 So, let's start. Next speaker is Shu and Shao and he's going to talk about tough anomaly and modular bootstrap. Alright, thanks for the invitation, it's my first time to the ICTP. So today I'm going to talk about Tohoof anomaly and modular bootstrap. And this is in collaboration with Ying Ling from Caltech. Can everyone hear me well? No? Okay. Is it better? Is it okay? Alright, I'll speak out. Okay, and I would also like to thank the organizer for putting me at a very sweet spot to talk about this project because the various ingredients of this talk will have been covered in the previous lectures and colloquial. So the talk start was about two major non-perturbative tools for non-supersymmetric quantum field theory. The first one we just heard the beautiful colloquial about is a program of conformal bootstrap. Is to exploit the consistency of CFD data to constrain local operator spectrum, sometimes even about operators living on the defect. The other program is to study the symmetry and anomaly. This was partially covered by Zohar's lecture, even though he didn't use the exact language of anomaly and symmetry, but we will see a lot of the consequences are similar. So the hoof anomaly is an obstruction to gauging a global symmetry G. And the anomaly is subject to matching, namely, the anomaly has to be the same when you check duality. And the anomaly also has to be the same if you start with the UV theory with a global symmetry G and flow to an infrared theory with the same global symmetry. So it's a powerful non-perturbative tool to check duality and give you confidence. So these are two amazing tools that have been used in the past to study non-supersymmetric duality and RG flows. So the motivation of the talk today is to combine the two. So the concrete question we are going to ask is, in a CFT with global symmetry G and the hoof anomaly, let's denote it by alpha, is there an upper bound on the lightest G-charge local operator? If there's such an upper bound, how does the bound depend on the hoof anomaly alpha? This smells very much like a weak gravity conjecture question, but not in the context of CFT. And if time permits out, we'll probably talk a little bit about the relation to weak gravity conjecture towards the end. And the concrete setup I have today is probably the simplest setup you can imagine. So let's study two-dimensional conformal field theory. And I'm going to focus on bosonic CFT, namely, CFT that does not require a choice of the spring structure. And let me take the global symmetry to be Z2. This is going to be a unitary Z2. So the Z2 is not going to involve spacetime action. It's an internal unitary global symmetry Z2. And the question we would like to ask is, is there an upper bound on the lightest Z2 odd operator in your two-dimensional CFT? Kind of a natural question, because if someone comes to you and says, hey, I have a CFT with a Z2 symmetry, then you might be tempted to ask, can the Z2, can the Z2, can the lightest Z2 odd state be very, very heavy, so that as a low-energy physicist, when you study the CFT on a circle, say, you don't really see the Z2 symmetry, because all the light states are uncharged under the Z2. And we are going to see that the existence of this bound on the lightest Z2 odd operator actually has a lot to do with the Tohoof anomaly of the Z2. So I'm going to spell out the result in the beginning. So the result we find is that if the Z2 has a Tohoof anomaly, namely, if the Z2 cannot be gauged, then we will find a bound on the lightest Z2 odd operator. The bound will be a function of the central charge for the CFT. On the other hand, if the Z2 is non-anomalous, namely, if the Z2 can be gauged, then you will not find such a bound. So the moral of the story is actually another manifestation that when the symmetry is anomalous, it's harder to hide it in the infrared. So in Zohar's first lecture, he mentioned that if we have a Tohoof anomaly in the UV, that tells us that the vacuum cannot be trivially gauged. It's impossible for the vacuum to be gauged and has a non-degenerate, unique vacuum. The gap phase, we know a lot about the constraints of Tohoof anomaly. Either the symmetry is spontaneously broken or there's a non-trivial TQFT reproducing the anomaly. The talk today is going to be about how the Tohoof anomaly constraints the quantum system in the gapless space. Namely, how does it constraint the CFT data? Shuang, sorry. Can I ask you a question? Yes, yes, of course. It's the fact that I'm a discernment to talk with a microphone. Yes. So is it obvious and actually should be the case that if you don't have an anomalous, you shouldn't get a bound? Because if you gauge it, you should be able to decouple all the non-gaging variant operators. So this is the expectation you should have, right? I would say this is the expectation, but it's not totally obvious because it could be that when you do the Z2 gauging, what you are instructed to do is to throw away all the Z2 odd operator. Right. But they don't have to be lights, right? You just discard them and you add some twist field backing. But the remaining theory should make sense. So this will be an operator with absolutely no Z2 odd operators and that should make sense as a theory. Oh, but in two-dimension, when you gauge a symmetry, you are instructed to add the twist sector. So the theory without the Z2 odd operator by itself does not make sense. Right, the classic example is the 2D Ising CFD. In 2D Ising CFD, there's a Z2 global symmetry. The charge state is the spin field. The 2D Ising CFD is self-dual under the Z2 gauging. That's the Kramer-Wenner duality. So when you gauge the Z2, you throw away the Z2 odd operator, which is the spin field. But on the other hand, you are instructed to add a twist sector. The twist sector is a disorder operator. It has the same scaling dimension as the spin field. And that's why the Ising is self-dual. But it's inconsistent as a Bosonic CFD to just have the identity operator and the energy operator. Yeah, but still the full spectrum, the twist sector operator will be Z2 even. Under the original Z2, yes. Yeah, yeah, OK. But it does not make sense to talk about the original Z2 once you gauge it because it ceases to be a global symmetry. But your intuition is