 Now we will have the second lecture from Katsuya Yonekura from Tohoku University, please, Katsuya. Thank you very much. So yesterday I talked about the difference between SU2 and SO3 and in particular, so an important fact is that this SO3 has non-trivial first homotopy group, which is given by Z2. So, please remember this fact. And also, I wanted to talk about the difference between SO3 and SO3. So let me quickly explain this difference. And for my purpose, the important fact is that we can construct new topologically non-trivial bundles if we use SO3, sorry, say, O3. So this can be seen by just a simple example. So let's consider S1, just a circle. Then this S1 can be constructed by gluing two ends of an interval. So we consider some interval, let's say from zero to one. And then we can glue these two ends and then we get S1. So this is a very simple construction of S1. And then let's consider fiber bundle over this S1 or more explicitly, let's consider vector bundle. So I take some vector space. This vector space can be just considered as R3, in this case. So let's consider some vector space on which is SU, sorry, O3-X. Then we can construct a non-trivial fiber bundle in the following way. So first we consider this interval times this vector space. So this is just a trivial vector bundle. This is just a direct product of this interval and this vector space. And then we take two points, sorry, we take two fibers. So one fiber is at the point zero and I take the element of the vector space here at V zero and I also take the point one in this interval and I denote the vector here as V1. Then we want to glue these two fibers and then in this gluing, so to construct a trivial bundle, we can just set the V1 equal V zero. Then that is a trivial vector bundle but we can twist the boundary condition. So in this identification between V1 and V zero, we can put some group element. So I forgot to write that this G is an element of O3. So we can use a group element as a transition function in this construction of the fiber bundle. And in this way, we get some bundle and this bundle is pathologically non-trivial if the determinant of this group is minus one. Then we get topologically non-trivial bundle on S1. So this is like a maybe it's a strip. So it's not essential that this is SO3. We can just consider, sorry, it's not essential that this is O3. We can just consider O1. Then in the case of O1, this V is just a one-dimensional vector space. And then we can use as a transition function just it's minus one. So this is just the construction of maybe it's a strip. And maybe it's a strip is topologically non-trivial. Yeah, you can just construct maybe it's a strip by papers. And then you can see that it's topologically non-trivial. So we can construct a bundle in this way and this G, this group element gives an element of pi zero. So this O3 has two connected components. One component is so the time. So this O3 has two components with determinant plus one and determinant minus one. And then so because it has two connected components, the pi zero of this O3 is given by Z2. So the homotopy group of this O3 is again non-trivial and I will use this fact later in my discussion of global anomalies. Okay. So the important point was that the homotopy groups of O3 and also the first homotopy group of SO3 non-trivial. Then by using these facts, now I want to explain some examples of global anomalies. But here I take the traditional approach to study global anomalies. I will discuss more systematic approach later in my lecture, but here I just try to study the path integral measure. And I will argue that the path integral measure has some ambiguity and that ambiguity is the anomaly. So I want to discuss anomalies which are related to pi zero of On and pi one of SOn. So I have talked about O3 and SO3, but that is not essential. We can consider any N, so we can consider On and SOn. Explicitly, I consider one dimensional system. So one dimension means that I just consider quantum mechanics. So I consider one dimensional, maybe I should say one plus zero dimensional. So I consider one plus zero dimensional Majorana fermion. So Majorana means that I take these fermi fields to be real. So I take N, real grassman fields and the Lagrangian is just keeping like this. So this is a very simple Lagrangian. Here, because I'm considering Majorana fermions, so this is, so usually here comes some pi bar, but because it is Majorana, this is just pi. So let's consider this Lagrangian. This theory has an On symmetry. So On symmetry is just described simply by rotating fermions by some orthogonal matrix. Maybe it's better to write it as G. So this G is just a element of On. So at the classical level, this Lagrangian has obviously this symmetry described by this orthogonal group. So we want to study partition functions of this theory. And before going to the partition function, which gives some anomaly, first let me mention the most basic partition function. So the most basic partition function is the thermal partition function. So this is just a quantum mechanical system. So we take Hamiltonian of this system. Actually, the Hamiltonian of this system is actually zero. But anyway, so if we have a quantum system then we take Hamiltonian and we take inverse temperature and then we take the trace of this exponential minus beta H over the Hilbert space. Then the basic fact is that this is described by the first integral on S1. And here this S1 is constructed by gluing to end of an interval. So I consider an interval from zero to beta. So this beta is inverse temperature. And then I glue zero and beta. Also another well-known fact is that the boundary condition is anti-periodic. So fermions obey thermal boundary conditions which is given by this. So this fermi field as the Euclidean time beta is equal to minus the fermi field at zero. So this is a well-known fact. So I just use this in my discussion. So this is the most basic case of thermal partition function. And then I introduce some background fields for the global symmetry and consider some new partition functions and then see the anomaly. First let me discuss the anomaly which is related to pi zero of O n. So let's take some element of O n. With determinant minus one or more explicitly I can just take this G to be the following diagonal matrix. I can just take it to be diagonal matrix with entries minus one and plus one plus one. So let's take this explicit matrix and then let's consider partition function which is twisted by this element. So because this theory has the O n symmetry this O n symmetry acts on the other side so this O n symmetry acts on the Hilbert space. So we can include this operator which realize this group element and then we can take the place in the Hilbert space. So at least naively we expect that we can do it but it turns out that there is some anomaly in this quantity. So to see the anomaly first let's notice the boundary condition. So it is described by pass integral on S one but now the boundary condition is twisted by the group element. So the first component of the fermion is now so it now satisfies periodic boundary condition so this is now plus. The reason is that from the sum of boundary condition we had this minus sign but from this group element we have another minus sign. So by taking the products of these two we get a plus here and other components just obey this anti periodic sum of boundary condition. We want to perform pass integral with these boundary conditions. So to perform pass integral first we take mode expansion we expand these fields in terms of Fourier modes and for psi one there is zero mode which I denote by this capital A so zero mode is possible with this boundary condition and also there is non-zero mode like this so this tau is the coordinate of S one so this is the Fourier expansion and this is the periodic boundary condition I believe there is an in the exponents and oh thank you yes yeah I must put in this yes thank you I expand other components so these components is j greater than or equal to 2 so this satisfies this anti-periodic boundary condition so they don't have any zero modes so they can be expanded in the following way so Fourier modes described by q, pi, i and minus one half so we get we get half integer Fourier modes so because it is half integer says no zero modes so we can expand the fields in this way in these expansions the important points are the following so first so in this case we can easily notice that non-zero modes appear in pairs so because pairs means so these modes is coefficient b n, c, n they have the same and opposite eigenvalues so this mode b has a class eigenvalue for the derivative operator derivative operator And this part has negative eigenvalue. So the absolute value of eigenvalues are the same in this mode, and their signs are different. Anyway, so they are paired. So non-zero modes are paired. And there is a single zero mode in this setup, so which I have denoted as capital A. So this is a zero mode. So here I don't explain the details, but it is known that this zero mode is topologically stable. So there is some kind of index seven. So this number of zero modes, mode two is a topological invariant, number of zero modes, mode two, in this situation is called mode two index. In this simple situation, you can easily check that this number of zero modes, mode two is really topologically invariant. So I mean that, so I chose this explicit matrix, which is a diagonal matrix, but we could have chosen other matrices with the properties that determinant is minus one. But however, no matter what matrix we use, it turns out that the number of zero modes, mode two is always the same. That is just determined by this determinant of the matrix. So this index, mode two index, is a topological invariant. And does it exist in, yes. I think there is a question from Pedro. Yes, can I ask a question? Yes. Oh, okay, sorry, this is very simple, but just to see if I get things clear. You're still working with Majorana fermions, right? Yes, yes. No, but because my question is, shouldn't you demand that BN and CN are either the same or conjugate numbers? I'm a bit confused whether they are actually a pair of, if really it's a free look of if they are connected somehow. Because the fermions being Majorana and BN real. Okay, so yeah, it's a bit subtle. Maybe it's not important, so maybe you want to go on because it's not the center of your discussion, but I was just confused. Yeah, I mean, if we, so in Lorentz signature, it's meaningful to impose that this field is here, but if we go to Euclidean signature, it's a bit subtle to discuss whether it is a real field or imaginary, so yeah, in, so if we work in Lorentz signature, then so these two are complex conjugates. Then, I mean, so we can just imagine taking the real part and imaginary part of then, then we have two variables. Okay, okay, okay. Yeah, yeah. Okay, thanks. Sorry, sorry for the interrupt. No, no, no, no. Thank you for the question. Hello. Yes. Can I ask you a question regarding these non-zero modes? Like, I mean, what does this pair mean in the physical, I mean, in the physics context? I mean, this is kind of mode expansion. I understood that, but physically what does it mean? Physically? Like in a condensed matter system, like. I mean, so I'm here, I want to discuss Euclidean parsing, maybe I'm now going to write down how they appear in the parsing integral. So maybe let me first write down. So if we consider parsing integral, then the parsing integral measure is given by these coefficients. So we want to consider parsing integral over this film field. And it is given by the product of these coefficients. So in the parsing integral, we integrate over these variables. Does this answer your question? Yeah, thanks, but I have another related question. Like, if you go back to the determinant of minus one condition, and can you hear me properly? Can you hear me properly? I think, but there is some noise. Okay, okay, sorry. So if I go to the condition determinant of G is minus one, I can always choose the determinant, the elements in some other way, like mostly negative. The pi one, psi one component is positive and things like that. So does not that put a constraint on the number of elements in the, appearing in G because for it to be determinant minus one, you must have an odd number of minus ones, right? In the element. That's right, yes. So does not that also change this integral measure you wrote like in terms of BN and CN? I was just curious. Maybe it sounds so simple, but I don't understand. Yeah, so in some case, I mean, so, yeah, for example, we can consider family of all three matrix, like this, minus one cosine theta, minus sine theta, sine theta, cosine theta. So we can consider this kind of matrix. And if theta equal zero, then this is minus one, plus one, plus one. So in this case, we get this expression. And then from here, we can continuously change this theta. And then, so at theta equal pi, we get minus one, minus one, minus one. So there are three minus entries. Then, so at theta equal pi, actually there is three zero modes. So originally these B and C are non-zero modes, but at theta equal pi, they become zero modes. So this B and C are now zero modes. So in this way, we can continuously connect from here to here. Okay, thanks, I understand. Yeah, yeah, yeah. What is the number of zero modes, mode two, all the way is conserved? Yes, okay, so, maybe I should, maybe I should eat. Okay, so this is the path integral measure for the fermion. And now we consider a symmetry which I denote by minus one to F. So let's consider this one. So this is what is called