 our school and we are happy to have the sort of lecture by Kazui Nikora. Okay, thank you very much. So yesterday, I discussed some examples of global anomalies in the case of one-dimensional fermions, but the method which I used yesterday was not so systematic. So today and tomorrow I'd like to discuss more systematic description of anomalies, in particular fermion anomalies. And they are now known to be described by what is called Atyapati Shinga Yetta invariant. So that Yetta invariant is a general formula for anomalies or fermions, and that Yetta invariant also contains the information of part of anomalies and also nonpart of anomalies. And it is believed that Yetta invariant is the complete description of anomalies or fermions. So I want to explain the physics argument to derive this general formula. So first, for that description, systematic description of anomalies, first I'd like to construct chiral fermions as H-modes or boundary modes of higher-dimensional theory. So first let me explain the setup. So we want to study anomalies of theories in D-dimensions. So I consider some D-manifold. So this W is a D-dimensional manifold. I also introduce some D plus one dimensional manifold. I will later consider massive fermions on this D plus one dimensional manifold. And then I realize massless chiral fermions on this D-dimensional space. Sorry, so I have to say that this Y is a manifold with boundary and the boundary is W. So this is the relation between this Y and W. And pre-eval I say manifolds. I also assume that I put the background fields of symmetries. For example, if we have U1 symmetry, then I assume that I put U1 gauge field here and also here. So it is always implicitly assumed that I put some background fields for the symmetries. Okay, so the situation is something like this. So we have a boundary manifold, which is this W and this bulk is Y. And also for later convenience, I take a coordinate which is orthogonal to this boundary. So I take this coordinate, which I call tau. So this is a coordinate which is orthogonal to this boundary. And I assume that this boundary is located at tau equal to zero. And this coordinate is defined only near the boundary. So this coordinate is not very defined inside this bulk, but it is very defined near the boundary as a distance from this boundary. And I also assume for technical simplicity that near the boundary, so I assume that there exists some number, Y, such that the region near the boundary is just a product form. So near the boundary, so the manifold is just a product and I also assume that the gauge field in this region is just a pullback from this boundary. So that means that the gauge field does not depend on this coordinate tau near this boundary. Maybe I should write it more explicitly. So gauge field does not depend on this tau and also the tau component of the gauge field is zero in this region, in this neighborhood of the boundary. So this is the basic geometric setup. And in this setup, I consider massive fermion, the bulk, P plus one dimension. And each Lagrangian is simply just the standard Lagrangian of Dirac Lagrangian. So this is a very familiar Lagrangian of fermion. Just for simplicity of presentation, I treat the case of Dirac fermion but it is also possible to describe Majorana fermions. In the case of Majorana fermion, we just need to take this Psybar to be roughly Psy up to some matrix. Then we can also discuss Majorana fermions. But just for simplicity, I discuss the case of Dirac fermion. And so because we consider fermions, we have gamma matrixes. And in particular, there is a gamma matrix in the tau direction which I denote by gamma tau. So this is gamma matrix in the direction tau. The matrix is of course satisfies the standard algebra. I mean in the orthonormal system of coordinates it's like this. So in particular, the square of this gamma tau is just equal to 1. So this means that this gamma tau has eigenvalues plus or minus 1. So I take this gamma tau and then I import some boundary conditions by using this gamma tau. So because we are considering manifold with boundary, we have to put some boundary condition on the boundary. And I choose the following boundary condition. So I impose the boundary condition which I call L that 1 minus gamma tau acting on psi at the boundary tau equals 0. This is 0. So this means that at the boundary I require that the fermi field is an eigenvector of this matrix gamma tau with eigenvalue plus 1. So that is the meaning of this boundary condition. So the components of this fermi field with eigenvalue minus 1 is set to be 0 at the boundary. So we can impose this boundary condition and one good thing about this boundary condition is that we can impose this boundary condition no matter what symmetry we consider. So this boundary condition always exists and this is consistent with any symmetry. For example, if we have some U1 symmetry or SUN symmetry or some symmetries this boundary condition is obviously consistent with these internal symmetries. And maybe it is not so easy to see but it's also consistent with some more subtle symmetries such as time reversal symmetries. So anyway, we can impose this boundary condition and I impose this one. James, I will I have a question. I was wondering so far you need not have taken D to be even or odd, right? So far the bulk and the boundary could be any dimension. Any dimension, yes, exactly. Yeah, and also, yeah, that is important because so part of a normal is exist only in even dimensions. Global anomalies exist in any dimensions. So, yeah, it's important to treat the general dimension. Thank you. Okay, now I want to study the behavior of this theory near the boundary. So near near the boundary so I want to study the direct equation so direct equation is so this quantity is zero. So this is a direct equation equation of motion and this can be rewritten like this. So we take gamma tau same we can write it in this way. Yes, in this way and here this Pw is defined as follows. So this Pw is basically this is a direct operator direct operator the money for W. So we can write this direct equation like this using the derivative in the tau direction and the direct operator in the W direction and also this master and I study the behavior of this fermion near the boundary in the large mass limit. So I take this M to be very large then it turns out that there is a localized mode near the boundary so there is a solution of the direct equation this bulk direct equation which is localized near the boundary for this solution to exist this mass parameter must be negative. So let me write down this localized solution so it's given as follows so this fermi-guid psi is taken to be a product some field chi times exponential minus m tau this chi only depends on the coordinates of W so the tau dependence is only here and so this chi depends only on the coordinates of W the boundary and also chi is assumed to be the eigenvalue of gamma tau with eigenvalue 1 so this is 0 and also this chi is assumed to satisfy this equation so this is basically the direct equation on the boundary then we can easily check that this is the solution of the direct equation so first of all it satisfies the boundary condition as a trivially because the boundary condition is that this quantity vanishes on the boundary but this is equal to minus gamma tau chi yes and I imposed this condition on this chi so this is 0 so it satisfies the boundary condition and also this solution satisfies the direct equation so the direct equation the bulk direct equation was given by this one I mean so if this vanishes then that means that this satisfies the direct equation so let's check this so this is 0 so from the derivative with respect to tau we get minus m from this Dirac operator on W we just get 0 because I assumed that this chi is annihilated by this Dirac operator so this is 0 so 0 and finally this last term this gamma tau is just plus 1 because I took this chi to be an eigenvector with eigenvalue plus 1 under this gamma tau so we have m so this is 0 so this this and that satisfies both boundary condition and the direct equation so this is the area solution and this is so localize near the boundary if must parameterize negative so by my convention this tau the bulk is in the region of the negative tau so that was my convention let's go back to this figure so tau was a coordinate which was orthogonal to this boundary and this bulk region bulk region is in the region of negative tau so that was my convention and then so this tau is negative and also this m must term is negative so this exponential of minus m tau behaves like this so this is exponentially localize near the boundary and this localization becomes better and better if this must parameter the absolute value of this must parameter becomes bigger and bigger so in the infinite must limit this mode is completely localized on the boundary but this solution exists only if the must parameter is negative so that is also important so this exponential minus m tau so this is localized near the boundary if and only if this m is negative and there is no such localized mode positive must parameter context of condensed matter physics this parameter region positive must parameter is called trivial topological phase and this region negative mass is called non-trivial topological phase and I will discuss a little more details later but anyway so here the important point is that the behavior of the theory is different depending on the sign of the must parameter so there are two different phases okay so we get localized mode on the boundary and this gamma tau can be regarded as a kind of generalized chirality operator on the boundary so this gamma tau anti-commutes with the direct operator on W so this is one of the basic properties which must be satisfied by chirality operator if we regard this gamma tau as a chirality operator then this chi is a massless chiral fermion on the boundary so this chi is chiral fermion because it's an eigenvector with eigenvalue plus 1 under this gamma tau and this is a massless fermion because it satisfies massless equation on the boundary so in this way starting from d plus 1 dimensional mass theory we get chiral fermion on the boundary for example if we consider d plus 1 called 5 so if we consider 5 dimensional arc then we have 5 gamma matrices gamma 1, gamma 2, gamma 3, gamma 4, gamma 5 then we can take this gamma tau to be gamma 5 so this is a well known chirality operator in 4 dimensions so this is the most famous case so I believe that this gamma 5 appears in any textbooks of quantum field theory so by this construction we can obtain chiral fermions 4-dimensional chiral fermions from the 5-dimensional arc massive fermion and in that case this gamma tau you really this gamma 5 is a very famous chirality operator in 4 dimensions but in any dimensions we can just regard this gamma tau as a chirality operator in some generalized sense okay and also we need to regularize this theory and it is convenient to use Pauli-Biller's regularization for this system so I mean I'm just considering just considering free massive fermions this fermion is free because I'm just taking gauge fields as a background field so I'm not integrating over gauge fields so in that case the most convenient regularization of fermions is Pauli-Biller's regularization and I take Pauli-Biller's mass to be positive because I don't want any localized zero-modes from Pauli-Biller's regulators so if I take Pauli-Biller's mass to be positive then there is no localized mode on physical Pauli-Biller's field so this condition is necessary for physically sensible theory so if we get some massless fields from Pauli-Biller's fields then that violates some physical principles such as unitarity so I need to take this mass parameter to be positive here I mentioned some four-dimensional example but let me also give another example to demonstrate that this construction is very general so by this construction if I take this P plus 1 to be 2 if we consider Majorna-Ferremium then I can obtain one-dimensional fermions which I discussed yesterday so I take Majorna-Ferremium with O-N symmetry so under O-N symmetry these fermions are rotated like this so this is the same symmetry which I discussed yesterday two dimensions I take the action of the Majorna-Ferremium to be like this place