correct. And I'll give a simple example to elaborate on that intuition. And please feel free to interrupt me during the talk. All right, so that's the result. And I think that's the main take-home message that the Tohoof anomaly actually dramatically changes your bootstrap bound. OK, so here's a quick reminder on what do I mean by Tohoof anomaly. So a global symmetry with Tohoof anomaly is still a perfectly healthy global symmetry in a consistent quantum field theory. This is not to be confused with the ABJ anomaly, where in ABJ anomaly, the axial symmetry is not really a true symmetry. The two concepts are related, but it should be distinguished. A global symmetry with Tohoof anomaly is healthy. It's just that you cannot gauge it. But if you don't gauge it, nothing is wrong. OK, so the outline of this talk is the following. So we have two parts. The first part is about anomaly. The second part is about bootstrap. So let's start with the first part. So in two dimensions, so first I will discuss how am I going to characterize the symmetry and its anomaly. Notice that we are talking about discrete symmetry and discrete anomaly here. So it does not show up in the correlation function of local operator. So we need a better tool to discuss them. And the tool we are going to use are the so-called topological defect. It sounds like a very fancy word, but it's actually not. So let's recall the continuous global symmetry case. If you have a continuous U1 global symmetry state, then by the Nerther theorem, we know there is an associated Nerther current. And as you do in the first course of quantum field theory, you take the Nerther current and integrate on the co-dimension 1 manifold in your spacetime. Sometimes it's taken to be the constant time slice. In this way, you constructed the Nerther charge. And the exponentiation of the Nerther charge is the topological defect associated to the U1 global symmetry. So topological defect is just a fancy word in the case of continuous symmetry for the Nerther charge. It's topological because in the continuous case, you have the current conservation. And current conservation tells you that if you deform your co-dimension 1 manifold in your spacetime by a little bit, correlation function is not going to change. That's the meaning of the word topological. Oops. OK. But in the case of discrete symmetry, I will not have a Nerther current associated to it. But still, there should be a co-dimension 1 topological defect, which will be denoted by L, that implement the symmetry transformation. By that, so let's say we are in two dimensions. Then co-dimension 1 means that it's a line. So I have a line in my two-dimensional CFT, and there's a local operator phi. By implementing the symmetry, I mean if I bring the line, if I deform the line a little bit to the left, nothing is there. The line is topological, so no correlation function should change. On the other hand, if I deform the line to the right and pass through this local operator, the correlation function will change by a sign. And if it's plus, that means the phi operator is z2 even. And if it's minus, that means the local operator is z2 odd. So this is the object we are going to play during the talk today. It is the topological line for z2 global symmetry. So this is just a summary of what I was saying. Whenever you have a global symmetry, there should be a co-dimension 1 topological defect. OK, so here I'm just repeating myself. The most elementary property of the topological line is that it's topological. So if you deform it, and if you do not pass through any local operator, no correlation functions change. The second thing we are going to talk a lot about is the defect-huber space. So given a two-dimensional CFD, you can try to quantize it on a circle, an S1. But with the topological line, you can insert the topological line vertically to the spatial circle. This insertion with the spatial S1 modifies the quantization. In the case of z2 symmetry, this is quite explicit. That just means that without the line, you quantize the theory by periodic boundary condition. With the line, you quantize it with a z2 twist. So as you bring your wave function, as you bring your field along the circle and back to itself, you just give it an extra sign. That's the z2 charge of your field. So the insertion of this topological line changes your quantization. Therefore, we are talking about a different huber space, which will be denoted as h sub l. This huber space is not the same as the huber space of local operator because of the change in the quantization. And this will be called a defect huber space. In the case when the z2 is non-anomalous, you can gauge the z2. And the z2 even part of the defect huber space becomes the twisty sector in the orbital theory. But when the z2 is anomalous, you cannot gauge it. But it still makes sense to talk about this modified huber space in the ungaged theory. And that's going to be our focus. Now, one nice thing about a topological defect is that since it's topological, it commutes with both viralsaurial algebras. So the huber space, this defect huber space, the states there should still be organized into representations of both the left and the right viralsaurial algebras. Now you can run the same operator state correspondence for the defect huber space. So a state you prepare in this modified quantization will be mapped to an operator living at the end of the line. This is not a local operator, but it's an operator attached to a line. And you should not freak out when we talk about such a non-local operator, because we talk about it every day. In QED, when we talk about electron, electron is not a local operator, because all local operators are gauging variant. The proper way to talk about electron is that electron is attached to a Wilson line. So electron is exactly of the same nature as this mu operator we are talking about here. And the mu operator via the operator state correspondence corresponds to a state in the defect huber space. Now comes to the thumb part of the story. There's a crossing relation for the topological line. Let me restrict to the Z2 case. So what do I mean by that? So imagine you are trying to compute a correlation function in your 2D CFT. The correlation