fermion parity. So this is a symmetry which just changes the sign of the fermions. So fermions goes to minus the fermion. So in theory, which contain fermions, this symmetry always exists. I would also consider it as a subgroup or when in the present case, but this path always exists in any theory which contains fermions. So let's consider this fermion parity. I think it says, chat, okay, no problem. Okay, so let's consider this fermion parity. Then the path integral measure changes the sign under this transformation. The path integral measure goes to minus the path integral measure under this fermion parity transformation. And the reason is that, so these non-zero modes appear in pairs. So the signs are canceled among them. So they don't contribute, but here there is a single zero mode. So from this zero mode, by performing this transformation, we get a minus sign here. Excuse me, Kazeya. Ah, yes. Question. So I've had difficulty in understanding that this, I understand that the non-zero mode appear in pair, but here this measure is an infinite product of this D. So we need to regularize this measure first, I think. Ah, okay, yeah. That's all right. Yeah, so I think there exists a measure regularization so that this sign doesn't change. You can take, for instance, DBN and DCN2, N minus one, and then the overall sign is still restored. I mean, you can regularize the instant way that there's no sign change. I mean, cutoff, the cutoff of this I, the super-screwed I. I don't think there's any regularization, quick. Yeah, because that's a significant product. So we need to regularize to discuss that is correct, that is correct. Yes, I'm kind of sloppy about regularization. Yes, I think there is some intrinsic structure that we need to put into this measure, like the charge conjugation or the complex structure or something. Yeah. Then we need to. No, no, I don't think so. I think, so we only have to impose this, the existence of this ON symmetry. And so maybe I should say that. So there's some difficulty in regularizing this theory. So one way to regularize this kind of Helmian theory is to introduce some polypillars field. Yeah, I understand this point. But I mean, we need to prove that under any regularization, it has this minus one change. That this is our goal, but one of PV doesn't mean that, I mean, yeah, you need to prove any regularization will reduce the minus one sign change rather than a spatial PV regularizer. Yeah, I mean, sorry, let me say that. So I try to say that this polypillars does not work in this theory because, so if we try to write down some must done in this theory. Yeah, let's break the symmetry. Yeah, yeah, that's right, that's right, that's right. So this polypillars regularization does not work. So there's no regularization which preserves this O and S symmetry. Yeah, yeah, so I mean, so I mean, we need to impose this charge conjugation symmetry. I don't think so. I think I only need to this O and S symmetry. Yeah, but yeah, but yeah, I mean, so yeah, here I'm just doing some traditional discussion and this traditional approach is not so systematic. Okay, so later in my lectures, I will discuss more systematic approach and then maybe we can discuss that. So a shorter question is that, should we assume that this Dirac operator, D over DT is invertible operator? I mean, F excluding the zero modes, is it? Should we assume that it's invertible? Invertible? D over DT, F excluding the zero mode. What is your motivation to ask that question? Yeah, if we assume that it's invertible excluding this zero mode, then we must regularize the measure so that B and C appearing symmetrically. Yeah, but this D over DT will transform B subspace to C subspace. So in that sense, then everything will be fine. Yeah. Okay, so yeah, so this, so the Dirac operator here is simply this derivative. Yeah, yeah, yeah. And this operator is kind of anti-symmetric matrix if we need to dimension out anti-symmetric matrix. Oh, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Okay, maybe we can dispute this point. Yeah, yeah, yeah, yeah, yeah. So, so, I mean, there could be ample time for discussion after the lecture. I mean, if you want to come back. Oh, I only have 10 minutes. Yeah, let's discuss in the discussion session. So, yes, anyway, so in this way, we can see the existence of anomaly. And this is very analogous to the actual anomaly. I think maybe you have learned about actual anomaly in standard textbooks. So actual anomaly can be understood by using triangle diagrams, but they can also be understood by using Fujikawa's method. And in Fujikawa's approach, the violation of actual symmetry by anomaly is described by some things similar to this one. And in the case of actual anomaly, the violation which comes here is described by the Appear Singer Index. But here, this violation is actually described by the mod2 index. So this sign factor is actually minus one to mod2 index. So in this way, this is very similar to the case of actual anomaly. Okay, and next, I'd like to talk about anomaly if it is related to pi1 or SON. And just for simplicity, let me just take n equal to because nothing changes in higher n. So just take n equal to. And I take some family of group elements parameterized by theta. So let's consider this kind of matrix. So here, theta is just a parameter and later we will smoothly change this parameter. But for the time being, let's consider this as just a constant parameter. So we consider this SON element and in this case, it is more convenient to use complex fermions. So I just take linear combination of fermions. Then this is just a theory with Dirac Lagrangian. This is just psi bar d d psi. And then I consider partition function with a twist given by this element. So this partition function is again given by the passing integral on S1. And the boundary condition is twisted by this group element in the following way. So this psi satisfies this boundary condition. So here, this minus sign comes from the sum of boundary condition and this exponential i theta comes from this twist by this group element. So the boundary condition is twisted in this way and psi is also twisted in a similar way. And under these boundary conditions, we can expand the field in terms of Fourier modes. So because of the twisted boundary condition, the mode expansion is given like this. So this is the Fourier mode expansion. So this minus one half comes from this thermal boundary condition. And also this theta over two pi comes from this one. So the Fourier modes are now shifted in this way from the integers. Then the passing integral is given by the product of eigenvalues of the Dirac operator. So in this theory, the Dirac operator is just this derivative operator. And so this partition function is given by the product of eigenvalues of the Dirac operator, which is just this quantity, sorry, sorry, sorry. I should write it as just proportional, right? I'm neglecting various factors. I'm focusing on the part which depends on this parameter theta. So this is the result of the passing integral. However, so there is still some sign ambiguity in this expression. So the reason is that so there are infinitely many eigenvalues which are negative. And we are taking the products of these infinitely many negative eigenvalues. So sign of this expression is not determined by this formal expression. So we have to fix the sign of this expression. And for that purpose, so I require some physical condition. So I choose the sign so that the partition function is positive at theta equal to zero. So this is a reasonable requirement because in the theta is zero, then this partition function is just a sum of partition function. And sum of partition function is just physically expected to be a positive definite. So we impose this condition. Then we can fix the sign. So the passing integral, the result of the passing integral can now be written in this way. So this is a manifestly positive at theta equal zero. However, let's smoothly change the parameter theta. So I change it from zero to two pi. Then at theta equal zero, all of the factors are positive. So I just required this condition. But at theta equal to two pi in this infinite product, one factor is actually negative. So that negative factor is this one, n equal one. So this one minus one half square minus theta over two pi squared. So this factor is negative at theta equal to two pi. And other factors are positive. So we conclude that there is some sign ambiguity in the sense that I mean, so this passing integral measure at theta equal to pi is actually minus the passing integral measure at theta equal zero. Sorry, can I ask what about the n equal zero factor? Can I make for? Isn't this also negative? Sorry, sorry, sorry. No, no, no, sorry, sorry. This product should be taken like this one. So sorry, yeah. Okay, okay, thanks. So, but I'm sorry that I'm running out of time, but let me conclude my lecture today. But this element g at theta equal to two pi is actually the identity element of SON because two pi rotation is just the identity. So, we expect that the passing integral measure at theta equal to two pi should be the same as the passing integral measure at theta equal zero. But actually we found that there's this minus sign. So this means that the passing integral measure has a sign ambiguity. And so this is a normally it is related to pi one or SON. So this is related to pi one or SON because this g theta, this is a function from S1 to SON. And here this S1 is parameterized by this parameter theta. So in this way we find an anomaly associated to this one. And so I have talked about this SON, but if we consider instead it's a double cover spin n group, so this spin n is a double cover of this O n, then this two pi rotation, two pi rotation is not the identity element of this spin group. So if we use this symmetry group, then there is no anomaly. So we can interpret either that the theory has this O n symmetry which is anomalous or we can interpret that the theory has a spin n symmetry which has no anomaly. So either the interpretation is of course okay. But for example, in the case of two-foot anomaly matching, it's better to have more anomalies then it's more convenient to use O n. On the other hand, if we are interested in gauge theories, then there should be no anomaly, so we have to use this spin n group. Okay, I'm sorry that I stopped my lecture here. Thank you very much. Okay, thank you very much Katsuya for your lecture. So let's thank the speaker. And now, so I think we should stop the recording and so we can enter into the... So how is everything at ICTP? Are people going normally to the office or? Pretty much I would say in the sense that the ICTP has been open for quite some time during the summer it was open. So I mean, of course you have to use masks and... But, yeah, the cafeteria was open. So yeah, I think it's where I'm... I would say going quite well. Here at CISA we had a little bit more restrictions but still, I mean, we have been coming to the office for almost a year, so... Okay, it's nice. It's not too bad. Yeah, probably it's traveling, which is still not really happening. Yeah, I will go to a in-person workshop in November in Rindberg, Munich. Okay, okay. An experience from so long. So that will be your first workshop after the pandemic? Well, there was already something at Simon's. You mean the summer workshop or something else? The summer workshop and also the meeting for the collaboration of categorical symmetries. Oh, okay, yeah. Also you were there? Yeah, I'm not part of the collaboration but it was a very nice event. Well, I guess that domestic travel in the US. Yeah, exactly, not the same. Like where I went by car, so it's not like traveling. Just wait 30 more seconds. People is now connecting. Maybe we can start the recording. All right, so we can probably start. So now we'll have the second lecture from Irana Varenswela from Harvard on aspects of the Swampland. So please. Okay, thank you very much. Thank you for the invitation again. And thank you everyone for coming back to the second lecture. That means that the first one was not so bad. So yesterday we dedicated a lot of time to introduce what the Swampland program is. What are the goals, which is to find what are the constraints that an effective theory must satisfy to be consistent with a quantum gravity completion. This is a very ambitious program, right? We are trying to understand what are the constraints coming from quantum gravity. And that's why we have to use many different approaches with all the techniques that are at our disposal, either street theory, black hole physics, holography and so on. But it's also very rewarding in the sense that if we can point out precisely what these constraints are, we can give new phenomenological implications for high energy physics. So then we started to discuss one of the most important ones, one of the most important Swampland constraints, which is the absence of global series. Okay, so this is the one that is better understood. And as we said, we have already some proofs in different contexts, like perturbative string theory or some cases, native CFT. And what I did yesterday is to try to explain carefully what we mean with global symmetry. So what is the difference between global symmetry and a gauge symmetry. And also what is the motivation coming from black hole physics or the absence of all these entries. Now, is there any question about this before I move on? Okay, in that case, the plan for today is to continue with this topic of global symmetries. So we are going to talk about generalized global symmetries, which is this generalized notion