manifold but in the context of elliptic differential operators this theorem was proven in 1963 by Sir Michael AA and Isidor singer who received the Abel prize in 2004 for their work moduli stabilization recently a paper was Published by basori which provides non-perturbative terms in the super potential and the combined effects of logarithmic Loop Corrections and two non-perturbative super

potential Kor 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 SII and adaa are coefficients that depend on various parameters of the theory and this Caligraphic V represents the internal volume of the compactification this potential exhibits a minimum at finite values of the volume modulus given as follows where w0 is the Lambert W function and P and Q are convenient parameterizations and U is a parameter related to the non-perturbative contributions 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 de space and stabilizes moduli Fields so why is this important Kurt well this is an important finding because it demonstrates the existence of stable de 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 as well as Providing insights into moduli stabilizations which connect String Theory vacua Landscapes with observable universe properties and particle

physics phenomenon known as string phenomenology Dark Energy dark energy is about the expansion of the 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 effective action by now you should be familiar with these symbols but for those who skipped around and want to refresher that capital G is the metric the dilon field

is fi the NS ns3 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 dynamically evolving over time alternative models in type 2A and 2B give different perspectives on dark energy and Cosmological evolution such as the presence

of extra brains and Orient folds which can stabilize the moduli fields and the interaction of fluxes and form Fields respectively is string theory that flashlight we need to illuminate the dark corners of the universe Ambit twister String Theory you've heard of Twisters but have you heard of Ambit Twisters what are they well they generalize Twisters by considering the complexified phase space Of null geodesics instead of manovski SpaceTime the Ambit twister space is a huge space that contains twister space as a Subspace Ambit twister string theory is a framework that uses both twister and Ambit twister

spaces to describe scattering amplitudes of massless particles the world sheet 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 Ambit twister strings so what's the difference their target space comprises the space of complex null geodesics rather than just regular SpaceTime the chy formula gives a compact representation for tree level amplitudes of massless particles expressed as integrals over the moduli space of punctured remon 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 Ambit twister strings to emerge decades later although Ambit twister String Theory simplifies scattering amplitudes encoding soft limits and colinear singularities it currently faces challenges such as limitations to the perturbative calculations a lack of understanding of the non-perturbative aspects and its applicability is mainly For massless particles non archimedian geometry there's another field called the ptic numbers so ptic numbers are defined as equivalent classes of Koshi sequences of rational numbers converging with respect to something called the patic

norm now just as there's nonukan geometry there's also something called non archimedian geometry the ptic numbers denoted as Q with a subscript P form the completion of the rational Numbers Q with respect to the ptic 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 ptic string theory was originated by volvic in the 1980s and it embeds the string World sheet into ptic SpaceTime using the adapted polyco action notice the ptic norm here this allows invariance under ptic reparameterizations and

vile Transformations ptic string amplitudes have factorization properties similar to their archimedian counterparts allowing for ptic analoges of veneciano and verosa Shapiro amplitudes tonic condensation occurs in the ptic setting giving a non-perturbative description of Deb brains the adelic product formula associating products of amplitudes with certain topological invariance hints at connections between ptic and archimedian string theories although this remains Wonderfully speculative another physical theory that involves the ptic numbers is the so-called invariant set theory by Tim Palmer which suggests that the Universe evolves on a fractal attractor more about this Theory coming up on toe shortly with

Tim palmer but also here's a podcast with Tim Palmer and Tim modeling enumerative 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 cab 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 simplec one and the B model so the complex one for topological string theories they give information on the modulized space of Kya 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 three-fold gromov wit and invariance generalized classical intersection Theory simplec modular symmetry in heterotic string vacua ishiguro kabashi and Uka recently examined the unification of flavor CP and U1 symmetries coming from simplec Modular symmetry in the context of heterotic string theory on Cal threefolds they found that these symmetries can be unified into this simplec groups modular symmetries of cabal threefolds 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 simplec modular symmetry they have S3 S4 T Prime and S9 non-abelian flavor symmetries on explicit toal orbifold with and without Resolutions on Z2 and S4 flavor symmetries on three parameter examples of cat threefolds this new result shows that non-trivial flavor symmetries appear in not only the exact orbifold limit but also a certain class of Cal threefolds This research is fascinating because it gives a different perspective on the unification of flavor CP and U1 symmetries Paving the way for more comprehensive theories in

string phenomenology allowing us to test String [Music] Theory layer seven congratulations on making 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 mistaken the Physical universe as being synonymous with reality this is a point that Hillary putner makes despite this understanding

String Theory gives you a bedrock at the fountain of reality the reality that can be established mathematically and logically let's get on with the iceberg no precise statement and derivation of ads CFT 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 lightlike boundaries understanding holography for lightlike boundaries as in the case of Kowsky SpaceTime differs significantly from Tim

likee boundaries typical of ads CFT the existence and nature of large and limits for theories that aren't gauge theories and for theories with less or no super symmetry isn't anywhere as developed either number five sub ads locality how do you understand the emergence of bulk physics at scale 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 gravitational dressing number seven 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 theomorphism and variance this is particularly relevant for operators that couple strongly to gravity number eight entanglement wedge Reconstruction so the conjecture that the boundary sub region