function involves local operator, as well as topological line. And you look at the little patch that's shown in this gray circle that's not very visible here. You look at the little patch here, and supposing the patch, you just have these two parallel topological lines. No local operator inserted. Then I can imagine I cut off this little patch from my correlation function and replace it with a different configuration. Note that the end point of this patch are identical on the two sides. But I just reconnect the line in a different way. You may ask, are the two configurations identical, or they differ by a sign? Namely, does the correlation function remain the same or change by a face under such a replacement? So let me hypothesize that the two correlation functions are the correlation function would change by a number alpha under such a crossing. Now there's a constraint on alpha, because I can do this replacement twice on the right figure to land on the original figure. So I will conclude alpha squared is 1. That leaves me two solutions. The first one is alpha equals to plus 1, and the other is alpha equals to minus 1. And here I claim, when alpha is plus 1, it corresponds to a non-anomalous Z2. It is a Z2 global symmetry that you can gauge. When alpha is minus 1, it corresponds to an anomalous Z2. It's a Z2 you cannot gauge. So this crossing relation of the topological line is a very non-lograngian abstract way to characterize the anomaly. And that's the thing we are going to rely on. And as you probably know, the Bosonic Unitary Z2 Anomaly in 2D is classified by the third group homology of Z2 with U1 coefficient. This group homology is Z2 value, and alpha equals to plus 1 corresponds to the trivial element here. Alpha equals to minus 1 corresponds to the non-trivial element here. And the previous condition that alpha squares to plus 1 is the co-cycle condition of the third group homology. You don't have to know all this fancy language to see that there are these two solutions. So here I just make the claim that alpha is related to the anomaly, but I didn't explain why. So let me explain why in the following slides. Okay, so let's say I have this crossing phase, alpha. But let me be a naive person. I'm just going to gauge the Z2 anyway. I don't care about all this to-hoof anomaly nonsense. I'm just going to do it. And if something wrong, I will be convinced that there's anomaly. Otherwise, I'm free. All right, so how do we compute observable in the Z2 gauge theory, namely in the orbital theory? Let's say I want to compute the torus partition function in the Ubi Z2 orbital theory. I put the word Ubi because when the Z2 is anomalous, such a theory does not exist. So this little box is supposed to represent the torus. So you identify the site here with the site here and you identify the site here with the site here. So the torus partition function of the orbital theory or the Ubi orbital theory is computed in terms of the partition functions in the un-gauge theory through this combination. The first two terms is usually called the untwisted sector. There's a one half and there's a topological line running here. And this sum and the one half here just means that we are projecting down to the Z2 even state in my original Hilbert space. And similarly, you introduce the twisted sector, namely the defect Hilbert space where you modify your quantization by vertical topological line. Again, you sum over two terms because you only want to keep the Z2 even state. So you have two sectors. For each sector, you do a projection. So in total, you have four terms. Let's look at the last term. This is the problematic one. The other three can be defined perfectly unambiguously but the last one has some ambiguity. That's why the quotation mark. There's a cross in the last term. You have to make sense of this cross. It does not follow immediately from our previous definition of the topological line, what this cross could mean. There are two possible resolutions for this cross. You can either resolve it this way or you resolve it in the other way. These two configurations make sense while this generally is ambiguous. But these two configurations are exactly related by a crossing relation, related by this move. And when alpha is plus one, these two resolutions are identical and therefore this configuration can be defined in terms of either one and you get the same answer either way. On the other hand, when alpha is minus one, the two resolutions differ by a sign and this is ambiguity in computing observable in the orbital theory and therefore signaling the anomaly. So this is the justification for why this alpha phase corresponds to the Tohoop anomaly or in other words an obstruction to gauging in two-dimensional CFT. There's a more advanced way to explain it but let me not do it. I think this is the most hands-on way to do it because if you ignore it, you try to compute the torus partition function in the orbital theory, it just fail miserably. Okay, so let me just recap. There's an interesting crossing relation where the topological line implements the Z2 symmetry. The crossing relation is classified by this alpha phase. When alpha is plus one, the Z2 is non-anomalous. That means you can gauge it. When alpha is minus one, it's anomalous so you better not gauge it. That's the first part. And the second part will be about how to use the thing we talked about to formulate it into a bootstrap equation. So as in any bootstrap program, there are two ingredients that you would like to have. The first one is the notion of positivity. The second is a crossing symmetry. Here in my context, the positivity will be the positive expansion of the torus partition functions onto the Viracero characters. So these Viracero characters will be the analog of the conformal blocks in the usual conformal bootstrap. And crossing will be the modular S transformation. The object we are going to talk about is the torus partition function. But let me remind you, this is in the ungaged theory because we want to incorporate both the anomalous and the non-anomalous case. So the ingredients will be the torus partition functions but possibly dressed with various topological lines. So this is the standard quantization of my 2D CFD on a cylinder