of symmetry that has appeared in recent years. And how this connects with completeness of the charge spectrum, which was another all Swampland constraint that now we understand in correlation to the absence of global symmetries. And also the co-ordition conjecture, which will also be a generalization of the type of symmetry that is not allowed in quantum gravity. Okay, so this will be the main theme, depending on how we're going in time. Maybe we start already discussing the towers of states that appears as the asymptotically means when we try to make a global symmetry very approximate. So then we will start discussing what happens if we break the symmetry a little bit, how much we can do it and how to quantify this in terms of the good gravity and the distance connection. Okay, so this is the definition of global symmetry that I gave you yesterday. Okay, it's the same slide. So we said that we need to have some local operator that is topological and that has some charged operators that transform non-trivial. Now, in order to talk about generalized global symmetry, so the first notion of that generalized symmetry comes from this P form, global symmetries, in which the only thing that would change here, sorry, I see there is a question in the chat. I have a general naive question, is this Schoenbrand only related to quantum gravity or happens in UV completion of any of these? Okay, these are good question. So already in quantum field theory, they are constrained and we need to satisfy to have a quantum field theory UV completion, okay? So, for example, like we know that there are positivity bounds, right? On the higher derivative terms and so on, that in order to guarantee that the UV completion is unitary and satisfies causality and so on. So it's not something only of gravity, okay? In general, in order to have a healthy UV completion, not unitary theory, you need to satisfy additional constraints. Now, the word Schoenbrand, when we mean when we refer to Schoenbrand constraints, we are restricting to those constraints that are correlated to gravity, okay? So, like we imagine that we already satisfy everything that we need in quantum field theory and then we wonder what is left, okay? So what if there are new rules, new constraints that can appear if the UV completion is in quantum gravity, okay? So all the Schoenbrand constraints should disappear if gravity is non-dynamical, okay? Okay, good. And, well, let me also add that it's also interesting how, I mean, people are also trying to connect all these positivity bounds and like the typical approach, not as you can get of bounding higher derivative terms and so on, we'll see that you can also use that in the presence of gravity to try to prove some of these Schoenbrand conjectures, like they will gravity, for example, okay? Okay, so I was, as I was saying, so the first most standard generalization comes from this before global symmetries, this generalized global symmetries, which the only thing that changes is that these operators now are supported on D minus P minus one meaningful because the operators that are charged under the symmetry are not point-like, but are localized on a P manifold, okay? So the charges now are not carried by particles, as usual for the sea of global symmetries, but are carried by P brains, okay? By extent objects, which are created by these charged operators. And the rest is the same, okay? Again, we need that this operator, the symmetry operator is topological and that act non-trivial and that it satisfies some group. So the analogous example of the SIFT symmetry of the action, but for a P-form global symmetry is to consider some gauge theory, a U1 for a P-form gauge field, such that, for example, one of the symmetries that this theory has is this P-form global symmetry, which is kind of the SIFT of the P-form gauge field, and you can compute as an exercise using Nether's theorem that the symmetry operator, which is going to be the integral of the exponentiation of the integral of the current, is just the integration of the field stress, okay? So this is what would be the current that you can derive from Nether's theorem. And then the charge operator, so you can do this as an exercise, it's a nice exercise, very simple. And the charged operators are like the Wilson surfaces, not the Wilson lines, that indeed not live on P-dimensional manifolds, right? And since this is a P plus one, that's why this has to be integrated in a D minus P minus one, manifold. Now, these global symmetries should be upstanding quantum gravity, so we can either break them or break them, or gauge them, okay? So how this typically happens? Well, for example, we can break it by adding charged states, okay? Why is that? Well, the easy way in this case that we have a phalagranian theory, is that the current that we are talking about, right? The conservation equation for the current corresponds just to the equation of motion for the gauge field, right? Because the current is just the gauge field. So this will be different from zero, right? So we will break the symmetry, the current will not be conserved, if we have some charge matter, right? So what appears in the right-hand side is the electromagnetic current, right? So this in general for a P-formate field is broken by P minus one brains, okay? The other option is that we can gauge it and this is done by coupling to a P plus one gauge field, okay? So how did we do it for the action? Then for the action, which is a zero-form gauge field, we couple the action of the current to a one-form gauge field. So now that we have a P-form gauge field, we have to couple it to a P plus one form gauge field, okay? So that we add this term in the Lagrangian, okay? This would be so, let me put the current. And this happens whenever we have like BF terms of charge-simon couplings in a string theory. I mean, all the charge-simon couplings that we get are always gauging and actually breaking the symmetry of the other one, okay? So the already charge-simon terms are playing the role to break, well, I'm taking care of breaking these symmetries. Now it could happen that they break it but they break it to a discrete subgroup. And the only way then to get rid of all the symmetries is to add charge states, okay? For all the representations. So what I want to show you is how then this relates to completeness of the charge spectrum. Any question about this? Okay, so completeness of the charge spectrum, I didn't write it here, but yeah, I mean, basically it's self-explanatory in the sense that it requires that there are states with all possible charges, okay? That are consistent with your quantization and so on. So you can find these statements together with the absence of global symmetries and the compactness of continuous gauge groups for example, in the paper of Bank-Cyber that have put the reference before in some place. And you can also motivate it by blah-blah holes but now the recent, like the most modern understanding is that this just comes from the absence of generalized global symmetries, okay? So that, for example, if you consider, let me now focus on a one-form gauge field. So we have the same theory that I told you before but just the simple ones, we have mass-worth theory and we said that the current just given by the field strength. So here I'm talking about the electric symmetry, okay? Then we will discuss magnetic ones. The charge operators are Wilson lines, right? So they live on these lines, on these one-dimensional manifolds, okay? And we said that the symmetry can get broken if we add charge states, if we add particles, right? How we see this in more detail? I mean, it's easy when we have the Lagrangian we don't have the current, we can just study conservation of the current but this formal range of the political operator is much more powerful because it allows you to talk about all these breaking of symmetries even if they are discrete and even if we don't have a current because we don't have a Lagrangian. So the idea is that what's going on is that since, let me write it here, since this symmetry operator, right, is acting non-trivially on the Wilson line, okay? This is just the definition that the Wilson lines are charged under the symmetry. The operator will become non-topological so the symmetry will be broken if the Wilson lines can end, okay? They can break. So why is this? Well, imagine that a little bit better. So we have this Wilson line operator, okay? And now we have this symmetry operator around it, okay? Since it's charged, if we shrink the operator, okay, since the operator is topological, we can deform it, okay? But then if we shrink it, this will give rise to the Wilson line again, just with a phase that is different from one, okay? Because we have this insertion of the operator. But if the Wilson line can break, imagine that it can break the Wilson line because I have some particles in my theory, okay? I have some local operators. Then I can shrink this topological operator, right? In such a way that it disappears, I mean, becomes trivial. And then I get just the Wilson line, which would be equivalent to say that this phase is one. But that's inconsistent with the fact that the Wilson line is charged, which means that this, in the moment with the Wilson line can break, the operator is no longer topological. You are not allowed to deform it this way, okay? All this is just to tell you what is the proper, the more sharp way to talk about breaking all the symmetries, which is requiring that the operators are non-topological, which happens if these Wilson lines can end, okay? Or can break. So keeping this in mind, if we add now a particle of charge N, this would mean that only Wilson lines of charge N because they are labeled by the representations of the group, only those ones can break, which means that still I will have some Wilson lines that can end, and therefore the operators that link non-trivial will remain topological, which means that I have a C and symmetry left. And this is something which is not easy to see just from the currents and the conservation equations. Okay, so in general, if I want to break all the possible one for global symmetries, okay, that actually belongs to the center of the gauge group, I need to have a complete spectrum. Okay, I need to add particles of charge one, and then be able to generate multi-particle state of all possible charges so that all possible Wilson lines can end. Okay, so this is true for continuous gauge groups, okay? Any question about this? If I'm not going to do this, yes. One question. Is the converse also true? If the spectrum is not complete, then you do have a global symmetry? Yes, for continuous gauge groups, it's an if and only if. Okay. Because as we said, if we add particles, yeah, should have put the double arrow. If we add particles of all possible charges, then I don't have one for global symmetries. And if I don't have one for global symmetries, it's because all the operators, all these, they are called hookup with an operator, are non-topological, which means that they cannot break. And this is valid for continuous gauge groups if I have this isolated gauge theory that I told you about, okay? If I start adding charge simultaneously and other things, then the thing becomes a bit more complicated because first it would mix electric and magnetic symmetries that I'm not considering now. And also there are other ways to break the symmetries with these charge simultaneously. So then it's not settled yet, okay? But if you have a pure gauge theory with charge matter for continuous gauge groups, this is an if and only if. Any things? Okay, this is a question in the chat. If an operator is non-topological, it means there is no symmetry in it. Then I think charge is not going to be fine. If this is correct, how do you distinguish both one and not one? It's just, I mean, it's like you get the result by contradiction. I mean, it's just, you assume that you have some symmetry operator and that you have words online that is charged under it. So this phase should be different from one. But then, since the words online can break, you can see that you can define the operator, shrink it to a point which would imply that this phase is one, which was in contradiction to the previous assumption. So something is wrong. And what is wrong is that the operator is no longer topological. Okay, this is the way to understand it so that symmetry is broken. Okay, another question. Considering the symmetry B minus L, if we assume this is gauged or broken, I think this would have implications as a fifth force or proton decay respectively. Is this correct? Yeah. So how can we deal with these implications? Yeah, indeed. So B minus L is the only possible candidate for an exact global symmetry in the standard model. And this means that according to quantum gravity, it should be either broken or gauged. So if it's gauged, it means that we have some U1 degree of freedom associated to it. And this degree of freedom in this new U1 has not been detected yet. It's a fifth force. And there are constraints on how small the gauge coupling has to be. So the experimental constraints coming from B minus L, I think is that the gauge coupling has to be smaller and 10 to the minus 23, which is very small. So it would be a very weakly coupled base theory. And then you can try to make to construct models of how this could happen, kind of string theory and so on. And I mean, it's very interesting. Also, it's a correlation also between this small gauge coupling and neutrino masses when you use the weak gravity connector, so I can talk about it maybe tomorrow if you wish. But this is one option