WEBVTT 00:00:09.000 --> 00:00:14.000 continuous spacetime symmetries. Take it away, Hinyaki. 00:00:14.000 --> 00:00:20.000 Thank you very much. Thank you very much for inviting me to give a talk. Thanks everybody for coming. 00:00:20.000 --> 00:00:25.000 Um, I want to discuss this paper that we. 00:00:25.000 --> 00:00:32.000 put out now almost half a year ago with Fabio, Nicola, Hotat, and Sakura. 00:00:32.000 --> 00:00:40.000 And it's about how to understand in the language of symmetry TFTs, which I will introduce. 00:00:40.000 --> 00:00:49.000 spacetime symmetries. 00:00:49.000 --> 00:00:55.000 So let me start by discussing what the symmetry TFT is. It's a kind of topological theory. 00:00:55.000 --> 00:01:01.000 That's useful for understanding symmetries in general categorical or generalized symmetries. 00:01:01.000 --> 00:01:04.000 Um, so let me try to introduce what those are. 00:01:04.000 --> 00:01:11.000 Let's go back to the standard case that we learned in textbooks or standard symmetry generators. Imagine that you have an. 00:01:11.000 --> 00:01:25.000 A billion continuous symmetric. And then you have a generator for the symmetry, which in the path integral language you can write as the integral of a sum d minus one surface. So, for example. 00:01:25.000 --> 00:01:31.000 In the standard case, you will choose this to be a space, or a space like slice. 00:01:31.000 --> 00:01:39.000 And then you integrate the nether current. In my conventions, the nether current is a D-1 form. If you want to translate this to more familiar statements. 00:01:39.000 --> 00:01:48.000 This would be something like the integral over a space of the charge. So when a specific component, which is typically written as J zero. 00:01:48.000 --> 00:01:52.000 PM implicitly acting with the star just because you'll do Macomb. 00:01:52.000 --> 00:01:57.000 So when you put these symmetry generators in the path integral. 00:01:57.000 --> 00:02:02.000 you can generate the action of symptoms already know how that works. 00:02:02.000 --> 00:02:11.000 So here J is another current. And inside the path integral, the another current satisfies the word identity. 00:02:11.000 --> 00:02:17.000 So, it's not quite conserved, it's almost conserved. Except that it gets some contribution. 00:02:17.000 --> 00:02:26.000 from the insertions of charged operators. So imagine that you put operators OK with charge QK. 00:02:26.000 --> 00:02:33.000 at points xk. So Dj fails to be closed precisely at this point by delta function. 00:02:33.000 --> 00:02:41.000 And how much it fails to be closed at those points is the amount of chart that you have at those points. 00:02:41.000 --> 00:02:50.000 So one of the fundamental observations in this field is that you can think of symmetry operators as topological operators. 00:02:50.000 --> 00:02:56.000 At least away from this delta functions, away from the insertion of the charged operators. 00:02:56.000 --> 00:03:01.000 So in mind that you have one such integral, something like this. 00:03:01.000 --> 00:03:08.000 And you deform sigma a little bit to some sigma prime. 00:03:08.000 --> 00:03:17.000 Now I want to compare the symmetry generator at Sigma and the symmetry generator at sigma prime, so I'm going to multiply by the inverse. 00:03:17.000 --> 00:03:29.000 Um, just because of this exponential form, you get the integral of… So, the exponential of alpha times the integral over sigma prime minus sigma in J. 00:03:29.000 --> 00:03:34.000 Now, because sigma and sigma prime are related via slight deformation, there will be some. 00:03:34.000 --> 00:03:40.000 D-dimensional manifold B. Such the boundary of B. 00:03:40.000 --> 00:03:52.000 is sigma prime minus sigma. So if you imagine taking sigma and deforming tilt a bit, it will be the kind of the inside with two boundary sigma prime and sigma. 00:03:52.000 --> 00:04:06.000 Now you write this as… Boundary of B, your price talks, you get DJ, DJ is this delta functions because of the word identity, and if you assume that none of the points, or you have inserted the operators are in B. 00:04:06.000 --> 00:04:16.000 Then you get the identity. So this means that in the path integral, this operator here is the identity or equivalently u alpha and sigma and u alpha in sigma prime. 00:04:16.000 --> 00:04:24.000 They give you the same results in the partition. 