with no topological line. And I get a Hilbert space H, that's the Hilbert space of local operator. And as is standard, I can decompose my torus partition function into, I can write the torus partition function as a trace interpretation. It's the trace over the Hilbert space of local operator. And it can be decomposing to the Viracero character, this chi H here with positive coefficient. Now we can address the configuration with some topological line. Let's first do it by wrapping the line horizontally along the spatial circle. This represent the action on your Hilbert space of local operator. The insertion of this line corresponds to putting a Z2 action in your trace. Now this guy is not positive definite because the coefficient here could be negative. So a Z2 even state will contribute positively to the partition function while a Z2 odd state contribute negatively. So this is the second player. And there's a third player, which is the defect Hilbert space we just talked about. So we can also try to compute the torus partition function of the defect Hilbert space. So now the trace interpretation is that we are tracing over a different Hilbert space compared to this one. And with no Z2 insertion. And as we were saying, the line is topological. So again, this defect Hilbert space should be decomposable onto the Viracero characters with positive coefficients. So that's the positivity statement. We have three torus partition function, the one without any line, the one with the horizontal line and the one with the vertical line. If you take the sum and difference, you get a torus partition function but with trace only over the even and the odd state. And that has a positive expansion on the Viracero characters. And for the defect Hilbert space, it's by itself positive. So these are the three positive torus partition functions. Now let's take a look at the defect Hilbert space here. If I perform a T transformation in SL2Z on the torus partition function here, I land on this configuration. If I do a T inverse, I land on this configuration. And the two configuration, as we were saying, are related by a crossing. And I might get a phase alpha when I do this crossing. This implies that the defect Hilbert space, namely this configuration, is invariant under T square if alpha is plus one. But you have to do it twice to go back to the original configuration if alpha is minus one. So the modular property of this torus partition function depends on the anomaly. And that's the key step on how this discrete Z2 anomaly enter into the bootstrap equation. So what does this mean? If I have such a, if the torus partition function is invariant under just T, that means all my local operators have integer spin. If it were invariant under T square, that means the spin has to be an integer or half integer. That would be the case for non-anomalous Z2. The spin here is the difference between the holomorphic weight and the anti-holomorphic weight. On the other hand, if the torus partition is invariant under T to the fourth, that will lead to a funny spin in my defect Hilbert space. It has to be a quarter plus an integer or a half integer. For local operator, the spin has to be an integer because when you circle two local operator around each other, you want them to be mutually local. But here we're not talking about local operator. Here we're talking about defect Hilbert space. And as you might recall, this state in a defect Hilbert space corresponds to operator living at the end of a line, of the Z2 line in this case. And they do not have to be mutually local with another local operator. Because as I bring a local operator state starting from here and do a two pi rotation back here, the local operator might get acted upon by the topological line. And therefore the operator mu does not have to have integer spin. And that's what we saw in the result here. In fact, when the Z2 is anomalous, you see that the spin of such non-local operator is never an integer. This is an interesting spin selection rule in the defect Hilbert space. Okay, so we spend a lot of time talking about the spin of a non-local operator. But eventually we would like to say something about our local operator. That's the results I show in the beginning of the talk. So how do we convert the information on non-local operator to information of local operator? The way to do that is to do modular transformation. So the logic is that we start with discrete-to-hoop anomaly. Naively discrete-to-hoop anomaly doesn't seem to constrain local operator data. Because it's discrete, it doesn't show up in correlation functions. On the other hand, through a rather elaborated argument, we saw that the non-local operator data is constrained by the discrete-to-hoop anomaly. And finally, we are going to use modular transformation to constrain the data on the local operator. So now let me just write down the crossing relation. For the ordinary torus partition function, it's well known that it has to be S invariant. So this just goes back to itself. But now if I put a topological line here, recall that this torus partition function has the interpretation as the trace over the local operator Hilbert space, because the line runs this way. But when you do an S, that becomes a torus partition function over the defect Hilbert space. But the two are related by a modular S transformation. And these are the crossing relation for the bootstrap equation. And let me just remind you that the anomaly enters through the spin content in this torus partition function over the defect Hilbert space. So this is just a summary of what I have talked about. So there are three torus partition function, the one without any line, the one with the horizontal line, and the one with the vertical line. These three are positive, and they go to each other under S in this way. And the anomaly enters into the defect Hilbert space torus partition function. So now with the positivity statement and the crossing statement, you can do the usual bootstrap to derive bound on operators in each sector. And as I emphasize here, the anomaly enters into the equation explicitly. And here is the result I displayed earlier that we found that, okay, so we can talk about bound in various sector. I can talk about bounds on