which is interesting for phenomenology. The other option is that it's broken. So that there's no gauge field associated to it. It's broken at some high energy scale. And then again, this is, now if it's broken, it's going to induce some operators, some higher dimensional operators that are breaking the symmetry are going to induce some proton decay. So again, you have experimental constraints about the separation for these operators. And you can start again playing the game of what is the scale at which you should break the symmetry so that you can detect this in the near future. It's quite high, I think it's around that scale, but it's also the other possible scenario. So it depends, I mean, it depends whether we will detect either this weakly coupled with theories or proton decay, then we can get information about whether it's broken or cached. Okay, any other question? Okay, so what I told you here is it's for continuous gauge groups, but one can do this more generally. And just so you know, for discrete and disconnected gauge groups, one can show and prove that for these pure gauge theories, this is still works, but one has to add here a more general notion of symmetry which involves non-invertible symmetries. Okay, so I'm not gonna discuss this in detail, but just for your information, in case you are interested in non-invertible symmetries like a new type, new notion of symmetry that has appeared very recently, that is exactly the same. Let me come back to this slide. It's the same that we had before, but with the difference that we don't require to have a group law. Okay, so the operators are not necessary, are not invertible necessarily. So this is another generalization and it seems it's important to be able to have a one-to-one correspondence between completeness of the charged spectrum and no global symmetries in complete generality. So this is the working progress here. You can check if you wish. Hey, then, Rach. Et al, this is a paper that we published this year. Well, you can read more about the topic, but it's a very active topic of research now in this one. Okay, and the last thing I wanted to say is that of course here I'm talking about the electric symmetries, but the same happened for the magnetic symmetry. The symmetry operator would be the integral of F2 instead of the hot-stool F2. So these are surface operators. This is for mass-welfth theory, okay? This is of the U1, its theory. And then again, right, the conservation equation is like the, of the current is the Bianchi identity. So this is different from zero. You have monopoles in your theory, okay? And no one can play the same story of having completeness of the monopoles related to the absence of all these magnetic one for global symmetries. Okay, any question? Okay, I see, I'm still confused. It's a question. It seems to me that the capability of the line is against the assumption of the political property. Will you clarify what are the assumptions and what are the conclusions? Yeah, yeah, so the point is, I mean, here, let me write it here. What I'm saying, no one for global symmetry, it means that there are no topological operators. And what I did is to show that this is equivalent to say that the Wilson lines can break, okay? So this is how this is shown. So Wilson lines can end, it means that you have charged spectra. So this is equivalent to this and that there are no topological operators like the symmetries are broken. So in order to show this correlation, you can see that if the Wilson line can end, then the operator cannot be topological because otherwise you could shrink it and reduce it, shrink it to a point away from the Wilson line so that it would mean that the Wilson line is not charged under it. So you have a symmetry to start with because there is nothing charged. So there is no topological operator in your theory under which the Wilson line is charged. So there's no symmetry left. But this is the one-to-one correspondence and one has to show. So of course you cannot have one without the other one. I mean, they come together, okay? Is it clear now? Yeah, so this would be when I say not topological operator, I'm thinking of the symmetry operator. And here, this is the Wilson line we are talking about, okay? Okay, so let's move on. So this is all for the completeness of the spectrum. The last thing that we are going to discuss before leaving this topic of flow of asymmetries is how to generalize this to cover the same classes. Okay, so as you can see here, like, I mean, the absence of flow of asymmetries, as I said, is at some point, I guess we'll stop calling it a conjecture in near future, I guess, because I mean, some cases has been really proven but the whole, the tricky part is try to understand precisely what is the notion of flow of symmetry that is not allowed in quantum gravity. So everything starts with just this usual ordinary symmetry, like zero flow of asymmetries, but now we can see that before flow of asymmetries are also not allowed. I mean, this is also captured by some of the proofs. And this relation with completeness of the spectrum suggests that non-invertible flow of asymmetries are also not allowed. I mean, this is not captured yet by any proof, but we can go further and try, and we can generalize this to cover these in classes, okay? So everything's about trying to determine precisely what is the notion of symmetry that should not appear, which comes, it's very correlated to all these developments in quantum theory that are also defined all these new notions of symmetry. So how we go to cover these in classes? Well, so let me do, I'm gonna do one example which we can understand and then show you how this can actually be characterized in terms of cover this. So imagine that you have some string theory compactify on some internal manifold to give some effective theory in, this is the non-compact space, D minus D. So if we have some gauge fields, we can integrate these gauge fields in some of the internal cycles and we'll have some flux, okay? Some fluxes. Now, this corresponds to a global symmetry. So right now we were discussing this magnetic symmetry, right? In which the charge, okay? So this is, right? Here the symmetry is just the exponentization of the conserved charge. So this charge is the integral of the gauge field. So these fluxes are actually given rise to these magnetic symmetries, which for a, if I am integrating, I may have a class of flux from a D form field strength. This corresponds to a D minus one for global symmetry, okay? Because it's the magnetic one, okay? So this is what would be the charge and the symmetry operator of the symmetry. As I said, it's just IQ. Now, these global symmetries, magnetic global symmetry, we have said that can be broken if we add states that are charged under it, which would be like monopoles. And now they are extended objects. So these monopoles are brains, okay? So the brains are the objects that are charged under these gauge fields. So if we add D minus one brain, this will break the symmetry, okay? Now, from the perspective of the lower dimensional theory, I mean, if I am compactifying on this XD manifold, this is a co-dimension one object, right? So it's a domain wall. So what is happening is that if I have, if I compactify on some internal manifold that have some flux Q one and another which has a flux Q two, the effective theory that I get, right? The two effective theories that I get can be connected by a domain wall, right? So these domain walls are changing the value of the flux. So here I have a domain wall. Okay, so the two EFTs that arise because of compactification with different values of the flux, with different fluxes can be connected by the domain wall, okay? So the statement that we are breaking this magnetic symmetry is equivalent to say that I have a domain wall in my effective theory that can interpolate between the two fluxes of Aqua, okay? This is equivalent of, is the physical meaning of broken, breaking these symmetries. Any question? Okay, so this is the standard thing. I mean, where if you work in a string theory, you are used to have always these domain walls that come from the machine reducing brains and they interpolate between the different fluxes of Aqua and that means that because of the presence of these brains and these domain walls, then these magnetic symmetries are broken. Now, this is something which can be characterized with cobalt dissymbol, okay? So more generally, you could wonder, okay, if breaking this symmetries is equivalent to have a domain wall that interpolates between the different Aqua that have different conserved global charges. So they are not conserved anymore because they can just change the charge from one value to the other. Is there any way to characterize this more generally? I mean, can I characterize to what extent two EFTs can be connected always by a domain wall so that I can always get rid of any possible global charge that I could have, okay? Any of these topological global charges. And the answer is yes, you can do it with these cobalt dissymbols. So more generally, I mean, this is just, I give you an example associated to a gauge field, but you can construct similar examples associated to gravity and more geometrical things. But in general, the political global charges of this type can be characterized by cobalt dissymbol classes, which is precisely the same amount of words that you can have a domain wall interpolating between the two theories, okay? So imagine that we'll have some compact manifold. Let's call it C and another one, let's call it D. And I have the EFTs that arise from compact defined string theory on this manifold. I have some EFT one and EFT two. And I want to ask whether I can have a domain wall interpolating between both of them, which is equivalent to us whether these two manifolds actually can be connected by a D plus one dimensional manifold so that I can deform one into the other. And whether this can happen, whether given two compact manifolds, whether you can connect them by a higher dimensional one is, well, it has an answer in cobalt dissymbol group because if two manifolds are connected this way in the sense that they are the boundary of another manifold of one dimension more, we say that they are in the same equivalence class of cobalt dissymbol theory, okay? So these are boredant or co-boredant to each other if their union is the boundary of one dimension, okay? So this is a mathematical definition that gives rise to an equivalence relation, okay? You can just define this equivalent relation. So two manifolds are boredant to each other if so they are in the same equivalence class if their union is the boundary of one dimension more. And this will precisely give me information of whether I can connect these two manifolds so that there's a domain wall interpolating between the two EFTs. So these forms a group. This equivalence relation can form a group which is this co-ordition group. And the different elements are like this difference equivalence classes, okay? Any question about this? There is something that is very useful in the context of anomaly and that I'm sure if today I couldn't watch it but today or tomorrow, I mean, I'm sure you will hear about it in the previous lectures. So that's an important notion there. Okay, so why is this important for this absence of probability? Well, the point is that now, thinking of all this topological global chart is, ah, sorry, is this a question? Should the domain wall be aligned on the diagram? Yeah, okay. I was making it like a bit thick but it's a co-dimension one object. So yeah, it's like a line. Yeah, it's a co-dimension one. So yeah, so thinking of this topological global symmetries, Jake and Conron formulated this co-ordition conjecture. Meaning that- There's one more question in the chat. Ah, sorry, thank you. So in what sense this is a group? What is the multiplication? Yeah, it's the union, right? So you can just, so the union of the different manifolds is the multiplication in this group. So that's, yeah, the two unions is still another element of the group. And the trivial element is the case in which a manifold is a boundary by itself. So I will discuss this in more detail now. This has a very geometric interpretation. It's an abelian group because everything commutes with each other. Doesn't matter how you join, how you produce this, you know. Okay, so the statement is that if we want, I mean, let me say what the conjecture is. So the conjecture is that all these co-ordition classes must vanish so that this group has to be trivial so that everything should belong to the same equivalence to a class, okay? To the same co-ordition class which is just the trivial one. Now, why? Because otherwise, if this was different from zero, we will have different classes, no, different elements that would give rise to conserve global charges because it would mean that I cannot transform one into the other. I cannot have a domain word that interpolates between one and the other. So I cannot get rid of the charges like a conserve global charge, my theory, and would give rise to a D minus D minus one form global symmetry, okay? Where the charges, these topological charges would be the different classes of this co-ordition group. So one example is the example of magnetic symmetry for a gauge field that I gave you. This is already incorporated here, but you can have more general examples just thinking geometric terms, no, in general. Like whether you can, I mean, in the end the question is whether you can transform at the form one charge into the other, whether you can break it because otherwise you would have