of psi i so the details are not so important but let me just write the equation for completeness and here this Y is a anti-symmetric matrix and gamma tau is matrix like this and gamma sigma is like this so I don't show it explicitly here but you can show that this Lagrangian is Lorentz invariant and because of this anti-symmetric matrix here so let's just take this action and then in this Massive-Ferremium theory there is a localized mode which is including coefficients the localized mode is given by this formula maybe I should define tau and sigma so we have two-dimensional spacetime which is Euclidean spacetime and so what I call the W is maybe I changed color so what I call the W is here and what I call Y is this half-plane then I obtain this kind of localized mode and by very simple calculation by just substituting this expression into this action it's an easy exercise so if you have time then please do it by yourself then after easy computation we get the following action so this is exactly the action of the one-dimensional Massive-Ferremium as we discussed yesterday so in this way we can realize this theory as a boundary mode of a bulk Massive-Ferremium and it turns out that this construction is very general so we can construct any ferremiums from higher-dimensional to one-dimensional Massive-Ferremium okay so this concludes the construction of boundary Kyler-Ferremium then I can now go to the discussion of anomalies of these Kyler-Ferremium so we construct the manifold Y with boundary W and I imposed the boundary condition which I denoted by L then we can consider the partition function of the theory on the manifold Y with boundary condition L so I denote this partition function as Z of Y comma L so this is given by the path integral of the Massive-Ferremium in the bulk sorry I have a question before we go on so I didn't really understand why we were doing this Pauli-Villard regularization before so why do we need to do this Pauli-Villard regularization what do we need to normalize? okay so I will explicitly use this Pauli-Villard regularization so maybe okay you will use it now okay and also another question regarding that so you said you had to take the mass of the Pauli-Villard field to be positive so there is no localized mode but I guess you could also just take a scalar particle no to do Pauli-Villard regularization and then there will be no localized mode right scalar field um no I mean so I think there is localized mode if the mass is negative so the Pauli-Villard field has the same action as this one so I mean so only the statistics is different but to show the existence of the localized mode I just used the Dirac equation and that Dirac equation also applies to Pauli-Villard field okay so I need to take the Pauli-Villard mass to be really positive okay yes yes if you if it follows Dirac equation yes yes thank you okay so I consider this function and for negative mass parameter the bulk is gapped so it has a large mass gap given by the absolute value of m and I take this absolute value of m to be infinity so this means that the mass of the bulk field is infinity so in the low energy limit there is no degrees of freedom in the bulk but on the edge or boundary maybe I should write boundary we have a chiral fermion the sky so the only low energy degrees of freedom of the system is this chiral fermion so there is no degrees of freedom in the bulk the sky on the boundary so physically we expect the partition function of the bulk theory this partition function is essentially equal to the partition function of this boundary chiral fermion so we want to we want to think that this partition function with boundary condition L is actually the partition function of this chiral fermion on the dimensional manifold so if we can really identify this partition function as a partition function of the chiral fermion then this equation gives the definition of the partition function of the chiral fermion so I want to take this this bulk partition function as a definition of this chiral fermion partition function but there is a program so this left hand side so this has a very good property this is completely gauging variant however this left hand side may depend on why so we want to define the partition function of chiral fermion on the dimensional manifold in a gauging variant way and gauging variance is guaranteed by this construction because so this left hand side is gauging variant I mean so that is true if we use Pauli-Piller's regularization I will later more explicitly use Pauli-Piller's regularization but Pauli-Piller's regularization guarantees that we can regularize this expression in a gauging variant way so this is nice we get the gauging variant definition of the chiral fermion partition function but now the problem is that this definition depends on the depends on how to take this bulk even though we are interested in the dimensions so this is a modern understanding of anomalies so this is an important point so let me write it so modern definition of anomalies is the following the question in the chat ah yes chat what gives symmetry so I didn't explicitly specify the symmetries but for example I mean so the fermions I am talking about can be can have any kind of symmetry so for example in this example these major fermions this action have this ON symmetry we can consider any such symmetries so I discussed it in very general but abstract way so I didn't mention the symmetries explicitly but so this side can have any symmetry maybe U1 symmetry or SUN symmetry or whatever symmetry you like and then I introduce background fields for that symmetry so implicitly the background field is here so it is a direct operator and it is given by the covariant derivative and this covariant derivative contains gauge field so we have gauge field for the symmetries and the gauge invariance means the gauge invariance under the gauge transformation of this gauge field so is it clear ok so ok so the modern formulation of anomalies are the following so first everything is formulated completely gauge invariant way so this is quite different from the old treatment of anomalies so in textbooks anomalies are described as a violation of gauge symmetries but here I'm formulating everything in a gauge invariant way so it is really important so there is no violation of gauge symmetry at all but the definition of the partition function or definition of the d-dimensional theory may depend on the d plus 1 dimensional bulk and this dependence of the theory on the d plus 1 dimension is the anomaly in the modern understanding so I so tomorrow I will talk more details about what kind of dependence I'm talking about so here I'm just abstractly saying that d-dimensional theory depends on d plus 1 dimension of theory but we can really isolate the dependence on d plus 1 dimension of theory and so this dependence on d plus 1 dimensions is described by the Atyapa Tadeh Shinga yata invariant and tomorrow I will discuss that point so maybe I should finish today thank you very much so let's thank Kazue for the lecture and now we can stop recording and proceed to the discussion session yes the question is whether you had only imagine that you had for example a charge n particle and not the previous ones or charge 3 so you can only construct multiples of 3 so that's what we mean with complete you need to have the charge 1 but how do I know if I have charge 3 or 1 because I can always redefine the constant so yeah sorry so this is why it depends like you need information about the first year in the sense that you need information on some of the monopoles okay so when we talk about completeness of the spectrum it's going to be completeness of the electric and the magnetic spectrum you also need the monopoles to break the magnetic symmetries and since you need both then by direct quantization yes you can define what is the charge 1 in terms of the other one so but then do you need to discuss these higher form symmetries can you just assume that there's monopoles and you have to do direct quantization then yeah like you need you also need the monopoles to break the magnetic symmetries because otherwise maybe you can get rid of the electric ones but you still have some magnetic ones for symmetry okay and then like you need both to get rid of all the symmetries and then precisely thanks to the fact that you have both like you can define what is your lattice right so the typical way to proceed is just like I mean originally you can have some gauge theory and you can define what is the lattice of gauge charges that is consistent with direct quantization and I mean if you have just a u1 it's easy because you just have the integers for both electric and magnetic more generally for other gauge groups and disconnected groups I mean you can have many different types of representations which are not so trivial but in general given some gauge theory you can define what is the lattice that you have and then the full lattice have to be complete all the representations including the monopoles including the magnetic ones okay so you need to introduce monopoles respect this no global symmetry yes okay thank you okay so we have spent two days talking about global symmetry because it's well you will see that many of the other things just come from refining these statements of how much we can break this symmetry so it's very a very central topic although as we discussed sometimes it's not so useful for phenomena because I mean we are just saying that we cannot have an exact symmetry but as long as we break it a little bit seems that everything is fine again so what we want to do now is to make this more precise and quantify how close we can get to the situation of restoring a global symmetry because this if we are able to do this this can have much more important implications for phenomena so that we gravity conjecture and the distance conjecture that I don't know maybe you have care of and that are formulated by different means actually now we can have an understanding of them also as quantum gravity obstructions to restore some global symmetry so I'm going to explain it this way okay I'm going to start asking like how close we can get to restore a global symmetry study what is the physics that appears there and then connect how this connects with the gravity and the distance conjecture so yesterday I started saying that restoring a global symmetry is going to be equivalent to moving in a modular space so in string theory all the parameters the continuous parameters are parameterized by vacuum expectation values of the scalar fuse so everything is dynamical and then we have some modular space which is the fielded space that's the different values that these scalar fuse can take and we can move in this modular space and get different effective theories depending what point of this fielded space we are because this will imply that we have different values for the parameter for the coupling, the masses and so on so you could then wonder okay if I can move in this fielded space what happens if I move in such a way that for example a gauge coupling goes to zero okay so and let me put a bit more bright that I can see okay yeah so I have some gauge coupling that goes to zero and this is going to be equivalent to moving I mean taking this limit like moving some limit in this modular space and then in the moment in which the gauge coupling becomes zero this is going to restore a global symmetry right because we turn off the dynamics of the gauge field it's like engaging the symmetry and we restore for example if we had some u1 h theory then we restore some u1 global symmetry zero form global symmetry okay now you would say okay but something has to be wrong now because we have said that global symmetries are not allowed in quantum gravity so how string theory deals with this well the point is that these points as I mentioned yesterday are pushed away to the boundaries of the modular space okay so they happen to be at infinite distance okay otherwise if they would be at finite distance it would be like restore a global symmetry in the middle and that would give rise that would be a contradiction with not having global symmetries so the point is that what happens is not that that point is not allowed like it's far away it seems to be the far away that as we are approaching the