the lightest Z2 even state. I can talk about bound on the lightest twisty sector state. But let me focus on the most weak gravity conjecture-like bound. Namely, is there a bound on the lightest Z2 odd state? Is there a bound on the charge operator? And we find that if the Z2 is non-anomalous, there's no bound. And when the Z2 is anomalous, there is a bound. And here is the numerical bound we obtained. So delta minus is the upper bound on the lightest Z2 odd operator. And the horizontal axis is the central charge of the conformal field theory. And anything below this blue line is allowed. And here, all these colorful points are the examples of the WZW model. And our bound is saturated by the SU2 at level one WZW model. That's the self-dual boson. And let me emphasize again, this is for the anomalous Z2. In the case when the Z2 is non-anomalous, there's no bound, namely the bound is infinite. Yeah, yeah, yeah. So it's the usual numerical bootstrap technique. So you write down the crossing relation. So now you have three different partition functions. And you try to organize the, maybe let me use the blackboard. So you have a vector Z. So these are the three torus partition function that has a positive verisoral character decomposition. And under crossing, they are related to each other by a non-trivial crossing matrix. This is a three by three matrix. This matrix can be read off by comparing these two expressions. And as usual, so this is more like a mixed co-relator bootstrap. So then you just move this term to the other side and expand both quantities in terms of the verisoral characters. And you know the coefficient has to be positive. And then you do use the linear functional method to derive a bound. Yes. Which anomaly, which gravitational anomaly? Oh, sorry. Yes, I assume C left equals to C right. Thank you. Can you repeat the question? Oh, sorry. Yeah. The question was, do I assume the gravitational anomaly to be zero? The gravitational anomaly that was referred to here is the left central charge minus the right central charge. And I do assume that to be zero here. I should have mentioned that. Thank you. Okay. Is there any other question? All right. So here's the bound. And let me just comment a little bit why I find this result interesting. And I think there are two points of view. First as a bootstrapper or as an anomaly. I'm not sure if this is a word, but let me just use it. For a bootstrapper, global symmetry generally help us target the CFD we want to bootstrap. Like in the colloquial, we just heard from Leonardo that if you want to bootstrap, for example, the 3D O2 model, you use the O2 symmetry a lot. And that constraints your bootstrap equation and help you hunt the theory you want to get. But simply saying that a CFD has a global symmetry is not the most refined information you can provide. CFD with the same global symmetry but different anomalies are quite different. So you'll be more desirable if you can specify the Tohoof anomaly for the global symmetry also in your bootstrap program. This is relatively easy to do if you are talking about a continuous global symmetry. Because for a continuous global symmetry, the continuous Tohoof anomaly will enter into the current correlation function. And that's something that shows up explicitly in your bootstrap equation. For discrete anomalies, that's obvious how to do it. And the result I showed you earlier is one way to do it, at least in the context of two-dimensional CFD. And as we just saw, even the very existence of a bound might depend on the anomaly. So it's obviously some important information you want to put in when you do bootstrap. So this will be my viewpoint as a bootstrapper. As an anomaler, I think it's also interesting because as we were saying in a gap phase, discrete Tohoof anomaly has dramatic consequences that we use all the time. But there's always a way out is to assume that maybe the IR is in a gapless phase. And people sometimes just throw their hands and say that, oh, it might be a gapless phase and it's some complicated CFD matching the anomaly. But what else universally or precisely you can say when there's an anomaly? And what we just showed is that discrete anomalies in a gapless CFD phase also has dramatic constraints on the spectrum of local operators. So this is an interesting result on how Tohoof anomaly constrained the gapless phase in a quantum system, not just a gap phase. So let me mention a few generalizations. Let me start with 2D CFD with U1 global symmetry. To specify U1 global symmetry in two dimensions, it comes with a holomorphic current and an anti-holomorphic current. And here the U1 symmetry is generated by the exponentiation of the integral of the current. And this is the Nether charge associated to the U1 symmetry with theta rotation. And in two-dimensional compact unitary CFD, if I'm given U1 global symmetry, we know that unitary bound actually separately implies that J and J bar are conserved. So you might often hear people saying that in unitary compact 2D CFD, if you have a U1, then it must automatically come with another. Because you might hear people saying that J and J bar separately generate the U1. But that's actually not true globally. If I tell you a U1 global symmetry, the best you can say is that there's two component currents with J and J bar components that generate such a U1. It is true that J and J bar are separately conserved, but they might separately generate an R symmetry instead of U1. The global form for the symmetry group is absolutely important for the following discussion. So here I'm going to assume that I'm talking about a U1, but not R. And notice that this U1 global symmetry is generally not holomorphic because it comes with a holomorphic current and an anti-holomorphic current. If you discard either of them, generically they generate an R symmetry, not U1. Okay. So again, we will argue the same qualitative conclusion. Namely, when the U1 is non-anomalous, there's no bound on the lightest U1 charge operator. On the other hand, when the U1 is anomalous, there is generally a bound. The conclusion is qualitatively the same as the Z2 case. Now let me comment on the previous work. So the authors of this two group of people have considered bound on the