EFDs that are disconnected. Okay, so the consequence of this is that all EFDs of the same dimension, okay? Here we are always comparing effective theories of the same dimension are connected by finite energy domain words because everything is the same class so everything can be connected. And this way I can get rid of all these topological charges and break all the symmetries. Any question? Okay, so the example, let me come back to the example we had of the fluxes, right? We have some internal fluxes. Now these, since the fluxes are quantized, right? Here I have some integer, no, that they have different possibilities which if you compute actually this co-ordition group taking the combi structure, you can see that this C which precisely is labeled by the different integers which corresponds to the different fluxes that you can have, okay? So if you just have some compactification with these fluxes it will mean that this co-ordition group is different from zero. So what's going on? Well, what's going on is that we are, we need to add something else. We need to add brains to break it. So once we add brains, if we add brains and you compute again what this co-ordition group taking into account the brains, you can see that it trivializes, okay? It's zero. So if we didn't know about the existence of brains you could also say that, okay, because I have all these fluxes since I need to trivialize this co-ordition group I need to have brains. So it predicts like the existence of defects that are breaking these symmetries. Now in some cases are obvious, we already know like we'll have brains in theory but other cases are not so obvious, okay? So for example, I put here one example if one studies the co-ordition group for zero dimensions, like if I don't compactify having a spin structure, okay? I also have fermions. One can see that it's different from zero. And this is precisely the co-ordition group that matters when you study type two string theory in ten dimensions, okay? If I have type two string theory in ten dimensions you can have this non-trivial co-ordition group so you will say, okay, how string theory is getting rid of this? Well, if I am in type two A the defects that are trivializing this co-ordition group are just all eight brains. See, this is like the co-dimension one object, the main wall that gets rid, oh, sorry, of this topological charge, okay? So nice, that's fine. But in type two B I don't have co-dimension one brains, right? So it's not known what would be the defect, the charge state that is supposed to break this global symmetry. So it has been shown that it should be completely non-super symmetric, which explains why we don't know it yet because all the objects that we know are always BPS objects and supersymmetric objects. So this would be some domain wall interpolating actually between type two A and type two B. So that's, I mean, they are theories of the same dimension so they should be able to be connected. And it's not supersymmetric and well, this is like an open issue in string theory not to find out what it is. So just to say that sometimes, I mean, this co-ordition conjecture, I mean, can be used to confirm the existence of objects that we already know, but can also be used to predict the existence of new objects if we believe that all the global symmetry should be broken. And so it predicts existence of new objects in string theory that are not supersymmetric. Okay. And this is what people are using also to constrain the possible compactifications and also the possible gauge groups because you can predict the existence of new defects and you can study anomaly from these defects and then put constraints on the possible pitch theories that can appear. So you can take a look at my lectures on the Swampland program and there you can find a lot of references about this topic, okay? So this is what comes together with this notion of string universality. So there's a bunch of people that are studying this in detail like this absence of global symmetries and anomalies and so on to try to determine what are all the possible gauge groups and theories that could appear at least in higher dimensions which is much easier, okay? And then check whether just by requiring the absence of global symmetries you can constrain what are the possible gauge theories that can appear and compare with the string theory. So compare whether everything that you can get indeed arises from a string theory to try to determine to what extent what we obtain from a string theory is representative of everything that should be consistent with quantum gravity, okay? So what is then we get everything that we could get and this is what goes under the name of string universality or string lamppost principle, okay? Any question? Okay, so let me make a last comment about this cover reason and then we will finish this. So yeah, I was asked also how this is a group so in order to be a group it's as important that you have this trivial class. So in this case, the trivial class is when a manifold is a boundary by itself so that if I have some this compact manifold I can shrink it to a point, okay? So that's why it's a boundary by itself. Now, from the perspective of the effective field theory, right? What this happened is that the effective field theory is ending. So here this domain wall corresponds to what we call an end of the war brain. I think there is a question. Does string universality imply string exclusivity? So it implies that everything that would be consistent with quantum gravity cannot rise from string theory like you can embed it in string theory. Doesn't mean that cannot arise in other ways but then you should be able to connect it to string theory. You should be able to deform it so that if there are other possible quantum gravities that are consistent somehow they should be connected to string theory as well. That's what would mean because you have all these domain walls that interpolates between all the possibilities. So everything is connected. Then there'd be other domain walls that connect for instance, string theory and end theory. So here it's about, when we talk about this topological global charges is always keeping the dimension fixed. Okay, so that's, yeah, you have two compact dimensions of the same type. Now, where you can connect them can be described by discoveries in group. So in that case, string theory and end theory are changing the dimension. So it's not described in the way I did it. Maybe we can find in the way of future to in the future our way to do it as well but not at the moment. And actually the expectation is that, okay. The idea is that when you have the same dimension everything should be connected by these finite energy domain walls in order to get rid of all these global charges. But if we change the dimension the expectation is that, okay, maybe we can still have these domain walls but they are going to be at infinite distance because, I mean, since we have to change the dimension we have to decompactify and this implies that the domain wall will be at infinite distance and therefore maybe it doesn't have finite energy. So that's the difference. Okay, so, yeah, so coming back to this trivial class so there is this particular class which indeed since we can string the manifold it means that the EFT ends, it's like the end of the war. And this is interesting because the this end of the war brain if now we can study the dynamics and it could happen that for example if we break supersymmetry it expands. So this is what give rise to the bubble of nothingness and stability that were formulated by Whitten 95, I think. Okay, so this bubble of nothingness and stability are again like end of the war brains but now imagine that I can nucleate this end of the war like a spherical bubble such that I have nothing inside. This is really the end of the war. So it's a hole in a space time that starts expanding if the dynamics is appropriate. Okay, so this will depend on certain energy conditions of whether it's energetically favorable to span but if they are satisfied which typically happens when you break supersymmetry then this bubble of nothing can expand and describe an instability of the vacuum. Okay, so this is interesting because if we are saying that we don't want to have all this topological global symmetry so they cover this in groups of bunnies it means that everything belongs to the trivial class. So any manifold that you could construct you should be able to shrink it to a point. Okay, so let me maybe emphasize so why this bubble of nothing is a regular smooth solution of Einstein gravity that can happen is because even if it seems very singular because you have a hole in space time like an end of the war we have this compact manifold is internal dimensions that are shrinking to a point at the wall of the bubble. Okay, so Whitten did this for this time for a cycle but you can do it more generally for a compact manifold. Okay, so if any compact manifold can be shrink to a point you can generate these holes in space time that if the dynamics is appropriate then they will expand and describe an instability. So if everything belongs to the trivial class everything can be shrink to a point so there is no topological abstraction in general to construct these instabilities, bubbles of nothing. Okay, so this is nice because I mean before these bubbles of nothing we're only constructing for cycles but now we know how to do it in general and like just using this formalism of cobaltism and we see that there is no topological abstraction if we don't want to have global symmetries in quantum gravity and therefore it's just a matter of the dynamics whether they will really describe an instability or not. So there's also a lot of research trying to understand what are specifically the energy conditions that you need to violate or satisfy to allow for this stability and whether they will always disappear whenever you break supersymmetry or something like that. So this connects with another conjecture maybe we can discuss a little bit about this the last day but there is this strong plan conjecture that claims that any non-suicy vacuum should eventually decay so that everything should be metastable at least and this is another conjecture, okay. And this was formulated in very without thinking of bubbles of nothing like by different means like actually thinking of the gravity conjecture and so on but now there is a new way to try to prove this conjecture by studying this bubble of nothing instabilities and try to show it since they are all seems to be almost topologically allowed whether we can also show that they will always expand one supersymmetry block. Okay, any question? Okay, so this is another example in which actually we'll see that all these one-plan statements are correlated that are related among each other. The last day we will keep track of all the relations that we are going to explore in these lectures and to try to use them. I mean, it's interesting to explore these relations precisely like to try to prove one conjecture coming from the other one as well. Okay, so let's leave this bubble of nothing instabilities that will eat our universe. Let's move on for the, let me, yeah, what I'm gonna do let me explain what I'm gonna do tomorrow and then we can move to the discussion, okay. So tomorrow, let me skip this. So tomorrow we are going to start discussing about what happens if we don't have exact global symmetries but only approximate ones, okay. So as we discussed yesterday, I mean, this is very nice because we can make very sharp statements but actually the absence of exact global symmetries is not necessarily very useful for phenomenology. I mean, still we can use all these co-ordination groups to constrain the possible VCO reasons one but in a more knife way, I mean, since we can break the symmetries very high this is not something that we can typically detect. So can we do better? Can we quantify how approximate the global symmetries can be? Okay, so for example, if you have some gates field and you send it to zero, you are going to engage the symmetry, right? So you are going to restore some U1 global symmetry. So this restores a U1 global symmetry. And then you could wonder, okay, what happens then? I mean, principle, that should not be allowed in quantum gravity. So I'm putting here that the gates coupling depends on phi. So in three theory, I mean, we can change the parameters of the effective theory and this is equivalent to moving in the modular space or the final space of my theory, right? So all the parameters are parameterized by scalar fields. So you can just think of moving this modular space such that you approach some point with some gates coupling goes to zero. So the parameter space is like the modular space because they are parameterized by the vacuum expectation value of scalar fields. And then what happens, so let me tell you what happens and what we discuss tomorrow and then I stop. And what happens is that all these points which you try to restore a global symmetry, they are pushed away to infinite distance in this field space, okay? So global symmetries can only be restored at infinite distance and tomorrow we will start with this and we'll start discussing what is the physics in these limits when you try to approach these infinite distance limits in field space. And we'll see what happens is that the cutoff of the effective theory goes to zero. As you try to engineer some global symmetry, okay? So this is what we will discuss tomorrow. We'll see how the cutoff of the effective theory goes to zero as the gate coupling goes to zero or as you approach more