point we are getting a better more approximate global symmetry and closer to be exact and by I mean this is the I'm gonna give you what is the conclusion of what happened when you study this in string theory but what happens is that the effective field theory is breaking down in a continuous way okay it's not only that that point doesn't make sense it's also that as you approach the point the cutoff of the EFT goes to zero okay so that there is always some energy regime in which the effective field theory makes sense but this is smaller and smaller okay so when you try to engineer a global symmetry the range of energies that you can describe gets shrink gets smaller and smaller and this is purely a quantum gravitational effect because by in quantum field theory I mean it's not only that global symmetries are fine but also that these limits in which you have some large field range or more intuitively when you have a very small gauge coupling are completely fine right like small gauge coupling is something that we like in quantum field theory because we have a perturbative gauge theory and there is nothing wrong with that like it's completely it's precisely the limits in which we have computational control in quantum field theory so quantum gravity I mean this idea of no global symmetry telling us that there are quantum gravitational effects that imply that precisely in this limits in which we have some principle computational control something has to go wrong the effective field theory has to break down there is new physics that must appear so the EFT let me write it so it's clear breaks down when approaching these points because there is an infinite tower this is what we observe there is an infinite tower of states that become light and massless at that point so since we are considering effectively theories that are weakly coupled to instant gravity having an infinite tower of states necessarily implies the breakdown of the EFT and it's a drastic breakdown it's not just that you can just add like integrate in a few states and you have a new quantum field theory description like we cannot have anything in the fields we click a couple of times in gravity like the scale at which quantum gravitational effects become important goes to zero so something drastic has to happen maybe you get a stream perturbative limit and you get stream theory or you grow extra dimensions we will see different possibilities that is a drastic thing coming from quantum gravity and yes I see this okay are these states always charged okay so if there is a gauge coupling that goes to zero there is a tower of states that is charged and we will see that in all the examples and what will be the conjecture is the weak gravity conjecture so that the mass is smaller than the charge if there is no gauge coupling you wonder more generally what happens when we approach a boundary of the modular space because maybe we restore a global symmetry which is not necessarily associated to a gauge coupling and then you will say okay then it's not necessary that the tower is charged in that case if there is no gauge coupling but we don't have any example of that like all the examples we have seen here so far I don't know why are have always a p-formed gauge coupling that goes to zero asymptotically so there is always a charged tower we don't know if that's a matter of the lamppost but yeah that's what happened so far but in principle I mean also what the distance conjecture would require is just that there is an infinite tower I mean that's an impulse that is charged okay so sorry if with a couple this effective theory from gravity we need to recover the description of the standard model for example in some scale right well the standard model is up all to gravity I mean yeah but only an example standard model but you recover an effective description of a system if we send the black mass to infinity so okay let me say what happens is that the mass of this tower over the black mass it's always in black units goes to zero okay so this is very important all the sum plan constraints are always in terms of the black mass such that precisely when you send the black mass to infinity I mean this becomes trivial and it could be that the mass of the tower is still very heavy it's just that this mass less in comparison to the black mass but if the black mass goes to infinity this has no effect the problem arrives when we couple effective theory with gravity yes okay thank you yeah this only happens is a quantum gravity so only happens when the eft is applicable to gravity and therefore the mass of the tower goes to zero only in terms of the black scale so it's only mars less when the black scale is finite okay so yeah this is what happens okay so we have this tower of states which implies that the cutoff of the theory goes to zero and this is how it goes to zero in all examples three theory we have and also what goes to zero according this is what will give rise to the weak gravity conjecture and this to the distance conjecture okay so that in terms of the field distance that do you travel in the field space because exponentially to zero and in terms of the gauge coupling if it is a gauge coupling it goes linearly to zero in terms of the gauge coupling so when the gauge coupling goes to zero the cutoff goes to zero any other question I have a question so as you said in string theory there are no parameters because all parameters really are expectation values of some field so this has some flavor similar to the fact that symmetries has to be gauged they cannot be global but it's not quite the same thing so is there a gravity conjecture that says that this should be also the case in quantum gravity or? the parameters are parameters by scalar fields yeah there are no there are no parameters there is a symmetry which is a minus one for global symmetry so yeah like a minus one for global symmetry the current is a parameter okay so the current is always like a you have to sum plus one so the minus one for global symmetry you have a current which is a parameter and then you can wonder whether this minus one for global symmetry whether we also have to impose that it's broken or gauged when this parameter, when this current is gauged it's equivalent to say that it's an action that is not the parameter so it's like the typical things when you have just stabilized some scalars with the fluxes then you get action monotomy this is the process of gauging this minus one for symmetries you want to say that it's broken then exactly it means that the parameter has to be dynamical it means that you have to be able to deform the charge so that it's not conserved and this process of deforming the charge like for example because you have to make words or that you can deform continuously the parameter because it's a continuous parameter it's equivalent to breaking this minus one for symmetries so the statement that there are no free parameters in quantum gravity is equivalent to say that there are no minus one for global symmetries okay it's something that we are something quite new and also this year we are trying to see if we can formulate this in a sharper way because it's a bit exotic with the minus one for symmetries but I really think that there is like a symmetry okay sorry René sorry in a stream theory there are some special limits for example conifer limit, large volume limit in the modular space if we try to visualize in this point one can learn something about this conjecture for example in this analysis yeah exactly I mean the thing is that mostly all the models we do in stream phenomenology like on the engineer the standard model or cosmology within stream theory they are always in one of these limits because we want to have computational control and these are precisely the limits where we go to large volume or weak coupling and so on and it's precisely in these limits where stream theory is telling us that there is also something some new physics that is going to appear so yeah I mean in those limits the cutoff goes to zero but we'll see that that's not only what happens I mean this tower of states can also have implications of what is the behavior of the case carvings or the scalar potential or the different quantities of your EFT as you approach the limit and this is what gives rise then to the other conjectures of okay in general the scalar potentials as you approach the limit has to have these features or interactions have to have these features and everything is a consequence of these towers becoming light so it's like you have when you try to do stream model and it's like on one hand you want to be close to these limits because it's when we know how to compute things but on the other hand we cannot be too close because otherwise the effectivity is going to break down and there is like some force pushing you in opposite directions but yeah we can learn for sure a lot about this stream model being there okay thank you so these are the with gravity and distance conjecture right now I don't I don't want you to pay special attention to the definition of the conjectures I'm just going to focus on these universal features that both of them have which is that we have these towers of states becoming masses as the gauge coupling goes to 0 or the field distance goes to infinity as we approach the boundary space and the cutoff of the theory goes to 0 in terms of the mass of this tower okay so first what we are going to do is we are going to see just some examples in the stream theory very easy examples of how this happens okay now and once we see these examples and we get an intuition of what these towers are and what's going on we will define more properly what the conjectures are okay that then claim that this feature, this pattern that we observe in stream theory is more general should be a quantum gravity phenomenon okay so we'll give examples to understand the pattern from stream theory and then we'll formulate the conjectures and see how especially for example the gravity connection can also be derived from thinking about black hole physics okay so I'm going to take very easy examples first like first in 10 dimensions for people that know about stream theory maybe this is a bit boring for people that don't know about it I hope this can help to get some intuition of what's going on so the let me take tie to a stream theory so all you need to know is that the Lagrangian the action that we have in 10 dimensions in stream theory has this form so we have the Einstein gravity term we have this scalar which is the dilatoll okay so this is what parametrize parametrize which is the stream coupling and then we have these are gauge fields so we have on one hand a two-fold gauge field which is a nervous bars field and one-fold gauge field which is a Ramon-Ramon and we have other gauge fields but I mean there's no need to write everything okay so we have different p-form Ramon-Ramon gauge fields and this is it's important we have always when we check these conjectures it's important that we have to be in the Einstein frame okay everything is formatted in the Einstein frame okay so let's imagine now we have only one scalar which is this dilatoll okay so we can move in this field space the distance when we move in the field space is the integral of the square root of the metric right so it's going to be logarithmic on this S on this dilatoll okay and we can take two limits one is the limit in which the string coupling goes to zero it's like the dilatoll goes to infinity okay and then the table of states that appears are just the string modes okay so when the string coupling goes to zero or the string excitations become light so we have string modes with a mass given by the string scale which is m-plank times gs to the one-fourth which in terms of the canonical normalize field in terms of the proper field distance indeed goes exponential right when you plug this here this is exponential of delta-5 so this is the stream perturbative limit and also since the gauge coupling of the two-fold gauge field goes as s gs the square root of s inverse the tension of the string is since it's a VPS object it's proportional to the charge the tension is the string scale square so indeed the mass okay goes as the as the gauge coupling okay the tension of this object goes as the gauge coupling which is