lightest U1 charge operator in two-dimensional CFT with U1 global symmetry. The motivation for their work is that they want to show the weak gravity conjecture in ADS-3 CFT-2. So having a U1 global symmetry in 2D CFT, in a 2D holographic CFT, means that there's a U1 gauge symmetry in ADS-3. And saying that there is a bound on the lightest U1 charge operator in 2D CFT implies that there's a bound on the lightest charge particle state in your ADS-3 quantum gravity. And therefore, if one can prove such a statement for 2D CFT, that would be a proof for the weak gravity conjecture in ADS-3. That was one of the motivation for these two group of people. But I just told you that there's no bound on the lightest U1 charge operator if the U1 global symmetry on the boundary is non-anomalous. So how is that consistent with the previous work? Well, if you read into these two papers, even though if you read into these two papers, you'll see that they assume that U1 is holomorphic. In other words, they assume that U1 is generated by a holomorphic current without an anti-holomorphic component. A holomorphic U1 is always anomalous. Namely, it can never be gauged. This is the usual chiral anomaly in two dimensions. So the result is consistent with our observation. So they are always in this case. However, I would like to comment that there's no bound if the U1 is non-anomalous. Namely, there's no bound if the U1 can be gauged. Let me just give one example. So let's consider a 2D compact boson example. This was mentioned in the discussion session in the morning. So we're in two dimensions and we want to consider 2D free scalar theory. In two dimensions, you can give the scalar radius. That means you identify the scalar operator with itself by a constant. This constant R is usually called a radius of the compact boson. R is an exactly marginal deformation of the 2D free scalar theory. So the 2D free scalar theory is actually very rich. It's not an isolated CFT. It's a CFT that comes with a modular space, a modular space of exactly marginal deformation. At any radius R, there are always two U1 global symmetry. These are the winding symmetry and the momentum symmetry. There's a precise way to describe them, but the intuitive way is just that your target space is now a circle because of the compactification. So there's a KKU1 associated to that circle. But there's another U1 corresponding to, in the language of string theory, a string wind around the circle. So we have two U1 global symmetries. Neither is anomalous. The winding U1 and the momentum U1 are both non-anomalous. In particular, that means they are non-colomorphic. But there's a mix to hoof anomaly between the two. It means that if you gauge the winding U1, then the momentum U1 will no longer be a global symmetry in the gauge theory and vice versa. Put it differently, you can consider the diagonal U1 of this two. This generally is not a holomorphic U1, but this will be an anomalous U1. In this simple example, let's try to see if there's a bound on the lightest U1 charge operator. So there are three U1s here, the winding U1, the momentum U1, and the diagonal subgroup of the two. The lightest winding U1 state is the winding mode. It's scaling dimension growth as you increase the radius of the compact boson. That's intuitively clear, because such a state is a string winding around the target space circle. So as you make the circle larger and larger, the mass grows. But then there's no bound. I can just take my radius R to be arbitrarily large and the lightest winding state can be arbitrarily heavy. And here I'm fixing my central charge to be one. So this statement is nothing deep. It's just saying that in 2D compact boson, just take the winding U1. There's no bound on the lightest U1 charge state. Conversely, for a momentum U1, the lightest charge state is the first KK mode. And its mass dimension, as we are familiar with, goes like one over R squared. Again, there's no bound. You just have to shrink the circle arbitrarily small and you can make the lightest KK mode arbitrarily heavy. But if you consider this diagonal U1 that has to hoop anomaly, then the lightest state is either the lightest winding state or the lightest KK mode. Its dimension is given by the minimum of these two. And there is an upper bound on the lightest charge operator with respect to this anomalous U1. So in this simple example, you immediately see the relation between a bound and the to hoop anomaly of the underlying global symmetry. But that's my comment on the U1. We didn't do much. We just want to point out that there's this winding U1. And the same qualitative result is identical with the Z2 case. Yes. I'm not exactly sure what does it mean to in... Well, sorry. Yes, I think that's correct, yeah. So yeah, sorry, you're correct. So the anomaly for U1 global symmetry is characterized by an integer. So let me call that integer, I don't know, n probably. And I think the bound will go like this. So delta has to be smaller. Delta is the lightest charge state. Has to be smaller or equals to C. There's a number. And I think this is only true for large C. C over n. Where n is the to hoop anomaly for the U1. So as you make it more and more anomalous, this bound becomes smaller. And when it's non anomalous, n is zero. So you do not get any bound. Lightest charge, in the holographic context, your lightest charge state will be very heavy unless n is also parametrically large, right? This is a different n than the n. Right, but C is very large in the holograph. But that's what you expect because in the weak gravity conjecture, there's a Planck scale in the statement of the inequality. And C is related to the Planck scale in ADS-3 CFD2. The dictionary will check out. Is there any other question? I'll talk about the second generalization. So we saw that the script of hoop anomaly has a very nice, has some interesting implication on the local operator for a 2D CFD. Does it generalize to higher space time dimension? So is there such an anomaly dependent bound on local operator in higher than two space time dimensions? The answer is no. So why? So it was shown by Jiuwen Wen, Xiao Gang Wen, and Edward Whitten that given a discrete unitary bosonic symmetry G and its anomaly alpha in D space time