generally this infinite distance space and what is the physics associated with which will be described by the weak gravity conjecture and the distance conjecture, okay? So we'll start with this tomorrow more slowly and then come to these more interesting, more phenomenological interesting conjectures which are the weak gravity and the distance, okay? So let me then stop here. Thank you for your attention again and let's see if there are further questions. Okay, thank you very much, Gerene, for this very simple lecture. So thank you, Gerene. Okay, so now we can stop the recording. We're going to a more informal environment, okay? Stop, so now we can start with the... Is it, is the recording stopped? Can the CTS confirm? It should be stopped. So let's go with the discussion session. Hi, so I wanted to understand cobertism better. So could you give like a simple example of a cobertism group maybe in low dimensions like where the boundaries, I don't know, some lower dimensional manifold. Yeah, so for example here, I mean, the easiest one is the circle, right? So if you have S1 by some circle and you study, yeah, what is omega one because it's a one dimensional manifold. So here you can see that this is C2 which means there are two possibilities. Well, let me say so first, if I don't have fermions, this is just zero in the sense that the circle, you can always rank it a point because you can always feel the circle, right? It's like the circle can be the boundary of this cigar and therefore the cobertism group vanishes, okay? Because the circle is a boundary by itself. Now, if you put for example fermions, then you have to study the cobertism group with a spin structure, something like stranger theory. And then you can see that the result is C2 because there are two possibilities, whether the fermions have periodic or anti-periodic boundary conditions in the circle. And then it's the same, like if they have anti-periodic boundary conditions, then it's the trivial class. And then again, you have this cigar which would be the null-bordism, like the cobertism manifold that is able to shrink the circle to a point. And this is like the easy example because it's just one dimensional. But then we need two boundary manifolds. I mean, okay, there is a trivial one where it's a co-ordinate to itself, but in general, we need two boundaries, no? So in general, same. So the point is that if you have two manifolds which are- You should definitely feel entitled just to unmute yourself and speak up. At any point, anything I say is unclear or if you have any comments or anything like that. So yesterday, based on an observation involving Euclidean wormholes, we speculated that a more correct version or perhaps we should say more general version of the ADS-CFT correspondence is one that relates a theory of gravity in anti-decider space. That is to say a theory of fluctuating metrics in asymptotically anti-decider space to a probability distribution. Okay, it looks like my tablet has frozen actually. Can you not see what I'm writing here? Okay. Yeah. Can you hear me okay? Yeah, I can hear you. It stopped at the ADS-CFT. Okay, give me one. This happened to me yesterday where suddenly my tablet got kicked off of the network. Yeah, hang on just one second. I'm gonna try and reconnect. But you guys can hear me okay? Okay. Well, why don't I- The sound is great. Yeah, it's weird. My tablet just spontaneously disconnected from the internet. Okay. I don't know. Talk amongst yourselves. Francesco, do you know any good jokes to keep the audience entertained? Not a good incentive. Do I know any good jokes? There were two cannibals eating a clown. One of them said, does this taste funny? That's my favorite joke. Everyone's heard that one already. Okay, I'm sorry. My tablet seems to be being very slow this morning. Okay, I apologize for this. Everyone can go get a cup of coffee. Hopefully this will be just a second. Okay, I might just have to try and restart. Okay. Why don't I do that? And I'll talk in words when my computer restarts. So yesterday, we speculated that there was a more correct version of the ADS-CFT correspondence where instead of a theory of gravity in asymptotic anti-decider space being related to a particular conformal field theory, we instead speculated that it was related to a probability distribution over the space of conformal field theories. And this way of thinking about the ADS-CFT correspondence in some sense just generalizes our more traditional version of ADS-CFT because you could always take a kind of trivial probability distribution where you just have a delta function on the space of conformal field theories. So for example, in this picture, N equals four super Yang-Mills theory is still dual to type 2B string theory on ADS-5 times S5. The only difference is now we're allowing ourselves to think about more general probability distributions aside from just the delta function distribution. So for example, what I'm going to be talking about today is I'm gonna introduce two different models for this averaged holographic duality. So in particular, I'll introduce a theory of gravity in two dimensions which is going to be related to an average over one-dimensional quantum mechanical systems. So in particular, this is a theory of gravity, JT gravity which is going to be described by an integral over a space of Hermitian matrices, each of which are interpreted as the Hamiltonians of some underlying quantum mechanical system. And the other model that I'll introduce and we'll sort of occupy most of our attention for the next two lectures on Wednesday and Thursday is a model where we integrate over a full space of conformal field theories, very simple conformal field theories but non-trivial conformal field theories nonetheless. And we interpret this as a theory of gravity in anti-dissiduous space. So in particular, in this setup, we'll be talking about a theory of two-dimensional conformal field theories and we'll consider a family of such CFTs with a finite-dimensional modularized space. We'll average over that space of CFTs and argue that it's related to a sort of three-dimensional theory of gravity, okay. And now it looks like my tablet has healed itself. So let's see if this works. Sorry we run into an issue. Okay. You never like it when you're a tablet. Yes, oh please, that's a great thing to do. Is the distribution intrinsic to the theory or we're just finding some examples? Good, so I think in this picture that I'm trying to advocate, a theory of gravity specifies a probability distribution on the space of conformal field theories. So for example, in the case of JT gravity, it turns out there's an incredibly specific probability distribution on the space of Hermitian matrices. That gives you an answer that is equal to the JT gravity partition function. Similarly, in the Norene example, there is a very specific probability distribution on the space of conformal field theories that gives me an answer that looks like a theory of gravity. Sorry, I just need to remember the meeting password. In particular, it is presumably the case that you could take one space of conformal field theories and then consider multiple different possible probability distributions on this space of conformal field theories, each one of which would correspond to a different theory of gravity. Any other questions while I get this all set up? Can I ask a question regarding this probability distribution? So you say that each CFT is a corresponds to a particular theory in JT gravity kind of thing, but what guarantees that these are one to one? I mean, is it possible to have like more than one CFT is correspond to the same theory of gravity and vice versa? Well, I mean, what I would say is that, so as far as we understand, one probability distribution on the space of conformal field theories corresponds to a theory of gravity. Of course, the usual understanding of ADS CFT accommodates this, different delta function distributions on the space of conformal field theories correspond to different string theory compactifications down to ADS, for example. And I see no need to have a different picture once we're talking about more general probability distributions. In any case, maybe now that my tablet is working again, I can just try and shift gears and proceed with the lecture as I was planning. But again, oh, that's not good. Okay, my tablet is suddenly disconnected again. Okay, I'm sorry about this. It's a Zoom problem. Zoom is just disconnecting. So can I continue? Yeah, please continue with your question. Yes. You said that there is no need. I mean, is this something because this is kind of more ambitious project to pursue or why there is no need? I mean, that will be nice, right? If we have like two CFTs dual to one gravity, I'm not quite sure what you're asking. Yeah, yeah, yeah. I mean, why there is no need to have two CFTs dual to one gravity theory? What do you mean? Why is there no need to have two CFTs dual to one gravity theory? No, no, you told me that, I mean, I mean, in view of the usual CFT duality, there is no need to have this kind of correspondence. But what I found that, I mean, if it is possible, then we can group the universal class of CFTs kind of, I mean, more general universal class of CFTs correspond to one gravity theory. That will be nice, right? I guess I'm not quite sure I understand your question. Okay, so what I was asking that, I mean, the probability distribution of two CFTs can it correspond to one gravity theory or vice versa? But your reply was that in view of the usual approaches to ADS CFT, that is not usually the case, right? Okay, do you mind giving me a minute to try and get my tablet connected? It's hard for me to both pay attention to you and to try and fix my tablet. Yeah, I'm sorry. So yeah. Okay, I guess I apologize for this. I keep getting messages that Zoom experiences an unknown error. Okay, sorry, I taught two classes so far using exactly the setup and given hundreds of lectures and never had this problem before. I think it's a Zoom problem. Sorry, this is not a particularly fascinating thing for everyone to be watching. You know, I think it is a problem with the internet on my tablet. It seems to just keep getting disconnected. Okay, we'll give this one more try and if that doesn't work, we'll go to plan B. I can always, you know, I have a million different, I have lots of different slides I can pop up so that we have something to look at while we're doing this. But this is a surprise. I mean, I think over the last year I must have given a hundred, must have given a hundred different lectures, seminars and so forth using this tablet and never had this problem. There is always a first time. There's always a first time, that's right. And I didn't mean to cut off discussion. It's just difficult for me to both simultaneously connect my laptop and pay attention to questions. Maybe can I ask you a very simple question? Sure. Should we not distract you too much? Yes. Yeah. I just wanted to ask you, I'm not too much expert in ADSFT. So is there a sharp ADSFT correspondence where the safety has a multiple disconnected phase space, a space time, but a sharp one which you can reproduce you starting from the blades? So I'm not quite sure. What do you mean? So what is the criteria that you would like for your theory? I would like to know if there is an ADSFT correspondence which is based on a string construction like NEPA 24, where the safety is not defined on RD or whatever space, but it's somehow defined in a multiple disconnected components. If you wish, we have a couple of safety in some sense. So one can always, for a given CFT, study that CFT on some disconnected space time. So here, we have to be careful about whether we're talking in Euclidean signature or Lorentzian signature. And the idea is that in Lorentzian signature, as we maybe mentioned at the end last time, or perhaps it was during the question session, in Lorentzian signature, black hole spacetimes, eternal black hole spacetimes have multiple boundaries. And there's nothing particularly complicated about that. They're ubiquitous. They would appear in almost any gravitational theory, sufficiently complicated gravitational theory. On the other hand, if you're talking about in Euclidean signature, one could always just define the CFT partition function in Euclidean signature. And there's nothing wrong with considering the partition function on a disconnected surface or on a surface that has multiple disconnected components. So for example, n equals four young mills. There's no reason why you couldn't study n equals four young mills on four manifolds that are disconnected. The problem is that one of the basic principles of conformal field theory, it's almost an axiom of conformal field theory, is that the partition function factorizes when you do that. And so presumably, if you were to study type 2B string theory on a space time in Euclidean signature that has multiple disconnected boundaries, there would be some reason why in the full type 2 string theory calculation, you wouldn't get a contribution from space times that connect multiple boundaries. The problem is that nobody knows how this works. Yeah, sorry, my question was on the explicit construction. So suppose you take n equal to four on any manifold you like with two disconnected components. My question is, do we have a type 2B background, which we understand to be the dual of this configuration from a stringy point of view, like or on a DS5 times S5? At this level, we do. Not as far as I know. Yeah. So for example, and I apologize, sorry for this. OK, now it says it's connecting. OK, I'm sorry about