the so the cutoff also goes proportional to the gauge coupling now we can take the other limit which is that gs goes to infinity or s goes to zero and what happens here is that I have these zero brains that become light right because they are equal the mass is the string scale over gs so we get that this is the plank mass gs to three-fourths and in terms of the canonical normalized distance again this is exponential and the same like I mean it's also proportional to the gauge coupling because the gauge coupling for the c one which is the gauge field under which the d zero brains are charged also goes to zero in the limit right so the mass of the d zero brains again they are VPS goes as the gauge coupling and this is what is this well this is very well known it's a decompactification limit in that way because what we get now is an MCOD description in 11 dimensions decompactification okay so these are the two limits that appear in that way and funny enough even if we go now to more complicated compactifications still everything that we find is always one of the two like either we find stream perturbative limits or we find decompactification is there any question can you repeat this this limit of the decompactification please the idea behind this yeah so what happens in day to day is that you have this tower of d zero brains that are particles they are d zero brain that are becoming massless in terms of the Planck mass and this tower of the zero brains I mean it's consistent with this context but the point is that when you take the limit what happens is that they behave as a kalusa Klein tower so the density is like a kalusa Klein tower and what is going on I mean how stream theory is resolving the stream to distance limit is that indeed one is growing an extra dimension because this is like the KK tower so that is becoming like and this is a well known duality in stream theory that what happens is that you get an M-theory description so when the stream coupling goes to infinity the tight way stream theory is not a good description anymore the perturbative stream theory is not a good description you have an M-theory description in 11 dimensions in one dimension more because you have opened up this this extra dimension because of the KK tower and that's the new description that appears that now describes the physics as well so tight way at strong coupling is equivalent to M-theory in 11 dimensions okay and this is a well known duality but you can also I mean it's a canonical example of why we have this tower of states whenever we take one of the limits yeah okay thank you okay so this is like a KK tower so let me now also explain another example of this the compactification so now we can take tight way on a circle so we get a new scalar which is the radius of this circle which is this one I call it phi here okay and it parametrizes this would be the radius of the circle radius of the circle and it also parametrizes the KK function for the KK photo the same time before that the dilatum was parametrizing the gauge couplings for the for the gauge fields so here again we can take two new limits regarding this new scalar this radius so we can take the it would be the decompactification like this so the radius goes to infinity and indeed what happens is that now we have a Alusaclan tower that become light so the KK modes go to zero we have in general dimension as 1 over r square 1 over 2 p r 2 d minus 2 okay this is you can find this formula in the literature and if you write in terms of the canonical in normalized field in terms of the proper field distance again the KK modes indeed go exponentially d minus 1 d minus 2 phi and the same the KK modes also are proportional to this gauge coupling so you can plug I mean as you can see here precisely the gauge coupling has the same exponential rate that we derived here okay because I mean this is like 1 over the gauge coupling square right so the the KK modes well without the square sorry the KK modes go as the gauge coupling calling the gauge coupling of the KK photo so again we have a tower that goes proportional to the gauge coupling and at the case exponential in terms of the field distance and this is the compactification limit okay but here we can take another limit which is that the radius goes to zero okay and what happens here well I guess many people already know this but let me say it anyway so what happens is that in street theory we also have winding modes so we can have strings that are wrapping this circle and therefore when the circle goes to zero they become tension less okay so we have new particles that are winding modes and that behave exactly the same as the KK modes when you replace the radius by the radius by alpha prime so the string scale over the radius so this is what is behind this t-duality okay in which you can have a description so we can either have that way on a circle or we can have that way on a circle with inverse radius okay and the tower of KK modes on one theory is equivalent to the tower of winding modes on the other one so from the perspective I mean what is going in this limit okay we get all this tower of KK of winding modes but this is also a recompactification limit okay because they are equivalent to a tower of KK modes in T2B so we are also growing an extra dimension okay any question? okay so all these towers of states I mean what I wanted to share with this example is that they are very linked to the existence of dualities okay so all the I mean if you study string theory one of the first things you learn in because it's one of the revolutions in string theory was the manifestation of all these dualities, dualities duality, duality with M-theory and so on and precisely the way to I mean in which these dualities are manifest is because of the presence of these towers that imply a new would clickable description of the theory so by understanding the nature of these towers like understanding the nature of dualities of the string theory okay so let me summarize a bit what we have obtained so there is always some tower of states that's some slides, exponential in the field distance proportional to the KK so if we start it in tight way right we could take on one hand the limiting with the string coupling goes to zero and that we recover the perturbative string theory I mean the quantum field theory breaks down full string theory to make sense of it or we can take the limit in which the string coupling goes to infinity and then we recover an M-theory description in one dimension more also if we compactify in a circle right we can send this circle to infinity and then we again recover tight way or we can send the circle to zero and then we decompactify to ten dimensions again but in tight way okay so this is the duality so the tower always implies a new weekly couple description sorry couple description of the theory and in all the cases the cutoff of the EFT goes to zero so it's smaller than the Planck scale okay so I like this because it's a very natural string theory okay like what we used to have these dualities all the time like from many years ago but still from the effective theory perspective from someone working in phenomenology it's actually very surprising that if you take one of these lemi like you send some gate coupling going to zero you are going to get a drastic new description of the theory like the cutoff is smaller the quantum gravity cutoff is smaller than the Planck scale so even if the Planck scale seems to be very high and it seems that it's not going to give you any important implication for phenomenology because it's very high and gravity is very weakly coupled that's not true in certain limits it's not true in certain regimes because the cutoff is actually much lower than what you would expect and this is something that we have learned from the very beginning string theory and I think we should take it seriously like it's a perfect example of something that is very natural in string theory but very surprising from an EFT perspective because now we are saying that effective theories with tiny gauge couplings are problematic and effective theories with large field ranges are problematic so we have an upper bound on the gauge couplings and the field range that we can describe within effective theory in terms of the quantum gravity cutoff so tomorrow we will see I will explain a little bit possible phenomenological implications because this can be used to constrain inflation dark matter and other things because there are many models in phenomenology which we try to have very large field ranges or small gauge couplings so you can use this information from string theory to tell you what is the cutoff and what's going to happen any question sorry when you said large gauge coupling is problematic do you mean just in terms of because it's non-perturbative or I mean small so very tiny gauge coupling very small one yes maybe I said it the other way around I mean tiny in the sense of small okay so what I'm going to do now so let's formulate these conjectures more precisely what is the sharp statement okay behind it's mainly based on this that are becoming light and then I will give you some more complicated examples okay to see that this is not just about 10 dimensional string theory okay so let me maybe start with the distance conjecture so the distance conjecture is just formulated in words this phenomena that as we approach some boundary of the field space we get some infinite hour of states that becomes light exponentially on the perfect field distance so the examples I gave you were very easy because we only had one scalar field but in general when we have a more complicated field space this we have to study what are the geodesic distances so this becomes exponential in terms of the geodesic distance as we move in this field space and the exponential rate is supposed to be an order one factor in these examples that we checked it's an order one factor like 1 over square root of 2 or things like this and is not specified by the conjecture so this is one of the open questions okay one of the main open questions of this conjecture is what is this exponential rate because in order to give precise phenomenological implications we need to say exactly what the exponential rate is and in recent years we have made a lot of progress understanding what is this exponential rate and there are lower bounds for it depending on the level of supersymmetry and the dimension of the space time okay so this is the distance conjecture so we have this double of states and therefore the cutoff of the theory breaks down exponential okay then we have the weak gravity conjecture which refers to the gauge theory we will have a gauge coupling and so the formulation what we call the electric formulation of the weak gravity conjecture is that you must have some electrically charged state you have a gauge theory with a charge to mass ratio that is bigger than the extreme charge to mass ratio of a black hole okay so we will see that this is motivated by black hole physics but it has the two motivations one is a black hole physics and the other is precisely this tower of states that are charged because in all the examples that we see when we take a weak coupling limit it is not only that we have one state but we have this tower of sub lattice of states of super-extremal states that are precisely equal to the gauge coupling when they are super-symmetric so in the examples I gave you before they were VPS states and that's why the mass was like the gauge coupling but in general could be that the mass is smaller than the gauge coupling smaller or equal and the charge and the charge is proportional to the gauge coupling okay so the idea is that the weak gravity conjecture is a bit milder in the sense that it just requires in principle some state that is super-extrema the weak coupling limit when the gauge coupling is small becomes stronger and implies a tower or a sub lattice of states becoming light so the EFT indeed breaks down in terms of the gauge coupling okay so the cathode has to be smaller or equal than the gauge coupling okay so I'm gonna show you some examples from string theory like string theory motivation of this and then once we finish with the string theory evidence we'll discuss the black hole motivation Erin can I ask so this Q