00:04:24.000 --> 00:04:28.000 So there was a starting point. This is a very. 00:04:28.000 --> 00:04:33.000 all observation, but Gyotoka pushed in Cyber and Willett 12 years ago. 00:04:33.000 --> 00:04:40.000 argue that this viewpoint of topological operators generalizes in many useful ways. 00:04:40.000 --> 00:04:52.000 Um, so the picture you described was something like this. We have a charged operator. Let's now put it inside some UK, and we surround it by the charge generator and we do the same calculations before. 00:04:52.000 --> 00:05:00.000 We act with UAlpha, which has this form. Now, sigma written here, is the boundary of some B. 00:05:00.000 --> 00:05:11.000 We get the integral of… e to the alpha… Integral on B. We get e to the alpha integral on B of dj. 00:05:11.000 --> 00:05:18.000 And because of the word identity, you can precisely in this phase, which is what you expect. The presence of a charged operator. 00:05:18.000 --> 00:05:21.000 is detected by the symmetry generator by a sin interface. 00:05:21.000 --> 00:05:25.000 Just put a portion out to the church. 00:05:25.000 --> 00:05:33.000 argued is that it's very useful to do this for any sort of, uh. 00:05:33.000 --> 00:05:39.000 operator, not just point-like operators. Here I'm drawing a… P-dimensional operator. 00:05:39.000 --> 00:05:46.000 So, if you want to do this game of surrounding the operator, and then filling it in and applying the word identities. 00:05:46.000 --> 00:05:53.000 The thing that surrounds and links the B-dimensional surface, it's a D minus p minus one. 00:05:53.000 --> 00:05:56.000 surface, where D is the total dimension of the spacetime. 00:05:56.000 --> 00:06:08.000 In the previous case, we have a D minus 1 surface, something that surrounds a point. In this case, when this is not the point, but p-dimensional, we get a D minus p minus 1 surface. 00:06:08.000 --> 00:06:17.000 You can apply the same logic other than this, there's a word identity that assigns a phase when you contract this. 00:06:17.000 --> 00:06:28.000 which is e to the alpha. Same SKU, the charge for… I'm going relative quick because I assume that many of you have heard a variation of this, uh. 00:06:28.000 --> 00:06:33.000 Discussion in many contexts, but if you have questions, please do interrupt. 00:06:33.000 --> 00:06:39.000 I'm more than happy to stop and… 00:06:39.000 --> 00:06:48.000 Okay. So… We saw that symmetries are… Any symmetry and standard symmetry gives a topological operator. 00:06:48.000 --> 00:06:52.000 The realization in the last few years has been that. 00:06:52.000 --> 00:06:56.000 It's useful to sort of make this connection stronger. 00:06:56.000 --> 00:07:05.000 And… define symmetry to be any topological operator in your theories, and then you have a topological operator, we say that we have a symbol. 00:07:05.000 --> 00:07:10.000 Now, the symmetry that you get out of this construction, not all topological operators are group-like. 00:07:10.000 --> 00:07:25.000 Not all topological operators are as standard as. The ones we are used to in the textbooks, but this happens to be useful. This is a useful organization that we can extend previous results. 00:07:25.000 --> 00:07:32.000 So, the ones I just described, the political operators of co-dimension P plus 1. So like the ones. 00:07:32.000 --> 00:07:36.000 Here, this could mention p plus 1, B dimensions. 00:07:36.000 --> 00:07:48.000 We call p-form symmetries. And that's because the objects that they measure are the charged objects are p-dimensional. So here I'm following the language in the original paper by Gayo Kapustin, Chaver. 00:07:48.000 --> 00:08:04.000 In particular, organized images I'm going to call zero form symmetries, because they measure point like operators that is 0 dimensional. 00:08:04.000 --> 00:08:18.000 All right, so that was fairly standard. And I imagine many of you have heard of the variation of this introduction. That's why I was being quick. What I'm going to describe now is something perhaps less well known, but I think very useful also. 00:08:18.000 --> 00:08:23.000 Which is that if you're going to talk out the set of topological operators in any quantum field theory. 00:08:23.000 --> 00:08:28.000 We have realized, since the last five years or so, 5, 6 years. 00:08:28.000 --> 00:08:33.000 that it's very, very useful to take some sort of holographic, if you let me use this word. 00:08:33.000 --> 00:08:40.000 perspective on these topological operators, and instead of talking about topological operators in D dimensions. 