this. It's hard for me to both answer questions and try and fix my computer at the same time. All right. I'm giving up on my tablet. Are you going to give me up? It doesn't work? Yeah, it's just been, I mean, it's been 15 minutes now of spinning wheels. I've already restarted it. Doesn't seem to help. Instead, what I would like to do, if that's OK with everyone, is pull up a set of slides. Somebody was proposing to use the blackboard behind you, but I don't know if that is a good idea. Can you see the blackboard behind me? OK, I would also need to dig up some pens. Yeah, I don't know. I mean, I do have probably not. It's probably not clear. You know, I do have my kid's iPad is kicking around somewhere. I could try and find that. That would be the most fun. No, but I mean, if you have slides, it's perfectly fine. You know what? OK. Yeah, I'm still getting an unknown error code one. Yeah, I'm getting an unknown error code on Zoom. You know what? Why don't I try? Can we all take a five minute coffee break while I go try and find my kids iPad and see if I can get that hooked up? I'm sorry about this, everyone. You know, I'll personally refund your admission fees. I hope they didn't charge any admission fees. OK, I'll be right back. All right, sorry about this. Unfortunately, it still doesn't seem. I can't seem to join the Zoom meeting here. Alex, what error are you getting when you try to join with a new device? OK. It says, I think it's there's a firewall or proxy setting that's not letting any of my devices join on Zoom. So when I try with my tablet, I'm getting an unknown error. When I try with my iPad, I'm getting make sure your connection to Zoom is not blocked by a firewall or proxy. I wonder if how I see T.S. people. It might be. Yeah. Is it possible that you that are you there? I'm here at Massimo. So the speaker, Alex Maloney, yeah, he's being blocked by a firewall and connecting out his previous device was not working. And now when he's found a different device, he can't connect it. So I think there might be an automatic blocking of these devices to our network. Yeah. I mean, to the Zoom pool somehow. Zoom is not. No, it's working fine on my computer. But whenever I try and connect either my Windows tablet or my iPad, I'm getting messages that I'm using the ID or the registration link. I'm using the meeting ID. Just meeting ID and password. Meeting ID and password. Yes, just the meeting ID and password. Yeah, it says. And there is no reason that. I'll try again. Second device cannot. Because you are not. There was no registering for this meeting is correct. No, there was a registration. There is a register. Are you using the same email for both? I mean, I'm just trying to join the meeting directly. No, Massimo, there was a registration because we're using UNESCO Zoom. But once you have the Zoom credentials, I think anyone can log in with it in theory. Let me check. It could be that because this is one UNESCO Zoom and it's not our usual Zoom. Yeah. Let me check if there is maybe could be that. You have to be unique. Can you register with another email also to have a double? Well, there's no, I don't understand. As far as I understand. Excuse me, can you repeat? You don't need the email to. Yeah, it doesn't. It doesn't use the email address. Log in. I just get log in credentials. You didn't have the registration to Zoom. Just check. Copy and paste. Now there was an invitation, the registration link. There was a registration link for Zoom. Yeah. I can enter in edit because we are joining the room. But is there a prop, is it not letting me use the same link to sign in from two devices? Is that the problem? I think that might be the problem. This could be the problem, yes. Yeah, so I see. So it doesn't want me to log in simultaneously from a tablet and from the computer that I'm using. Yes. That doesn't make sense because Irene was using two devices. Yeah, and I was doing this yesterday and it worked fine. But yesterday you used both tablets in the computer. Yeah, yeah. So I didn't change any preferences or details on the meeting, so it should be like yesterday. Yeah, I'm not quite sure what to say. I'm neither able to join through the iPad or through my tablet. Shall we go with plan C, Alex? Yeah, I'm gonna try this one more time. Sorry about this. Well, the only difference to me seems like before you joined, like before the recording started and now it's after that my business is somehow related. I post the recording anyway. Okay, now I'm getting... All right, let's give this a try. Okay. You entered with another device? Yep, now it's letting me on. Okay, that's good. Which mic you use on the second device? I just put the name Alex. Let's see if that works. It says waiting. Let me check. Ah. No, there is no waiting room. It says Connecting. Okay. I give you the code for the device. Maybe it can help. Let's see. All right, it seemed to be... Okay, here I am. Okay. It's okay. Now I just need a notepad. So anyone who uses iPads, is there must be a notepad application? If there's nothing you stole, you should be able to just use the notexa that should allow you to write on it. That I can use what? Just the generic Apple notepad. It should allow for handwriting. Like, it's not amazing, but... It's not great. Okay, that's fine. But most of the other ones are paid for. Yeah. You guys can see my iPad. Great. Good. You guys can see this okay? Yeah, there might be a long delay. Yeah, good. Okay, great. Okay. Okay, okay, this will be fun. Uh, okay. So where were we? Can you read this okay? So we speculated, okay, broadcast to Zoom has stopped due to attempted to start invalid broadcast section. Uh-oh. Yeah, we don't see... You guys can't see anything. It's frozen. Yeah, I got kicked off Zoom again. It's possible that it's something to do with my wireless network here, but I'm just surprised because now it seems to be working again. Don't get me on these kind of issues so far. Now we can see our tablet iPad desktop. Okay, good. Now it seems to be working. You guys can see this okay? Okay, we'll see how this goes. Okay, it's an adventure. We're all in this together. Okay, good. So we speculated, and let me know if this thing crashes at some point or if you can't hear me anymore. We'll see how this goes. So we speculated that a more correct version, or we might wanna say more complete version of the ADS-CFT correspondence is one that relates to theory of gravity in asymptotically ADS space to some probability distribution over the space of conformal field theories. So in this picture, a quote unquote fancy theory of gravity, and this would be one that is completely UV complete. And in particular a theory of gravity where we can completely understand the microscopic physics of black holes. That should correspond to some sort of delta function distribution on the space of CFTs where this theory of gravity is dual to an exact conformal field theory. That is to say a single conformal field theory. On the other hand, the problem with such fancy theories of gravity is that they are nearly impossible to study precisely in the bulk. And so instead, if one studies a simple theory of gravity, perhaps such as quantum general relativity, this would correspond to some non-trivial probability distribution over many conformal field theories. And this has the advantage that it is not a complete UV complete theory. So it's not UV complete. There is no sort of complete understanding of black hole micro physics. But nevertheless, we can understand for example, the path integral theory of the theory. Exactly. So can I ask, I guess, for the last one? Please, please, please, please. With this non-UV completeness and not complete understanding of black hole micro physics, how do you see this in the CFT then? Because in principle, okay, you need to take a distribution now, but in principle, I would imagine I would be able to complete anything in the CFTs, right? So good, good question. So the point is that one of the characteristic features and this is something that we'll get into in a minute, one of the characteristic features of these quote-unquote simple theories of gravity is that they don't have a discrete spectrum. So for example, physics in ADS is like physics in a box. In particular, the spectrum of the theory, including the spectrum of black holes is discrete. And so for example, if you want a picture of a black hole in anti-decider space, you should think of it as a black hole in a box whose size is the ADS radius. And this black hole is in equilibrium with its own Hawking radiation. So you sort of think of it as being an equilibrium with its Hawking radiation. So in any complete theory of quantum gravity in anti-decider space, you would expect a discrete spectrum of states. And indeed in any conformal field theory, at least in any standard compact conformal field theory, the spectrum is going to be discrete. So in particular, the spectrum of operators, which is the same as the spectrum of states because of the state operator correspondence is going to be a discrete spectrum. So in any sort of single compact conformal field theory, you're going to get a discrete spectrum. But all of the attempts to obtain a discrete spectrum directly from a gravitational path integral seem to have failed with a few very specific exceptions that we'll get to in a minute. And all good things must come to an end and the screen now has disappeared again. And I seem to have gotten kicked out of this Zoom session. I am not quite sure what to do here. So we could, I don't know if this is a problem with my internet here, with the devices that I'm trying with the way that Zoom is configured at ICTP or what? So none of the other speakers have had this issue as far as I understand. Is that correct? Yeah. I mean, also in your case yesterday it was fine. So this is quite... Yeah. I mean, I can just try and reconnect because that was working for a minute. Otherwise, I mean, other possible options could be either if you have some slides, we could go through them or we could do some discussion in which we could maybe you introduce some ideas without formulae. Yeah. I mean, I suspect, I mean, okay. I'm sharing my screen again. I strongly suspect it's gonna work for a few minutes and then stop again. I have a sneaking suspicion. It might have something to do with my internet here. Where was the notes application? I mean, what we could do is this appears to be working right now. So why don't I proceed? Go ahead. Yeah. And then we'll see how it goes. Okay. Go ahead. Yeah. Because what else can you do in life? Right? Okay. So yeah, I don't wanna waste... I don't wanna spend too much more time mucking around with tablets and stuff like that. So let's just talk physics and if it stops working I'll just pull up some slides from a talk that I gave. And then we can have a discussion and then we'll get this all sorted out for our lecture tomorrow. So... But what I really wanna talk about from the physics point of view is that there is a real precedent for this in condensed matter physics. When we study disordered systems. Okay. And it seems to have frozen again. You guys can't see this. Okay. Now it seems to be continuing. It's flowing. Okay. It's flowing. It's flowing. It's slow. You know, I wonder... Okay. I'm gonna stop talking about this but I suspect it's the fact that it's something to do with my internet at home which is unfortunate. Okay. In any case, when we studied the physics of disordered systems in a condensed matter setting, what are we doing? Well, we're typically studying a very complicated system. So for example, you could imagine studying something like the Ising model where we have a bunch of spins that are interacting pairwise. Okay. So here I've written down the Hamiltonian of the Ising model where we have a bunch of spins that are interacting via some couplings that I've called Jij here. Now in the traditional Ising model, these Jijs would all just be some constant. This is an exactly solvable system that you all probably know quite well. But if we were studying a more complicated system, then those Jijs could be complicated numbers that are essentially random. And if these Jijs are some very complicated numbers, then this would be a very difficult problem to solve exactly. And in particular, if the Jijs are sufficiently complicated, this might be spin glass, for example, in that case, this would be described by the famous Sherrington Kirkpatrick model or the Edwards Anderson model. Okay. And now my tablet has disappeared again. But in that case, just to complete the thought, in that case, the individual Hamiltonian that I wrote down just a second ago is very difficult to study exactly. And so the approach that condensed matter theorists take is rather than trying to study an individual Hamiltonian with very complicated coupling constants that are essentially random, you treat those coupling constants as stochastic variables, as random variables. And you consider the ensemble of Hamiltonians where those coupling constants are all pulled from some statistical distribution. And the remarkable thing is that these ensemble average theories are often much easier to study. And in some cases are, in a sense, exactly solvable, even though the individual Hamiltonians that make up the ensemble are almost impossible to study. Okay. Some people are suggesting on the chat window that I use the whiteboard. Unfortunately, I don't have any markers for the whiteboard. So that's not gonna work. So just to complete the thought, however, in this analogy, what we're imagining is that a theory of gravity, a simple theory of gravity described by a path integral is something