over M of an extremo black hole right you only trust this until the plank mass so do you or do you have to make some kind of assumption of the corresponding state to an extremo black hole as you go below the plank mass an assumption of whether the state is below the plank mass or no I mean so suppose I mean you have this inequality in terms of the charged mass ratio of an extremo black hole as derived by Einstein Maxwell theory but you only trust this when the mass of the object is bigger than the plank mass you mean the black hole like this extremo value yes but the idea is that okay finish you know maybe below the plank scale the corresponding state which is an extremo black hole maybe that corresponding state as you go below the plank scale has quite a different charge to mass ratio we don't know yeah okay so when I say the streamer black hole I mean let me put it here it's a black hole with a very large charge and very large mass so this is a concrete order one factor that is fixed because it's the the charge to mass ratio of a very large black hole yeah so that's that approach is some particular value and this is the value that enters in the conjecture so what you require is that you have particles with a charge to mass ratio bigger than that value now it's true that small black holes can receive corrections and can have a charge to mass ratio that is bigger or smaller than this value and people are studying I mean we can maybe I will mention it tomorrow but people have studied these corrections over small black holes with the hope or the goal of showing that the corrections by if you want to preserve genitality causality and so on are always such that the charge to mass ratio of small black holes is bigger becomes bigger than the one that is very large black hole so that small black holes themselves are already satisfying the wood gravity conjecture so it's a way to prove a very mild version of the wood gravity conjecture in which small black holes are the states that are satisfying the conjecture it's a very mild version which wouldn't be useful for phenomenology because it's just about small black holes and the expectation is that maybe when the gauge cappings are long and large and you are in the middle of the moderate space maybe it's just about small black holes that are satisfying the wood gravity but then when you take a wake up limit you really have a tower of steel that is becoming light so they are not black holes I mean they are becoming massless so it's I mean what we observe from a street view is that there is something stronger than just black holes satisfying the wood gravity conjecture that is realized but I guess what I mean is particularly surprising behaviour of the charge to mass ratio below the plank scale because as you say people compute these corrections for small black holes but obviously they can't go below the plank scale now suppose the state below the plank scale you know has an almost vanishing mass and yet can carry a lot of charge and that state can exist we just can't calculate it how does that change the story in any way? I mean the only thing that happens is that if you have a small black hole with a very large charge to mass ratio then large black holes can decay to this small black hole so if you just want to require I mean the motivation for the wood gravity is just that the black holes can decay as we see then small black holes are already enough to guarantee that large black holes can decay but you still need to guarantee that these small black holes also can decay and in the end you need to have also some particle where the small black holes can decay so it doesn't really change much it doesn't really matter I mean in the end I think you're going to need some state could be very heavy but some state to which all the black holes can decay okay let me think and sharpen okay yeah I mean we can discuss it again the motivation okay yes so there was another one that I didn't see before I don't know one way I can reply it anyway so there was some question I didn't see which is what is the problem with tower of light states the problem is that the Planck scale I mean since they are coupled to gravity the Planck scale gets normalized so the quantum gravity cutoff to zero like gravity becomes so you can if you have infinitely many states with regard to instant gravity instant gravity completely breaks down like the the quantum of this quantum gravity is like the Planck scale gets normalized and the proper quantum gravity scale goes to zero completely so the problem is that the EFT with regard to instant gravity breaks down I mean quantum gravity sound effects become important okay and the question now is will gravity can be or other concepts such as wormholes or white holes or black brain with this structure yes so a black brain yeah we'll see that this is this is can be generalized and will be generalized to any black brain like any P-formed it's happening so we require that we have a P-brain states that are electrically charged and whose charge is bigger than the tension of the object okay so it works the same which can compute it it even works for actions and that's why people use it for implication for inflation like the action is like a zero form it's filled and therefore you need to have instant that are electrically charged and has a an action that is smaller than the charge which is one of the big constant and wormholes or white holes are kind of related to these actions and because you could say that the wormholes is like the equivalent of the black hole for the particle so it's like the fact that wormholes can break because you have instant this kind of the equivalent notion that black holes can decay into particles um and white holes have never seen anything like that okay okay so in the coming time which is not much I think I'm just gonna outline some of the other examples in stream theory okay because I don't want you to get the impression that this is just about like 10 dimensional stream theory and perturbative and the compactification limits like why we are so confident about these conjectures is also because even when going to very complicated compactifications and very weird limits and give general proofs that we get these towers of states so I want to outline I mean briefly what these examples are and tomorrow we will start so today is about stream theory tomorrow we will start with the black hole motivation and yeah the black hole motivation some of the refinements and the and phenomenological implications and in the remaining time we can discuss a little bit other conjectures and the scalar potential okay so let me just give you some more examples so first of all you could say okay here you are just taking like radius and compactifications very easy things let's say something more complicated let's take a calabio 3 fold and let's move in a direction that has nothing to do with the compactification or calusa crane towers and so on so for example in the complex structure of a calabio 3 fold you have it's a product of the hyper multiplies and vector multiplies so this is a 4D and equals 2 theory and we can move in the complex structure modular space vector multiplies and what we can prove there in complete generality is that the tower of bps states become light and the infinite distance singularities of this modular space and this I mean they are serving as a guide to uncover some new mathematical structures that are behind the compactifications that are allowing these conjectures to hold in a very non-trivial way so there is now a lot of connection with mathematics of the compactification that is coming from studying these conjectures so let me just outline how this is satisfied so what happens is that at this infinite distance singularities there is a monodermy transformation for the current values of the standard model theory we have what is the estimate of the cut-off if we assume coupling to gravity yeah so you can take the electron that is charged and the gauge coupling of the photon the point is that it's quite large in Planck units so it's only two orders of magnitude three orders of magnitude below the Planck scale so actually the cut-off according to the gravity conjecture is very high and that's because the electron is very super extreme like the mass is much much smaller than the gauge coupling in Planck units so for this I mean the standard model satisfies the conjectures by a lot let's say okay now when we try to add now beyond the standard model proposal it becomes more interesting because typically we would very quickly couple things or like things for that matter and then everything becomes much more constrained okay yeah so let me outline this so what happens at this at the singularities of the modernized space is that you have some monodermy transformation which is the so this scalars so here we have some complex scalars this is a complex manifold so we have some let me call it SI plus AI so we have some non-periodic scalars some action so the symmetry of the actions are these monodermy transformations okay and this monodermy transformation can be of infinite order or finite order so infinite order means that there is no integer such that by going around the singularity you come back to the same situation okay so this is what means infinite order and what we could show using some Milpot and Orbit theorem of algebraic geometry is that the infinite distance singularities always are required to have an infinite order monodermy and then by using this theorem this monodermy can be used to populate an infinite tower of VPS states because the charges transform under this monodermy so since it's of infinite order we populate an infinite tower and using I mean I don't have time to explain but using these theorems you can show that in general for any Calabillo the tower of VPS states will behave exponentially in the field distance where this alpha is bounded by this monodermy by some properties of this monodermy transformations so that in a Calabillo 34 it's always bigger than 1 over the square root of 6 okay so this is just an example of a more complicated manifold in which one can show in general that you always have these towers of states and the last one and I finish let me just say it very quickly you could also consider like you say okay but this is just about closed strings what about open string gauge fields like can I somehow engineer situations so here with the gravity right we want that the mass is smaller equal than the gauge coupling in blank units can we engineer a situation with open strings such that the gauge coupling is fixed but the mass becomes very large so that I will violate this conjecture so that they never become masses and you could think of naive situations in which you could try to do it like you have for example DC brains some compact manifold and the gauge coupling of these DC brains is given by the square root of Gs so if you keep Gs fixed the gauge coupling will be fixed but the mass will have open strings in between and the mass is given by the distance between the brains so you could say okay so let's move the brains very far away so that the mass becomes very large and we'll violate this I don't have towers in the state becoming light and they are not proportionate but they are bigger than the gauge coupling the problem is that if you try to make this very big very large since the space is compact because we want to keep the plank mass finite at some point I mean you cannot make this as large as you want and precisely you can see that the mass when you try to make the mass very large like by making this distance large the plank mass also goes to infinity in such a way that M square over the gauge coupling over the plank mass in plank units this is always smaller than one so there is no way to engineer this in a compact space and also even considering that there is a lot of work we put the reference of Lee, Lerche and Bayesian in which they try to engineer situations in which you keep the overall volume fixed because you make some cycle very large and another cycle very small and try to engineer situations that would violate these connections but then when you do that there is still so when you try to make some cycle very large the gauge coupling goes to zero but then there is another and the plank mass