dimensions, there is a D-dimensional TQFT carrying this symmetry n anomaly. Notice that this is not an anomaly inflow. The TQFT is in the same dimension as the symmetry n anomaly. It's not an inflow mechanism. Okay, this is a proven result. How is this related to my previous question? When the space time dimension is greater than two, this TQFT is constructed by Wen, Wen, and Whitten as a unique vacuum. So it's a theory with trivial local operator but non-trivial anomalies. And therefore, if I'm given a quantum field theory, a non-trivial quantum field theory in D greater than two with some global symmetry G and anomaly alpha, I can just tensor product with this TQFT and modify the anomaly of my quantum field theory. Since the TQFT has a trivial vacuum and no other local operator, I did not change the local operator spectrum of my quantum field theory. So in higher than two space time dimension, I can change the anomaly of a quantum field theory without modifying the local operator spectrum. And therefore, there cannot be any bootstrap bound on the local operator that depends on the discrete unitary bosonic anomaly. Another way to put it is that in D greater than two, there's a TQFT which has trivial local operators but non-trivial anomaly. So anomaly has no implication on the local operator spectrum because the local operator is completely trivial here. In D equals to two, there's something accidental. The TQFT is constructed by Wen, Wen and Witten. Always have degenerate vacua. Namely, we are always in the spontaneous symmetry breaking phase. So if you remember Zohar's lecture, there are three options if you have to hoop anomaly. The first one is spontaneous symmetry breaking. The second is TQFT. The third is in a gapless space. But in two dimensions, the first two options are the same because all the non-trivial TQFTs have degenerate vacua. Therefore, always in the spontaneous symmetry breaking phase. So in two dimensions, such TQFTs has non-trivial local operators. And when you tensor it with the QFT, you modify their local operator spectrum. So you cannot modify the anomaly without changing the local operator spectrum in QD for QFT. But there might be other anomalies that cannot be carried by TQFT in higher than two dimensions. Opal symmetry anomaly cannot be carried by TQFT. So you will be interested to study how those anomalies constrain the local operator spectrum in quantum field theory in D greater than two. Okay, so let me still have a few minutes. Let me just say a little bit about, what about the non-anomalous case? The non-anomalous Z2 seems like a love child that we have forgotten since the beginning of the talk. So we know that there's no bound on the lightest Z2 odd, oh sorry, and I'm going back to the Z2 case. So we saw that there's no bound on the lightest Z2 odd operator if the Z2 is non-anomalous. This is the plot for the anomalous case that I showed before. So what can we say for non-anomalous Z2? There's an older disorder bound that we found. So we found that there is a bound, not in the odd sector itself, but in the unions between the odd sector and the defective space. This sounds a little bit random, but I think it has a nice physical interpretation. The lightest Z2 odd operator in many contexts can be thought of as older operators that detects the spontaneous symmetry breaking of the Z2 global symmetry. And the lightest defective space operator can be thought of as a disorder operator. So what our result is saying that the older and the disorder operators cannot both be too heavy in a CFT with a non-anomalous Z2. There was a paper by Michael Levine two weeks ago. He showed a similar, he showed an analogous statement for a gap 1D spin chain. He showed that in a gap 1D spin chain, you cannot have a non-zero older parameter and a non-zero disorder parameter. I'm not sure exactly how to translate that result for the gap phase to the result I have for a gap less space, but it's quite tantalizing that they are related. So for example, in the 2D icing CFT that was mentioned in Leonardo's talk, the older operator is the spin field. It's a local operator, that's the Z2 odd. The disorder operator, which is usually denoted as mu, it's an operator, it's not a local operator. It's an operator living at the end of the line. This is the simplest example of the older disorder operator. And what we showed is that they cannot both be too heavy. Here's the numerical bound we found and it's saturated by the theory of freefer males, and in particular by the tensor product of two icing model when C is one. Okay, so let me get to the conclusion. So in this talk, we asked some kind of weak gravity conjecture question in CFT that is there a bound on the lightest charge operator? And in 2D CFT, we saw that the bound dramatically depends on the to-hoof anomaly of the global symmetry. In particular, there is a bound if the symmetry is anomalous, but not otherwise. And we tested this observation in the case of Z2 and U1 global symmetry. And this is a manifestation on how the discrete-to-hoof anomaly even constrained local operator in a gap-less space. So let me comment on a few outlooks. The first one I already mentioned that it would be nice to generalize how anomalies constrained local operator for anomalies involving space-time action. So perhaps the time reversal anomaly. And also for anomalies that cannot be carried by TQFT, they might also have similar consequences on local operators. For example, the continuous anomaly for say a 4D continuous group. And lastly, let me just make some rather speculative comment. So I mentioned that previous work on constraining the lightest U1 charge operator was partially motivated by the weak gravity conjecture in ADS-3 and CFT-2. But we saw that through this simple free boson example that there's no bound on the lightest U1 charge operator if the U1 is non-anomalous. So is there an interpretation of this from the weak gravity conjecture or from ADS-3? There is kind of an interpretation and let me say it at the very end of my talk. So first we have to ask what does to-hoof anomaly mean in the context of ADS-CFT? So I have a boundary 2D CFT. It has a global symmetry and the global symmetry has a to-hoof anomaly. If you want to couple this global symmetry