00:08:40.000 --> 00:08:49.000 We're going to talk about the topological field theory in one higher dimension. And we call this topological theory, the Sim TFT or symmetric theory. 00:08:49.000 --> 00:08:54.000 So there have been many papers that have discussed this symmetry theory. 00:08:54.000 --> 00:09:07.000 And here's a selection. One thing I want to… Highlight, because I thought it was a nice convergence, is that some of these papers are condensed matter papers, some of these papers are string theory papers, and some of these papers are not papers. 00:09:07.000 --> 00:09:16.000 And everybody sort of agreeing. sort of reaching this… Description, which seems to be really useful. So what is MTF? 00:09:16.000 --> 00:09:19.000 Well, it's a description of the symmetries of a theory. 00:09:19.000 --> 00:09:26.000 In which, instead of working with a d-dimensional theory, you build a sandwich, like the one I'm describing here. 00:09:26.000 --> 00:09:36.000 So, I'm extending the D-dimensional picture to D plus 1 dimensional picture. I'm adding one extra dimension, which is the horizontal one in this picture. 00:09:36.000 --> 00:09:45.000 And the local degrees of freedom. Live on the right boundary, in this picture, which I'm calling P tiller. 00:09:45.000 --> 00:09:56.000 So you should think of this as the local degrees of freedom of your quantum field theory. Imagine that you are doing n equal to 4, so this would be the genomes and the. 00:09:56.000 --> 00:09:58.000 Gage bosons and the scalars are equal to 4. 00:09:58.000 --> 00:10:05.000 And these all live in this gapless bundle. Then there is a five-dimensional topological theory. 00:10:05.000 --> 00:10:13.000 which is known as the symmetry theory, which admits these local degrees of freedom as edge modes. 00:10:13.000 --> 00:10:20.000 And this symmetry theory also admits gapped boundaries, topological boundaries. 00:10:20.000 --> 00:10:21.000 Yeah. 00:10:21.000 --> 00:10:26.000 All right, Yaki, the T tilt. The right boundary doesn't have… the right boundary theory doesn't have to be gapless. 00:10:26.000 --> 00:10:37.000 It doesn't have to be, no, sorry. I have in mind construction of Gabbler's theories through this construction, a complete write. I could do the whole thing up. 00:10:37.000 --> 00:10:38.000 It could be like, yeah, the whole thing could be cut. 00:10:38.000 --> 00:10:43.000 It could very well be gapped. For example, you can understand the different Yang Mills theories for. 00:10:43.000 --> 00:10:44.000 for different. global forms of the gauge group, just using this picture. 00:10:44.000 --> 00:10:50.000 Correct. 00:10:50.000 --> 00:10:56.000 Yeah, yeah, yeah, definitely. 00:10:56.000 --> 00:11:11.000 Um, bye. So… As Gurya said, for example, if you want to understand the different global forms of many theories, I will give an example in a second, but Dan Mills, for example, the thing you would do in this picture. 00:11:11.000 --> 00:11:19.000 is to choose the left boundary gap, although there is completely right, the red one covers a bigger gap. 00:11:19.000 --> 00:11:30.000 So you can choose in different ways and all these different choices, they give you the different global structures for the quantum field theory that you are. 00:11:30.000 --> 00:11:33.000 I think it's also important to say that rho is topological. 00:11:33.000 --> 00:11:46.000 Mm-hmm. Yes. Sorry, I… Yeah, yeah, sorry. Yeah, Andrew. 00:11:46.000 --> 00:11:58.000 Alright. So, in this kind of description, if we wanted to construct the four dimensional theory, the language that I'm drawing is a 5-dimensional picture. There's one extra dimension, which is the horizontal one. 00:11:58.000 --> 00:12:07.000 And we typically speak of… collapsing the sandwich. This is the topical theory that's not really very meaningful, but I think it's a good way of. 00:12:07.000 --> 00:12:17.000 Get an intuition for what's happening. So we bring these two boundaries together. And we obtain an effective 4-dimensional material, three dimensional image. 00:12:17.000 --> 00:12:25.000 This topography theory, we have some topological operators in them. And once they hit the left boundary. 00:12:25.000 --> 00:12:29.000 They may or may not survive, depending on the elephant. It might become trivial or not. 00:12:29.000 --> 00:12:33.000 So some of them will become trivial, some of them will survive faster. 