remains fixed this is another option then there is another cycle that goes to zero to make this possible and then there are towers of states associated to these other cycles so again the gravity conjecture and the distance conjecture is always satisfied so you can find these examples in more detail in my lectures I will do it but the upshot is that somehow string theory is always finding a way to satisfy this and the next goal is try to understand why this is a general feature of quantum gravity I mean why it should happen this is just string theory examples but why this should happen in complete generality and this is what we will try to discuss tomorrow also thinking of black hole physics sorry that I run out a bit of time and let's stop here and see if there are more questions let us thank Reem for the lecture as always to stop and interrupt me and ask questions feel free to enter a question in the chat window or just pipe up, unmute yourself and let me know if anything I say is unclear or if you have any comments or questions, anything like that so in my lecture today and tomorrow what I really want to do is flesh out two main examples of averaged holographic dualities and we've spent a lecture and a half or so motivating this setting up some groundwork and what I'd really like to do today is just dive into the detail so the first model that I want to discuss is JT gravity which is a two-dimensional theory of gravity and the action for this theory is quite simple so it involves a metric and a standard Einstein-Hilbert term in two dimensions the coefficient of the Einstein-Hilbert term is Newton's constant but it's called S0 for historical reasons generally speaking it's because JT gravity arises as a dimensional reduction from some higher dimensional theory down to ADS2 and if you have some for example ADS2 cross S2 near horizon geometry of some black hole that coefficient S0 would represent the entropy of that black hole or be related to the entropy of that black hole there's a second very important term in the action involving a scalar field phi and then there are various boundary terms that are important but about which I won't say too much since at this point I'm trying to just give you a slightly more general overview but the basic idea is that the equation of motion for phi essentially acts like a Lagrange multiplier to set the curvature of our two-dimensional surface to be negative and constant so what that means is that this is a theory of constant negative curvature surfaces so for example if we wanted to compute some partition function of the theory so here we're talking about a theory that's defined in Euclidean signature and you could imagine computing for example the partition function of the theory where our boundary is a circle with some length that I have called beta because of course the circle will be interpreted as a Euclidean time circle and the length would be interpreted as the corresponding inverse temperature then the way that we would compute this is by summing overall constant negative curvature surfaces whose boundary is a circle of length beta so for example one contribution would be a disc the next contribution would be a disc with a handle glued in and so on and so forth and the point is that the higher genus terms in this expansion are suppressed by factors of that Einstein action e to the s not root g r appearing in the JT action gravity and because we're studying a constant negative curvature geometry root g r is just given by the Gauss-Benet theorem so it's just the Euler character of the two manifold and so these will be suppressed by factors of e to the minus s not times the Euler character so when we compute gravitational path integral and think about a genus expansion then the higher terms in the genus expansion are going to be suppressed by these factors of e to the s not times the Euler character so for example the disc has Euler character minus one so this would go like e to the s not every time you glue in a handle every time you increase the genus the Euler character drops by two and you get an e to the sorry there an e to the minus s not for this term here and so on and so forth similarly if you were to compute more complicated exertibles say for example if you were to compute the path integral with a pair of circle boundaries then the leading term would involve a cylinder and then you would have sub leading terms so for example a cylinder with a handle glue down and again these would be suppressed by factors of e to the minus s times the Euler character so this would go like this term this first term would be order one because the Euler character is zero this term would be down by a factor of e to the minus two s not and so on and so forth and the basic idea of the correspondence between JT gravity and some sort of matrix integral is the usual idea that we always have in the back of our head when we're thinking about callography which is that we think of the genus expansion as the large n expansion in a gauge theory but of course here we're talking about a two-dimensional theory of gravity so that should correspond to some kind of one-dimensional gauge theory that is to say just as some kind of matrix integral or some kind of matrix model model. Can I ask a question? Please. I'm just confused by the definition of JT gravity. You said that it's a theory of constant negative curvature surfaces right? But if you have a constant negative curvature surface you need a genus to be greater than zero right? So if you have no boundary but remember for example the disk has a boundary and so you'll get an e to the s not because the genus is minus one. Yeah I mean so for example I wrote down the hyperbolic metric on the disk and that would be an example of a contribution to this path integral. You know there's some bells and whistles that I'm suppressing here so for example the boundary terms are very important and in fact a lot of the non-trivial work that goes in here involves carefully keeping track of these boundary terms so for example you might think that you know the constant negative curvature metric on the disk how complicated could that be but there actually is a little bit of work that one has to do in calculating this disk amplitude and all of that work comes into the fact that you have to pull this boundary into finite distance and imagine all possible boundaries of specified length that are embedded in this disk and then integrate over these possible boundary wiggles, these possible boundary in the disk and that's where the sort of non-trivial calculation of for example the disk amplitude or the cylinder amplitude comes in in this JT gravity theory it's not a terribly difficult calculation so for example it was considered by Stanford and Witten where it was shown to be one loop exact one can compute it exactly but nevertheless there's a little bit of a calculation there for the sake of time and to keep things relatively simple I'm not going to focus too much on the details of that boundary calculation but the main punchline is that this theory of gravity is a 2D theory of gravity it has a simple genus expansion and we're going to try and interpret that genus expansion in terms of some kind of matrix integral in the usual way that in higher dimensions you might interpret a genus expansion in terms of some large gen expansion in a gauge theory sorry Alex small diffusion so when you say that the boundary theory is a matrix model so wouldn't that be a zero dimensional gauge theory let me tell you exactly what I mean let me write it down and then you can then you can you can tell me exactly what your question is but it's really a matrix integral that's perhaps what I should say better sorry Alex because I just quickly ask you a question that can clarify something following up on the previous question so here you're in two dimensions and also you're in Euclidean signature so you're considering this pattern to grow over a pattern to grow over a matrix which match onto some boundary because you're in two dimensions that you don't have to worry about so first of all like you were saying in the previous lecture are you integrating over real matrix here and is it because you're in two dimensions that you don't have any problems with negative modes and I'm integrating over real matrix let me talk in a little bit more detail about how exactly I'm going to think about that let me proceed and I'll say a few more words about how we think about that integral over matrix so strictly speaking you would have to worry about some very complicated gauge fixing procedure for example in the paper of Saad Shankar in Stanford that's gone through in some amount of detail but actually I think there's a very simple and intuitive way of thinking about that integral over the space of matrix which we'll get to in just a second so maybe I'll proceed and you guys can then repeat your questions in a few minutes if something is still unclear but really the question is what kind of matrix integral should we be talking about here and the idea is that because we're studying a one-dimensional CFT that is to say a one-dimensional quantum mechanical system I want to think about this as an integral over possible one-dimensional Hamiltonians and a Hamiltonian of some quantum system is just some N by N Hermitian matrix and the this expansion is supposed to be the one over N expansion of some Hermitian integral over Hermitian matrices these Hermitian matrices being interpreted as the Hamiltonians of some one-dimensional of some quantum system so in particular the statement of the duality statement that I'm not going to prove to you but I will sort of indicate to you how we should think about its proof that the observable that we would compute is the JT gravity path integral so this would be loosely speaking an integral over this space of constant negative curvature matrix weighted by an appropriate Euclidean action and you want to imagine that we're integrating over manifolds M two-dimensional manifolds whose boundary is the disjoint union of a bunch of circles of lengths beta one all the way up to beta M so they're going to be an integral over a space of manifolds where you have N boundary circles and then this turns out to be equal well I should really put a squiggly line here equal in a certain sense to a matrix integral so here this is going to be an integral over N by N Hermitian matrices times some measure that is interpreted as a probability density on the space of matrices and then this is going to be a function of those lengths that I called that I called beta and in particular it's going to be the integral with this probability measure of N finite temperature partition functions of this matrix integral okay so when we say that we have an averaged holographic duality this is exactly what we mean that there is an equality up to that little squiggly sign that I'll explain in a second between a gravitational path integral and an average over Hamiltonians of quantum mechanical systems for some Hilbert space of size M and the length of the boundary circle is literally interpreted as the temperature of this quantum mechanical system so here we are literally computing the expectation value of some endpoint function of the partition function of a quantum mechanical theory and here this angular bracket denotes the expectation value with a particular probability measure so this is the statement of the JT gravity matrix integral duality a few comments are in order the first is I should tell you exactly what I mean by this squiggly sign so the squiggly equals means that these two expressions on the left and the right hand side of this integral agree at all orders in 1 over N which I remind you was e to the s0 on the gravitational side where s0 was basically 1 over Newton's constant so this is an agreement at all orders in perturbation theory in the matrix model but an agreement that is true at all orders non-perturbatively that is to say what we would call in an instanton expansion on the gravitational side they do not agree at finite N in part because I don't even know how to define the left hand side of this expression at finite N and so if anything we might use a matrix integral of this sort as a definition of JT gravity at finite N and that would involve including what we would think of as doubly