to a gauge field in ADS, to cancel the anomaly, you have to add a Chern-Simons turn in ADS-3. But if you have a to-hoof anomaly, that means there's a Chern-Simons turn in ADS-3. As we learned from Zohar's lecture yesterday, a Chern-Simons turn in three dimensions will make the photon massive. When the photon is massive, charged particles are not confined because a massive photon cannot mediate long-range interaction. Since the charged particles are not confined, you can probably run the weak gravity conjecture argument through the charged black hole emission. On the other hand, if the U1 global symmetry on the boundary is non-anomalous, that means there's no Chern-Simons turn in the bulk. So in the bulk, you have a Maxwell-Matter theory in ADS-3. And also as we learned from Zohar's lecture yesterday, if you do not have a Chern-Simons turn, there's classical confinement in three dimensions for a Maxwell theory. So charged particles will be confined in the absence of Chern-Simons turn. And therefore you cannot run the weak gravity conjecture argument. And maybe that's why it's consistent with the CFT statement that there's no bound on the lightest U1 charge operator. This argument is flawed because I assume there's a single U1 gauge field in the bulk. If I have multiple U1 gauge fields, I have to take the mixed Chern-Simons turn between them into consideration, which I did not. And that's why I say the comment is speculative. So I think it would be good as a future direction to discuss the interpretation or implication on weak gravity conjecture. And thank you. Thank you very much. So just to be, just to understand it concretely, so this anomalous, basically in modular, if you have some, it looks like asymmetric orbifold. It reminds me of asymmetric orbifolds which don't satisfy a level matching or some motto condition. Are these Z2 anomalies that you're talking about some kind of chiral asymmetric orbifolds which are not guaranteed to be modular and variant? Is there, it looks like there must be a relation of these anomalies with some level matching condition or some motto condition not working out. Well, I think it might be a little bit, well, Alice is confusing to me to think about level matching in this context because we're not necessarily doing Warsh's string theory. So level matching, I don't think it's very natural from a CFT consideration. But you want to get a modular invariant. It's not related to that. I mean, for example, if I just take a free boson, Yes. Presumably if I try to consider an asymmetric orbifold by some Z2 symmetry, then it will not lead to a consistent conformal field theory, even apart from thinking about string theory. Yeah, it might be related. I think if we cannot lead to a modular invariant partition function, that means the Z2 is anomalous. But I wasn't, I think the asymmetric orbifold will be one example of such Z2 anomaly. I just wasn't phrasing in that language. Okay. And I think this language should incorporate that particular case. I think the notion of asymmetric is particularly confusing in the case of Z2 global symmetry because you have to say what do you mean by asymmetric. Asymmetric usually means that you do something different on the left and on the right. So you have to talk about what does it mean by holomorphic Z2. But that could be a dangerous notion because, for example, we can take the SU2 at level one WZW model. If you look at a left SU2 current algebra, you can say, oh, let's talk about the center Z2 of the left SU2 current algebra. That seems like a holomorphic Z2, right? It's the center of the left SU2. But in the SU2 WZW model, it actually acts in the same way as the right center. And therefore, the two Z2 are identified. So in that sense, that Z2 is not really asymmetric, but still anomalous. Let me just phrase it differently. So let me just say that again. In that particular example, it's actually the example I show here, the saturating example here, this guy. This is SU2 at level one WZW model. It has a Z2 center symmetry. It's not asymmetric. That Z2 acts in the same way on the left and on the right, yet it's anomalous. So I would say asymmetric over it might be a special case of Z2 anomaly. I wonder if there is, do you know if there might be some interpretation of this bounds in terms of inflow, TQFT? Like for example, the charged operator should end on some line operator of the inflow. I mean, of the TQFT in three dimensions which cancels the anomaly. Yeah, let me comment on that. So let's go back to the spin selection rule. So let's comment on this one quarter. So there are a million ways to get the number one quarter, but let me show one way to get the number one quarter that's related to Zohar's lecture yesterday. So if you remember Zohar's lecture yesterday, he was talking about U1 at level K, 3D transiment theory. He beautifully explained that the spin of the nth anion in 3D U1 transiments n squared over 2K. Now let me take K equals to two and n equals to one. And this becomes one quarter. I think there's some way to, yeah, okay. And I claim that's the same one quarter over there. I think the reason, let me answer Pavel's question. I think the answer is the following. So if I have a Z2 anomaly, the 3D SPT is A cup, A cup, A. A cup, A cup, yeah, A cup, A cup, A. And now we can try to, once we couple to the SPT, we can gauge the theory. But the gauge, the 2D, 3D system. If I had not coupled to the 2D boundary, the 3D SPT with A cup, A cup, A will give you the 3D digraph written theory, the Z2 3D digraph written theory with the Z2 twist. And that theory is U1 at level two, tensor product with U1 at level minus two. So you have a spin one quarter anion and a spin minus one quarter anion. And I think once you gauge this 2D, 3D system, so you see that the spin here could be one quarter plus an integer and minus one quarter plus an integer. So I think once you gauge the 2D, 3D system, half of them will be attached to the spin one quarter anion and the other half will be attached to the spin minus one quarter anion. And this spin selection rule must be there before you have to have consistent braiding between the anions. That's my interpretation. I didn't work it out explicitly. That's questions. If not, let's thank Shushan Gege.