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 sub region entanglement while the causal wedge is the bulk region causally connected to that boundary sub region 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 fuzz balls and the micr structure of black holes The fuzzball Proposal in string theory suggests that black holes possess a micro structure 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 Deb brain bound States it's an attempt to describe the near Horizon geometry using the Dual conformal

field Theory to Translate that attad the fuzzball conjecture replaces the mysterious core as well as the edge of black holes with information storing strings the Beck and Stein Hawking formula agrees with the degeneracy of these fuzzball states accounting for the micro states 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 micr structures and micro States to be specific the micro structure generally refers to the arrangement of string excitations that comprise the black hole's interior while micro states 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 holees entropy now how do you Generalize these fuzz balls 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 background Independence and

challenges 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 witon but why is this such a confounding conundrum well it's because string theories perturbative roots demand a predefined background however Kurt what if you incorporate the pon tensor derived from the poov 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 smolen books of string theory by the way a podcast with Lee smolen was just released about a week ago check the description or click subscribe to get notified pure spinner formalism in super string

theory and alternative to traditional Raymond nevu Schwarz and green Schwarz formalisms exist and they're called Pure spinner formalism so what makes this psf different the formalism employs what are Called Pure Spinners quote unquote which are a special class of spinners being selfdual 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 direct 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 a new work published in just 2022 which by the way is only a blink of the eye in this field antonas lacome and Leon Tes presented

a cosmological inflation scenario within the framework of type 2B flux compactifications what makes their work different they used three magnetized D7 brain Stacks the inflation is associated with a metastable desitter vacuum and the inflation is identified with the volume modulus the authors proposed 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 detailed 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 consider 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 DE 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 SpaceTime time isn't fundamental but comes from something pre geometric in condensed matter systems Elementary excitations

in a lattice such as spins and cubits 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+ 1dimensional SpaceTime as the emergent gauge fields and gravity are realized via fusion and braiding of anionic excitations in the system anons Are exotic quasi particles in two-dimensional systems the emergent gauge group structure relies on Anon 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 SpaceTime 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 discreet 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 field and hence the structure of the Kor potential and super potential the super potential and Kor 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 super potential and Kor potential structures the Kor potential is herian and modular invar 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 louon Den dri 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 analoges you can do so by following the cell decomposition theorem In string theory considering the modulized space of Kia manifold more explicitly on screen here where C YN represents the set of

all Calo nfolds in O minimal structures caligraphic o and m subal graphical denote the corresponding modui space this is brand new research and the best paper I found is by Grim on taming the landscape of effective theories that is using o minimal structures to explicate the swamp land string Universality string universality is the conjecture that every consistent quantum gravity Theory corresponds to the vacuum of some string theory or a string theory compactification it's based on the Fairly braggadocious belief 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 super symmetry we can symbolically represent this conjecture as follows Where qg is the space of all consistent quantum gravity theories and the caligraphic 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 end point 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 super symmetry 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 there's also the absence of anomalies which you can think

of as some Quantum consistency even though I don't like that word and then there's stability which you can Think of as the absence of runaway or uncontrollable Behavior 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 smolen 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 brain can be seen as generalizations of strings as you well know given that you're now at layer 7even 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 Hammer off's idea that the same Mechanism responsible for quantum gravity is twinl responsible for Consciousness it's known as orchestrated objective

reduction and we've covered it here on this podcast with hammer off 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 Ed to conjoin string theory and Consciousness questions of Consciousness such as the hard problem and the So-called problem of other minds are explored on this channel theories of everything this podcast here but what isn't researched are the roles of these extra dimensions and compactified spaces on different conscious experiences is there something like a dilaton field

of qualia here's what Ed Witten says on the topic 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 and of how we perceive anything including physics and that has to do I think with the Mysteries that bother a lot of people about qu 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 For ourselves makes us extremely uncomfortable especially because of our Consciousness which seems to clash with that idea consider Carl yung 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 fountain of the world there were hundreds of years of philosophizing with imprecise language and models of the times about nature such as thees a presocratic Greek philosopher who suggested that water is the origin of all things and the load Stone has a soul he had an early cosmological model attempting to explain The nature of the earth and its position in the universe so these can be seen as primitive nurse crops in the Daniel dennit sense necessary for the development of the more articulated mathematical laws 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 m 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 paparian science what about what lies outside in principle observation like singularities what about obser ation 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 all right 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 being on a more correct path than some of the Alternatives 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 neoy in this all and if someone wants to collaborate with me then please comment the word collab c o 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 prall Colin AE 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 it is it is I'll be honest but I do have to prep for this I do have to get my outfit and I do have to shower and all of that even if I'm just recording 5 Seconds of some extra material but anyway you're not here to learn the secrets of my exquisiteness 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 Wolf's Theory I ran out of time I also wanted to do ASM Totic safety and what

it means to have negative dimensions of space I also want 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 willr on his theory of everything it's on screen here if you're interested there are two as wi's 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 funer there's no connections

that I have unfortunately I get bitter about it because some 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 umic 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 onetime 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 2 hours maybe 2 and A2 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 Kurt jungle. org and that has a mailing list the reason being that large platforms like YouTube like patreon on 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 Quinton 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 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 gain from rewatching lectures and podcasts I also read in the comments that hey toll listeners also gain from replaying so how about instead you relisten on those platforms like iTunes Spotify Google podcast whichever podcast catcher you use and finally if you like to support more conversations like this more content like this then do consider visiting Patreon.com jongle and donating with whatever you like there's also PayPal 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 tow 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