00:12:33.000 --> 00:12:43.000 So in this way, you can generate topical operators in a d-dimensional theory, starting from a set of topological parameters in d plus 1 dimension. 00:12:43.000 --> 00:12:50.000 So here's one example. If you take n equal to four superjam means in four dimensions with gauge algebra SUN. 00:12:50.000 --> 00:12:56.000 This completely fixes the Lagrangian, and once you fix, I get tau, the complexity of the coupling. 00:12:56.000 --> 00:13:00.000 But as a quantum theory, you still have choices to make. 00:13:00.000 --> 00:13:11.000 Typically, this is described as the choice of global form for the gauge group, so the algebra is UN and meets a global form factor gauge group, which is just UN. It could also be PSUN. 00:13:11.000 --> 00:13:20.000 And here, I should also mention that there's an extra set of data, some sort of discrete theta angles that were fully classified by 107 Tachikawa. 00:13:20.000 --> 00:13:29.000 Will not need to go into those, but… There's a bit of extra exercise to do sometimes, even if you fix the global form of the screen. 00:13:29.000 --> 00:13:33.000 But once you specify this finite set of data, you have specified what the theory is. 00:13:33.000 --> 00:13:39.000 One way of classifying or to understand the distinction here is that. 00:13:39.000 --> 00:13:44.000 Different choices of global form have different one forms of interest. 00:13:44.000 --> 00:13:50.000 In this picture. Um, the way you obtain the different forms, so SDN or PSUN. 00:13:50.000 --> 00:13:58.000 is by choosing different roles. different topological boundaries for the symmetry, topography. 00:13:58.000 --> 00:14:09.000 In this particular case, if you want to classify all the different global forms for SGN, the thing you need to pick for the symmetry TFT is some discrete. 00:14:09.000 --> 00:14:11.000 Gage theory, in particular, I said N gauge theory. 00:14:11.000 --> 00:14:14.000 I'll give the Lagrangian for this theory in a few slides. 00:14:14.000 --> 00:14:21.000 But what you would do if you want to understand this theory, is put the local degrees of freedom for an equal to 4 on the right. 00:14:21.000 --> 00:14:38.000 put the set engage theory, and then you classify all the possible boundaries, all the possible topological boundaries for symmetry. And in this way, you will reproduce precisely the classification of harmonies. 00:14:38.000 --> 00:14:45.000 Any questions? 00:14:45.000 --> 00:14:58.000 Okay. So… This construction might remind you of most of the talk will be to try to connect this closer to holography for a specific class of cynicals. 00:14:58.000 --> 00:15:03.000 So far, I think it's fair to say that we understand. 00:15:03.000 --> 00:15:08.000 quite well, I would say. In the case of discrete symmetries, like this. 00:15:08.000 --> 00:15:13.000 Set in symmetries that would appear in this case. In this case, these theories differ by. 00:15:13.000 --> 00:15:20.000 The one-form symmetry, which are set to 10 symmetries, typically. 00:15:20.000 --> 00:15:28.000 We simply understand things better, but in the last couple of years, we have started understanding how this picture generalizes to. 00:15:28.000 --> 00:15:40.000 Um, to continuous symmetries. And the same sort of picture is still true, is believed to be true if you complicate the symmetry theory a little bit, and the way I will describe it. 00:15:40.000 --> 00:15:45.000 But the same picture seems to… seems to… 00:15:45.000 --> 00:15:53.000 Now, everything, pretty much, that has been understood has been so far on internal symmetries. 00:15:53.000 --> 00:16:03.000 And what I want to discuss is our paper, which is a first attempt, trying to understand how continuous spacetime syntries, such as translations of rotations. 00:16:03.000 --> 00:16:11.000 Could fit into this picture. It's not a given that you could do this, but I think it does, at least. 00:16:11.000 --> 00:16:18.000 And here I want to emphasize that I'm dealing with continuous space-time symmetries. In particular, I want to think about the park. 00:16:18.000 --> 00:16:22.000 spacetime symmetries that are connected to the identity. So translations. 00:16:22.000 --> 00:16:38.000 I connect to entity or location connectivity. That still leaves the problem of incorporating discrete spacetime symmetries, such as term reversals or reflections, into the same TFT, and here I'm much more lost. Here, I don't know what to do. 00:16:38.000 --> 00:16:50.000 I think it's a very important problem. But it's not clear to me that this… discrete spacetime symmetries, such as time traversal. 