much like the averaged theory that is used to study the Sherrington-Kirk-Patrick model, whereas the UV complete theories of gravity are the individual instances of the ensemble, the individual Hamiltonians, which have some very complicated coupling constants, and these are the things that make up the ensemble. These are the things like N equals four super Yang-Mills theory. And so, for example, the fact, if you took a spin glass, that's an actual chunk of material, which has got a Hamiltonian and a Hilbert space and all of that good stuff. And it has a discrete spectrum. It has a discrete spectrum of states, but just a spectrum that's very, very hard to study. And so, if you want to instead go to the more coarse, you know, ensemble average description of the system, then that's a description where you sacrifice something. You sacrifice an exact knowledge of the microstates of the system. You sacrifice a discrete spectrum and all of that stuff. But the reward that you get for making that sacrifice is that you then get a system which is exactly solvable. Okay? You get a system where you can, you know, compute course features of things like the density of states or the statistics on the density of states exactly. And so, the way that I think about these ensemble average theories is that they're like the post-average version of the Sherrington Kirkpatrick model or the Edwards Anderson model. And so, what we're doing here is we're doing essentially a conformal field theory approach a version of the same physics of disordered systems that condensed matter theorists study that, you know, for example, you know, it's a good week to talk about spin glasses, right? Because we have Parisi giving a colloquium at the end of the week, right? And so a lot of the inspiration, I think, for this particular point of view of the ADS CFD correspondence comes from this condensed matter analogy. So we have a couple of different ways that we could proceed. Let me ask you the following question. How readable is this? This is not very readable. Because one thing, because I do have written notes, is that readable at all? Is that backwards? It looks backwards to me. Not backwards. That is, if you hold it close, it's, it is actually readable, yeah. It is actually readable. Okay. I mean, this is like, yeah, I don't know. Back in the olden days, when people gave lectures on blackboards, you didn't have to deal with things like Zoom crashing in the middle of your seminar. So it is readable. Okay. So maybe what I would suggest is we could, I could proceed, yeah. I mean, is this insane? It might be insane, but that doesn't mean it's the wrong thing to do. What I could just do is just read through some of these things and then take breaks to answer some questions. Does that, does that sound reasonable? Someone suggestes I can turn off the camera to free the wifi line. That's possible. I suspect that's not going to help. Okay. I could scan them and post them on the Slack. I was planning on doing that anyway. Okay. But maybe at this point, you can tell us in word and then if there is a problem. What did I tell you? I mean, we spent 50, it's been 50 minutes. It's been 50 minutes. I don't want to continue, I don't want to continue this process indefinitely. So maybe what I would suggest is that we turn the rest of this meeting today into a question and answer session and then we'll try very hard. What I'm gonna do is go to my office and try and give my lecture tomorrow from my office where hopefully all of these problems will be resolved. But I would suggest that we just turn the rest of this lecture today into a question and answer session if that's okay. Sorry again about this. I mean, I swear I've taught three classes and given God knows how many seminars using exactly the setup and this is the first time it's happened. Good. So please, please. Yeah, yeah, yeah. Okay. Now I can answer questions because I'm gonna be devoting my full attention instead of trying to fuck around with my tablet at the same time. Okay. And then maybe we can stop the recording at this point since it's gonna be a... Yeah. If the recording of this vanished down the memory hole and was never posted on the internet, I would not necessarily object. Okay, there was a question. Can I go or is it... Yeah, yeah, yeah, please. Okay. So the goal is not then now. So the goal is not now to start, again, really understanding really the microscopic states of this black hole physics. It's more to have some effective description, find the distribution which resembles what we know should happen in black holes. Yeah, yeah. I mean, I think the thing that is remarkable about this set of examples that we're considering here is that they reproduce a hell of a lot of the microstructure of black holes, even though they don't reproduce the discreteness itself. And this is a point that I think was probably originally made by Schenker and collaborators in their paper on black holes and random matrices, which is that even though when we talk about trying to reproduce the micro physics of black holes, the gold standard would be to reproduce the exact spectrum of states, a discrete spectrum of states. But that is a very tall order. And I think a lot of the progress has been made by realizing that there are a lot of things that you can understand about black hole physics without trying to necessarily reproduce the exact discrete spectrum of microstates in some model of gravity. So for example, I think the original motivation for this might come from Cardi's formula, which is a universal formula obeyed by conformal field theories in two dimensions, which reproduces black hole entropy in terms of the asymptotic density of states of conformal field theory. But it's a density of states, it's not a discrete spectrum. And any individual conformal field theory of course will have a discrete spectrum, but it's only in some coarse grain sense that Cardi's formula is reproducing that. So Cardi's formula is reproducing an approximate density of states at high energies, whereas any individual conformal field theory will give you an exact microscopic density of states. But I think what Schenker and Friends sort of very, very clearly articulated is that you can go much farther than just talking about the density of states. You can talk about the statistics of microstates. So the density of states would be something like some smeared version of the density row of the number of states. But you could also talk about things like the two point function of the density of states or the three point function of the density of states or something like that, which would be some average or some smeared version of the square of the density of states or the cube of the density of states. And so for example, one of the things that I think really motivated this field is the recognition that there are gravitational saddle points that reproduce things like the two point function of the density of states, not just the one point function of the density of states. So for example, if you think of a black hole entropy as being described by, you know, the semi-classical black hole entropy is gonna be an average of the density of states at high energy. You could also look at the average of the two point function of the density of states, which is telling you not just about the density of states but telling you about, for example, the level repulsion between the density of states. And so for example, this is something that we encountered yesterday when we talked about this very simple Euclidean-Warmel solution in two dimensions. We decided that the cylinder amplitude, so for example, in that very simple model of 2D gravity that I was starting to describe yesterday, we had a disk partition function, which is giving you the trace of E to the minus beta H, the average value in our interpretation, but you also had a cylinder amplitude that was telling you the variance of trace of E to the minus beta H. What is that? That's the two point function of the density of states. And that thing will encode level repulsion, okay? So for example, a very famous feature of probably one of the defining features of chaotic systems is that they exhibit level repulsion. So if you have an interval system, then typically the states of the system all pile one on top of one another because the states live in representations of some gigantic symmetry algebra. But on the other hand, if you have a chaotic system, then typically all of the energy eigenstates are non-degenerate and they'll repel from one another. And so for example, this wormhole amplitude is telling you about the statistics of the two point function of black hole microstates. And so I think one of the crucial insights is that even though you have a gravitational theory that might not reproduce the exact micro physics, that is the exact discrete energy spectrum, nevertheless you can still reproduce more refined features than just the density of states. And so I think this JT gravity exhibits that very clearly. U1 gravity related to the Norena ensemble exhibits a version of that. That is a system where you can even reproduce some of the microstructure, exact micro physics of these black hole states from a gravitational interpretation. Although it's still an ensemble average theory. Good. There's a question, Upamanya. Yeah, thank you. So if I understood correctly earlier today in response to one of the questions, you had suggested that different members of the ensemble of which ensemble of CFTs on which the probability distribution is defined, there could be different bulk duels for the different CFTs. So I was wondering whether it would be appropriate to say that the gravitational theory which is dual to this ensemble of CFTs is also dual to some ensemble of other gravitational theories. Or is there a two-digit agreement? I think that's inevitably what you're led to. We think of a simple gravitational theory as some average of many fancy gravitational theories. And that is of course, it resonates with the sort of effective field theory approach where you might have one low energy theory of physics with many different UV completions. And indeed, I think that's exactly the right point of view to take on this subject. I think the thing that I want to articulate clearly though, which is different from the standard effective field theory approach is that usually an effective field theory, you don't include, you don't quantize, you don't study non-perturbative effects and effective field theory. You don't study a high order quantum effects and an effective field theory. But these are all sufficiently simple theories of gravity that you can actually sum up a full series of non-perturbative effects and get an exact, and match that with an ensemble average of CFTs. In the JT gravity example, you can compute all perturbative effects and all non-perturbative effects. And the theory is still missing the what people call doubly non-perturbative effects in order to get an exact sort of UV complete theory, but you're almost there. And similarly in this Narene theory, when one studies the ensemble average of these free boson conformal field theories, you get something which includes all perturbative effects and all non-perturbative effects of a theory of gravity. Now it's a rather exotic kind of theory of gravity. It's not really a theory of metrics or it's not, I think, best understood as a theory of metrics, but rather is something closer to a gauge theory, but still it has many of the characteristics of this area of gravity. I have one related question. The different members of the ensemble of CFTs. So what features, I mean, at least what features do they have or should they have absolutely in common? For example, should there, I mean, let's take ADS-3 gravity, for example, should their central charges lie between some particular range or is there any restriction that forbids them from having two different, I mean, having some order of magnitude difference in these central charges? So I just want to... The answer is nobody really knows. Okay, so if you ask me what I think is true, I think it is likely that three-dimensional pure gravity and ADS should be understood as an average over all two-dimensional conformal field theories. I don't, what I don't know is what the correct probability distribution on that space of theories is, although I think based on the analogy with some results, okay, I think there's some plausible guesses for what that measure is, but I don't think, for example, we know exactly how the coupling constant of the theory of gravity, which would be the Newton, you know, the ADS radius and Newton's constant, the dimensionless coupling should map on to the central charge of the dual ensemble of CFTs. So the standard answer that you would guess based on the Brown and O formula is that the central charge is related to the ADS and Planck units. But I don't know that that is completely correct. I mean, I think it could also, that may be something that is true on effectively at large central charge in the semi-classical limit, but it's not obvious to me that it shouldn't be modified in some way, you know. So for example, I think it's perfectly plausible that the ADS radius in Planck units is not the central charge, but instead is something closer to, you know, a chemical potential conjugate to the central charge or something like that. That's something that shows up, for example, in symmetric product orbitals and in some of the examples that we understand a bit better. Good. Thank you. Thank you. Sorry, that answer was getting into some technical stuff that we haven't really covered in the lectures yet.