non-perturbative effects that are non-perturbative on the matrix model side like e to the minus N but would be doubly non-perturbative they would go like e to the e to the 1 over Newton's constant on the gravitational side and I don't want to go into too much detail about how this is derived but let me tell you a little about why this works so why does this work? sorry quick question you said s0 is can be associated to some entropy is that why it's exponential is always 1 over an integer so how you know y is e to the s0 is 1 over an integer this isn't really I mean this is no I mean we're working in a large N expansion where the fact that N is an integer is really invisible so I think it would be a little unfair to think of our gravitational theory at least as defined in the way that we have done it here as knowing anything about the integrality of an animal I think that would probably be stretching it a little too far but should that be to the minus s0? yes I'm sorry I was a little sloppy about that thanks good thank you any other questions? well let me I mean I think as a way of kind of addressing some of the questions that came up I just want to indicate very briefly about why this works so why does this work? what is an integral over a space of constant negative curvature matrix so let's imagine that we're trying to do this integral over matrix on a manifold of some specified topology so it turns out and this is a famous fact in mathematics that if you have some constant negative curvature surface so it could either be a sorry that my window keeps resizing there it could either be a compact surface or non-compact surface so let's imagine that we have some non-compact surface so here I've drawn a contribution to what I would call z of s1 so it turns out that there is a finite dimensional space of negative curvature matrix on this manifold and the integral over the space of such matrix is going to be the volume of the modular space of these constant negative curvature matrix on this manifold so in particular one way of characterizing modular space of constant negative curvature matrix is and here maybe I'll make my life a little more interesting to draw a more complicated case so let's draw something with two handles so the basic geometric fact is that you can always think of a constant negative curvature manifold as being a sum of a bunch of pairs of pants here what we imagine doing is representing the surface as a bunch of pairs of pants glued together and there is a unique constant negative curvature metric on a pair of pants provided you specify the lengths of the geodesics at the cuffs of this pair of pants and so because of that there is a finite dimensional space of such matrix so the coordinate system on the space of such matrix would be given by a length parameter for each one of these cuffs so here for example I've drawn 1, 2, 3, 4, 5 different cuffs those are those little circles and the coordinates on the modular space of surfaces would be a length parameter for each cuff along with an angular parameter because whenever you have two cuffs you can glue them together with some kind of relative twist so the cuffs of this space would be 5 complex dimensional or 10 dimensional space and so there would be some complicated 10 dimensional integral that you would need to do in order to compute this modular space volume now in principle such modular space volumes are quite complicated to compute and indeed there are no known closed form expressions but it turns out that although they have not been computed exactly these modular spaces obey various recursion relations and in fact there are various different recursion relations that have been discussed in the literature many of which were motivated by some of the relationships between these and matrix integrals but for our purposes the relevant recursion relations were written down by Mirza-Khani about 15 years ago and it turns out that these recursion relations turn out to be equal to the equations that are obeyed by the partition functions of matrix integral of matrix integrals and so depending on whether you're a mathematician or a physicist you have different names for the recursion relations obeyed by matrix integrals if you're a mathematician you call these loop integrals loop equations if you're a physicist you call them Schringer-Dyson equations or if you're a normal person who's neither a physicist nor a mathematician you call them integration by parts that entities of some rather complicated matrix integral and the basic observation then is that because these modular space volumes that appear in the computation of these gravity path integrals obey the same recursion relations as those in a matrix integral and one can also check that the starting point of this recursion relation which happens to be this disc amplitude and cylinder amplitude that I wrote down above obey the same starting point as the elements of these recursion relations that is what allows us to sort of prove this duality between great JT gravity and a matrix integral one comment that I should make is that I have not written down the probability density over the space of Hermitian matrices and the reason I have it written down is that there's no simple closed form analytic expression for it so p of H this probability density is rather complicated to write down and it is usually discussed as being defined only implicitly through what is known as the spectral curve of this matrix integral so one could study it perturbatively we could plot it for you in certain approximations but it's not something that is easy to write down it is not for example just a Gaussian distribution over the space of Hermitian matrices but rather it's something more complicated so hopefully I'm not sure this I think schematically answers some of the questions that were asked earlier there are a lot of details that go into this calculation that I'm glossing over here so for example one thing that is not at all clear I think from what I have said so far is what the induced metric on the modular space of constant negative curvature surfaces is with respect to which one has to compute this volume so for example when mathematicians talk about volumes of modular spaces they usually have a particular volume element or symplectic structure on modular space that they're thinking of that's the vae Peterson modular volume element one calculation that one has to go through and check is that when we talk about the integral over constant negative curvature metrics on this space that the induced volume element on this space of metrics is the same one that the mathematicians know and love and that's something that one can check but it's a little bit non-trivial because when we think about doing an integral over a space of metrics we're really thinking about some infinite dimensional integral when we talk about the modular space of constant negative curvature metrics that has been turned into a finite dimensional integral and so you need to be rather careful when you figure out for example the the Jacobian factors that arise when you zoom down to this finite dimensional integral fortunately it all sort of works exactly as you would expect you know one way of proving this is to rewrite JT gravity not as I have done above where I wrote it in terms of metric and a scalar field the dilaton instead it turns out one can alternatively formulate it in a language that's a bit closer to a gauge theory so it's a BF theory and then one definition of this V. Peterson symplectic structure is that it's the natural thing that you get from BF theory or from Tren Simon's theory if you're working in one higher dimension so I don't think it's necessarily super instructive to go through that in detail but I thought I would just mention it so this I think is I think I didn't just follow but so you say this holds this Z second compute on both sides if you go up a little bit to all orders in N but then at the same time you also said that this probability distribution you only know in certain regimes because it's very difficult to write down well we know the probability distribution is typically I mean there is an exact probability distribution but it is not written down, it's typically not presented as a function of the Hamiltonian but rather it is defined implicitly through a different object that is known as the spectral curve which is a gadget it's a gadget that you can use to compute these matrix integrals recursively in a one over N expansion so I don't want to if you don't know what a spectral curve is don't worry because it's not like you know it's just some technical tool that we use when we calculate large N matrix expansions of this type good but basically it's a different way of encoding this probability distribution that is particularly well adapted to the one over N expansion I think that's probably the best way of thinking about it essentially the spectral curve encodes the same data as that disk amplitude great, are there any other questions? Yes Alex can I make your question? Please Do you hear me correctly? Yes, yes, yes, please So maybe this related with the topic you wanted to discuss in the previous lecture but from the gravity point of view this object that you have defined on the left hand side JT of several temperatures is it something from the gravitational point of view is it natural to propose and a connected question related to that is the following Can I, you mentioned also in the previous lecture that when you have these canonical examples of ADS 5 versus 5 and you have sort of like P of H, like the distribution becomes like a delta or something that is in a strict but can I now grab for example several copies of this N equal 4 super young mills let's say in spheres I put several spheres and then I like induce this ensemble I put several copies try to do like a gravitational computation and these copies become entangled or connected or something So first of all I think we should be careful about the word entangled here entangled is a word that we talk about in Lorentzian signature when you're talking about states in Hilbert space So if you have a Lorentzian wormhole, that is to say a black hole then in the standard ADS CFD picture of an eternal black hole that is represented as an entangled state between two states in Hilbert space Here what we're talking about is something very different where we have Euclidean wormholes which lead to a probability density on the space of theories not an entangled state in Hilbert space just to make sure that we're all on the same page this is not entanglement this is a probability distribution on the space of theories here the probability distribution is that function I called p of h and so the space of theories is literally the space of quantum mechanical Hamiltonians represented as N by N Hermitian matrices that's a space of Hamiltonians that we're literally integrating over and then if I hand you a Hamiltonian what is the observable what is the only thing that you could compute well a Hermitian matrix is up to unitary equivalence completely characterized by its spectrum so the only thing that you could compute the observables that extract all of the data about this probability distribution unitary invariant data of this probability distribution are just trace of e to the minus beta h along with its various correlation functions the things that I called expectation value of z of beta 1 of z of beta n z of beta is literally trace of e to the minus beta h for this probability distribution and then your question was really is this the natural thing to compute on the other side and I would say that if you think about you know how we defined a theory of gravity in ad s is that it is a theory where we integrate overall metrics in the interior with some specified boundary conditions out at infinity and so the only thing that you can compute in a gravitational theory is a partition function that is a function of those boundary conditions out at infinity with one-dimensional boundaries the only boundaries that you can have are circles and the only data that you could ask about is the length of that circle so this z of beta 1 up to beta n from a gravitational point of view is kind of the only observable or it is the sort of you know maximal observable that you could imagine asking for in a