00:16:50.000 --> 00:16:59.000 How they fit in the same picture, if they fit at all, certainly not. 00:16:59.000 --> 00:17:02.000 Okay. So yeah, that's what we do in the paper. 00:17:02.000 --> 00:17:08.000 The sympia, one of the things I didn't emphasize, but one of the things that makes it quite useful. 00:17:08.000 --> 00:17:16.000 is that even if it's reminiscent of holography. It does not really need to have a holographic tool. 00:17:16.000 --> 00:17:24.000 topography here in the bulk is, in some sense, an abstract representation of symmetries, and it does not necessarily have to come from. 00:17:24.000 --> 00:17:32.000 But whenever you have a holographic dual, there should be a way of relating sim TFT. 00:17:32.000 --> 00:17:41.000 with the holography, the holography picture. In holography, you can still talk about symmetries in the park. 00:17:41.000 --> 00:17:45.000 And this symmetry should be there. in a way that's compatible with MPL. 00:17:45.000 --> 00:17:50.000 So I'm going to start with cases that do have equivalent. 00:17:50.000 --> 00:17:57.000 Right, so I was just speaking of the SUN and equal to 4 theory, 4 dimensions. 00:17:57.000 --> 00:18:06.000 And as I said, Haroun7 classified all the possible global forms, so SGN and PSUN, and many others. 00:18:06.000 --> 00:18:12.000 Um, so how would you understand all these different global forms, holographic? 00:18:12.000 --> 00:18:20.000 And this was answered very soon after the finished paper by Whitten in 98. 00:18:20.000 --> 00:18:30.000 And what he observed was that if you look at the action of type 2B on S5, so ADS5 versus 5 with 10 units of plaques on the S5. 00:18:30.000 --> 00:18:38.000 If you reduce this action on VS5. You have to really be careful because of the shows like this and whatnot, but the effective action that you get in. 00:18:38.000 --> 00:18:44.000 In ADS 5 is of this one, two pi i n. 00:18:44.000 --> 00:18:54.000 times the integral of ADS5, which I'm writing as X5, simply because… Unit topology to extinguish the global form, so you probably want a small generalization of ADS5. 00:18:54.000 --> 00:19:08.000 Or B2H3C2. He argued that these effective action dominates the analysis of large differences. Now, there's an important subtlety here that was pointed out by Belva Moore later. 00:19:08.000 --> 00:19:12.000 So you have to be quite careful with single stones. 00:19:12.000 --> 00:19:15.000 I'm assuming that we are forgetting all the singleton. 00:19:15.000 --> 00:19:23.000 I'm assuming the existence of boundary conditions. Uh, for which is a singleton dash notification. 00:19:23.000 --> 00:19:35.000 This hasn't been done in complete details. You know, if you look at the boundary conditions that you have to put in ADS. 00:19:35.000 --> 00:19:53.000 In this system, with this Ladrangia. You'll realize that, um… The choice of boundary conditions is in one-to-one correspondence with the classification of global forms in a hierogeneous hybrid and particle. So within the SUN and PSUN case and in this paper. 00:19:53.000 --> 00:20:00.000 Years later, we reproduce the full classification upon gene SUM, but you know. 00:20:00.000 --> 00:20:05.000 Classical knee transplant. 00:20:05.000 --> 00:20:11.000 Um, and the way you do this is you look at the boundary state, and the boundary conditions in ABS. 00:20:11.000 --> 00:20:27.000 By quantizing this BF theory. using the radial direction as time, so you… the Hamiltonian quantization of this theory, and the boundary condition will be a specific state in the Hilpert space of this theory. 00:20:27.000 --> 00:20:30.000 So what you can argue with this paper in detail. 00:20:30.000 --> 00:20:41.000 Is that the Hilbert space of this theory precisely gives you the choices between SUN and PSU. 00:20:41.000 --> 00:20:50.000 So… One of the things you will notice and one of the things that motivated the Syn-PF deconstruction was that. 00:20:50.000 --> 00:21:01.000 This Lagrangian here is precisely a discrete setting gauge theories, a representation of the discrete set 10 theory. 00:21:01.000 --> 00:21:09.000 So, if you forget about everything else about holography and just keep this Lagrangian and classify it to political boundary conditions, you will still be able to reproduce all the global forms. 00:21:09.000 --> 00:21:22.000 And in doing this, you can… You can obtain a construction that is valid also when n is small. We don't need n large for being able to quantize all the. 00:21:22.000 --> 00:21:32.000 to quantize this theory. So the sympia fees in a way distillation of the aspects of ADS/CFT. 00:21:32.000 --> 00:21:38.000 Which, um… To have to deal with global structures, I think, as one way from standpoint. 00:21:38.000 --> 00:21:45.000 In the cases where you have a holographic motion. 00:21:45.000 --> 00:21:54.000 So… What we then argued is that… 00:21:54.000 --> 00:22:02.000 If somebody wants to ask, please let me know, because I cannot actually… I can actually introduce it. 00:22:02.000 --> 00:22:16.000 So Witten's argument shows that the sim TFT comes from the near boundary behavior of B2 and that's what he argued. Again, this is effective hasn't been very carefully resolved. 00:22:16.000 --> 00:22:24.000 Let's proceed. You might ask you might ask in other examples where you have a holographic description. 00:22:24.000 --> 00:22:34.000 Um, can you sort of have the same picture? That is, can you extract the same TFT from the holographic data in some way? 00:22:34.000 --> 00:22:38.000 And in other examples that we know, although we don't have a general proof. 00:22:38.000 --> 00:22:44.000 The same TST seems to arise from the limiting behavior of the Park theory near the conformal bundle. 00:22:44.000 --> 00:22:56.000 That is, if you look at ADS very, very close to the conformal boundary of ADS, and you keep the… Extremely large distance behavior, which tends to be some sort of TFK. 00:22:56.000 --> 00:23:06.000 Then that thing should be the simplicity. And there has been checked in many examples, apart from Wittens. Here's two that I. 00:23:06.000 --> 00:23:12.000 find particularly nice. certainly the only ones. 00:23:12.000 --> 00:23:21.000 Um, so one of the things that makes these categorical symmetries store interesting is that the symmetries that we obtain don't have to be group-like. 00:23:21.000 --> 00:23:31.000 Um, and whenever we have symmetries that are not group-like, typically we call them non-invertible, so this is a… An exciting new class of symmetries that we have been playing with. 00:23:31.000 --> 00:23:41.000 And you might ask, where are… This non-invertible symmetry that people have found in quantum field theory in the holographic tool. So you can find. 00:23:41.000 --> 00:23:47.000 For example, 4-dimensional theories with a holographic dual that do have non-invertible symmetries. 00:23:47.000 --> 00:23:50.000 How did these things appear in the holographic one? 00:23:50.000 --> 00:23:58.000 These papers, for example, it was argued. that non-invertebral symmetries appear in the holographic dual, that is in string theory. 00:23:58.000 --> 00:24:05.000 From the physics of the brain in… In certain geometric structures. So D brains are, uh. 00:24:05.000 --> 00:24:24.000 And when you put them close to evaluate. Um, one of the things that made me interested in this whole story is that I was trying to… Understand what it was the symmetry structure for certain theories for which no Lagrangian is known, in particular. 00:24:24.000 --> 00:24:30.000 for the National Superconformity theories. n equal to 3, so torque supercharges. 00:24:30.000 --> 00:24:35.000 And… Any quantity? 00:24:35.000 --> 00:24:41.000 And for this series, we don't have a Lagrangian, but we do know what the holographic dual is. 00:24:41.000 --> 00:24:47.000 So, by sort of refining this idea of extending this idea of Whitten. 00:24:47.000 --> 00:24:58.000 to the horrific duals of n equal to 3. We could work out what the symmetry structure of these theories was. 00:24:58.000 --> 00:25:03.000 Which is quite interesting because we know very little about this theory. They seem to exist. 00:25:03.000 --> 00:25:11.000 But they are quite mischief, so we don't have a good handle on them. 00:25:11.000 --> 00:25:16.000 So, I think it's fair to summarize that for all the cases that we understand. 00:25:16.000 --> 00:25:24.000 Uh, whenever the… You know, you have an anomaly-free zero-form internal symmetry that's sort of the most vanilla thing that you can construct. 