theory of gravity that includes only a metric view where gravity had more interesting degrees of freedom in the bulk let's say that you're integrating over other local fields as well you could introduce sources for those fields on the boundary then you would have more interesting kinds of observables to compute but this is a very simple theory of gravity where the object that I have written here is the only thing to compute and there is it's the only thing to compute on the gravity side and it's the only thing to compute on the matrix integral side so my question was a question about what happened how should we think about this in comparison to theories when we only have a single Hamiltonian and I think that's a very interesting question so one question that you could ask is what would happen if we tried to change this probability distribution so that instead of having some probability distribution supported everywhere all over the space of Hermitian matrices we had just a delta function that's a complicated question some things can be said about it but roughly speaking the idea is that as you go away from this probability distribution the gravity theory starts looking less and less like a simple gravity theory whose path integral we can compute exactly as we have done in this example and it starts looking like something more mysterious from the bulk point of view and we'll see this again for example you asked about type 2 string theory so for example you could ask about what is the partition function of type 2B string theory where I have multiple boundaries in Euclidean signature and the answer is that if type 2B string on ADS5 is dual to exactly n equals 4 yang-mills then it must factorize exactly the issue however is that we have an independent bulk calculation of that in type 2 string theory we certainly don't know of any Euclidean saddle points that connect two boundaries that are disconnected say S4s or something like that in Euclidean signature so there is certainly no saddle point contribution but that is not to say there couldn't be other non-saddle point contributions to the path integral but that is a very important lesson of this JT gravity theory because one of the things that is important to note I didn't dwell on it here is that this is a full integral over spaces of metrics not just those that are solutions to the equations of motion I didn't really write down the theory in full gory detail but it turns out that the geometries that I'm integrating over here are not just solutions to the equations of motion but they are an integral over the full space of metrics amazing answer thank you very much it's an amazing question good great I have a question since on the JT side things are defined only in the 1 over N expansion it's just one level of non-properity effects does that mean that the probability distribution that would make this identity work is really only defined in the 1 over N expansion I think that's exactly right I think that's exactly the correct point of view the spectral curve defines this probability density in a 1 over N expansion and so if you really wanted to define this at finite values of N you would have to define a probability distribution as a function of N that goes over to this thing at infinite N and my understanding is that there is not a unique choice for such a thing and so that there could be many potential UV completions of JT gravity in that sense another good question maybe I'll answer one more sentence which is that in the next example that I'll consider there is at least a plausible more or less unique choice of probability distribution at finite we can argue about how unique it is but there's a natural guess yes the other question is so in ADS CFT usually on the frequency side we have a CFT so here is there some constraint on this probability distribution in order to have a CFT in some sense this is really so one famous fact about so are these individual Hamiltonians appearing here Hamiltonians of conformal field theories and the answer is that they can't be because what is a conformal quantum mechanical system so a conformal quantum mechanics system is one that has a Hamiltonian that sits inside SL2R and a straight exercise a great exercise for anyone who hasn't done it is to show that any such Hamiltonian has a continuous spectrum so it's impossible to have a Hamiltonian that sits inside an SL2R that has a discrete spectrum so what that means is that you shouldn't think about these Hamiltonians actually as individual CFTs so for that reason this JT gravity matrix integral duality fits in not in standard ADS CFT so people sometimes call nearly ADS2 nearly CFT correspondence great question it's related to the fact that the precursor to all of this JT gravity stuff was the SYK story where it was noticed that a particular disorder quantum mechanical system in one dimension the SYK model which is a theory of four fermion interactions with some disordered coupling constants pulled from an ensemble this is not a conformally invariant theory but it is a conformally invariant theory in the infrared so it looks like something that could be dual to an ADS theory of gravity of low energies but not exactly in fact it's because of these issues that we try to consider other examples which are maybe a little closer to the spirit of the ADS CFT correspondence so the second example that I'd like to talk about maybe for the rest of the time today is a different example where we average over a space of CFTs and in particular if you wanted to try and construct a version of this average ADS CFT duality where your theory of gravity is a little bit richer than just gravity in two dimensions you might want to try an average over two-dimensional CFTs so that you have a three-dimensional theory of gravity and there are various reasons why this might be interesting so one reason is that theories of gravity in two-dimensional ADS are extraordinarily simple and there's a sense in which they don't contain black holes I mean there are ways of thinking about ADS to itself as a black hole say in Rindler coordinates but in three-dimensional ADS gravity there are genuine black hole solutions there is an ADS Schwarzschild black hole whose entropy you could hope to count and whose degrees of freedom you could hope to explain so a richer example might be an example in higher dimensions and two-dimensional CFTs and three-dimensional theories of gravity seem like a great place to try this one reason being that two-dimensional CFTs have an infinite symmetry algebra the algebra of Vero Zorro symmetries and so they're much easier to study than CFTs in even higher numbers of dimensions so a different and sort of more ambitious goal might be to try and formulate a version of this average ADS CFT duality where we integrate over a space of two-dimensional CFTs so to begin let's ask how we would do this so what would we need in order to average over a space of two-dimensional CFTs so to begin let's try and remember what is the data that defines a 2D CFT so a CFT is defined by a list of operators and in particular a list of primary operators along with some data so in particular you need to specify the spectrum of the theory so their dimensions and spins along with a set of coupling constants and the way that we characterize coupling constants in conformal field theories is in terms of operator-product expansion coefficients or 3-point coefficients so here we would call these Cijk and roughly speaking they're the 3-point functions of these operators which tell you how if you take 01 and 02 and expand it in a basis of operators 03 what those expansion coefficients are and so this set of data is the set of data that defines a 2-dimensional conformal field theory or indeed a conformal field theory in higher dimensions although the spins would not be numbers there but instead representations in any case if we wanted to define an average theory of gravity by averaging over a space of CFTs then we would really need three things the first thing that you would need to do is determine the possible values of this data delta i, ji and Cijk then you would need to find some probability distribution on this space of data and finally you need to compute averages and compare to some theory of gravity so these would be the three steps in order to define an ensemble of CFTs and this really generalizes the notion of a matrix ensemble now if you have a matrix like the Hermitian matrices we were discussing in JT gravity a Hermitian matrix is determined entirely by its spectrum and so we don't have to worry about these three-point function coefficients essentially when you integrate over when you determine a Hamiltonian you're specifying some set of eigenvalues and then you would just need to define some probability distribution on the space of eigenvalues so for Hermitian matrix a matrix integral step one is done it's just a set of eigenvalues but for conformal field theories of course even step one in this process is very very complicated and in particular step one has a name it's the conformal bootstrap problem so even step one in this procedure is a very difficult thing to carry out you would be rich and famous to the extent that physicists get rich and famous if you could solve step one of this procedure so even at the beginning it seems like this is a very difficult thing to try and carry out and so in order to make our lives easier and to have a hope of solving this what I'm going to do is what we always do as physicists when we're confronted with a problem that we can't solve what I'm going to do is I'm going to add symmetry to the problem until we can solve it and in particular what I would like to do is add as much symmetry as I can to the problem while still keeping this problem non-trivial and it turns out that when we do so we're going to be able to carry out all three of these steps explicitly so we'll be able to solve the bootstrap problem well we won't solve the bootstrap problem we'll cite people who solved it many years ago we'll define a probability distribution on the space of conformal field theories and then it turns out that we'll be able to compute averages and just as in the matrix integrals described above the most general average that you could compute was an average of trace of e to the minus beta h to some power we'll be able to compute the average of an arbitrary observable in this theory and we'll see that it does have simple interpretation in terms of ADS gravity so in particular what we will do is we're going to consider CFTs with some central charge that I'll take to be an integer that I'll call D and with as much symmetry as possible consistent with that central charge now in a two-dimensional theory the way that we characterize symmetry a chiral algebra the reason being that the conservation law for a current in two-dimensions D mu J mu is equal to zero is just another, you can always write that in complex coordinates as D bar J sub z is equal to zero that means J sub z is holomorphic and J bar is anti-holomorphic so we characterize symmetry algebras in two dimensions in terms of an algebra of chiral operators and anti-chiral operators so in particular the biggest algebra that I could consider that is consistent with this central charge is just to have a bunch of U1 currents so a bunch of dimension 1 chiral operators and it turns out so that there is a finite dimensional space of such theories so it's dimension D squared with a let us say canonical or natural probability distribution and finally it turns out that there is a beautiful mathematical tool that one can use to compute integrals over the space of theories with this probability distribution and again this is not something that we physicists invented this was known to mathematicians some time ago where it was quite popular in number theory and it goes under the name of the Siebel Bay formula so maybe though rather than trying to dive into any technical details right now I should pause and see if there are any questions and in fact I believe if this was a 50 minute lecture let's stop here but of course I'm very happy to answer questions or ask questions to anyone who has any thank you so let's thank Alex for the lecture thank you now we can stop recording and go to the discussion session I'll learn sorry we just wait a little bit ok