00:25:24.000 --> 00:25:30.000 given by a group G. And then the same TFT is on BF here. 00:25:30.000 --> 00:25:37.000 In the case of discrete symmetries, this is something you can prove in the case of continuous ones. I think there are good arguments for this. 00:25:37.000 --> 00:25:49.000 Here, A is a connection in G. bees and form valued in the joint of G. 00:25:49.000 --> 00:25:55.000 So this is a very simple topological theory with Group G. 00:25:55.000 --> 00:26:00.000 And in all the cases that we know, this is the form of the BFC. 00:26:00.000 --> 00:26:12.000 Again, here I'm simplifying some things. So if you have anomalies, so… Your symmetries are more complicated, you have to complicate this a little bit, but that's okay. 00:26:12.000 --> 00:26:19.000 So, when we started thinking about the space isometries, the first thing that you could imagine trying. 00:26:19.000 --> 00:26:26.000 Is, well, every other example that we know of this type, we want to analyze some sort of BFU. 00:26:26.000 --> 00:26:30.000 So let's try to do BF theory for the spacetime symmetric. 00:26:30.000 --> 00:26:34.000 And let's see if this works in the way that that was. 00:26:34.000 --> 00:26:39.000 It was going to be the conjecture I'm going to test. 00:26:39.000 --> 00:26:46.000 And surprisingly. This seems to work well. The extent that we can test it, it seems to make sense. 00:26:46.000 --> 00:26:48.000 Quick question. Are you… 00:26:48.000 --> 00:26:53.000 Going to ignore the global form of the spacetime symmetry group? 00:26:53.000 --> 00:26:55.000 Since you're representing it in this… 00:26:55.000 --> 00:27:04.000 Yeah, yeah, yeah, exactly. So everything I'm going to say is going to be about the algebra, if you want to. I like to think of it as a. 00:27:04.000 --> 00:27:17.000 Transformations continuously connected today. And that's what I was mentioning before, I just don't know how to think about time traversal in this framework, for example. 00:27:17.000 --> 00:27:24.000 It's a very good question. 00:27:24.000 --> 00:27:31.000 Okay, so then… I think the first. 00:27:31.000 --> 00:27:39.000 thing that comes to mind when you're faced with this proposal is that, well, if I'm talking about the space-time symmetries in holography. 00:27:39.000 --> 00:27:45.000 I'm supposed to know how space-time symmetries act in holography. It has something to do with gravity in the box. 00:27:45.000 --> 00:27:49.000 Maybe you can think of ephemorphisms that don't vanish at infinity. 00:27:49.000 --> 00:28:01.000 This does not quite look like a BF tier. So where's this laying? Where's this hiding in gravity? 00:28:01.000 --> 00:28:07.000 So… Let me explain how that works in the one example that we understand. 00:28:07.000 --> 00:28:15.000 As usual in Adscft, you want to relate correlators. 00:28:15.000 --> 00:28:19.000 In… in the CFT, for example, correlators of operators are in this form. 00:28:19.000 --> 00:28:25.000 to partition function of a string theory with some boundary conditions. 00:28:25.000 --> 00:28:40.000 And you would ideally want to make sense of formula like this after some suitable regularizations, something that connects the result you will calculate in CFT for some observable with the partition functionality. 00:28:40.000 --> 00:28:46.000 So this is the basic form. In particular, for the case of zero-form symmetries, we know that the bulk. 00:28:46.000 --> 00:28:56.000 representation of a zero form symmetry is one form gauge field. So an A mu dynamical A mu. 00:28:56.000 --> 00:29:01.000 And, if you want to compute the correlation function CFT of. 00:29:01.000 --> 00:29:12.000 This kind of operator, so some nether current. A couple to some background for the symmetry, which I'm calling A0. So here, A0 is the background for the symmetry. The zero point symmetry. 00:29:12.000 --> 00:29:16.000 What you would like to do is compute the partition function of a string theory on ABS. 00:29:16.000 --> 00:29:29.000 with boundary values. So BD here is the boundary. The boundary values, which are given by the background of the plane. 00:29:29.000 --> 00:29:38.000 Alright, so let's translate this to the language I have been using of symmetry operators. A symmetry operator is an insertion. 00:29:38.000 --> 00:29:48.000 on the CFT at integral of e to the alpha integral of some d minus one dimensional manifold of the current. So it pretty much of this form. 00:29:48.000 --> 00:29:55.000 If I choose my boundary. value for H0, the background field, if you like. 00:29:55.000 --> 00:30:00.000 to be the form of a delta function. 00:30:00.000 --> 00:30:09.000 So this is already a bit different from the kind of thing one does normally in holography. We tend to prefer to work with a smooth backgrounds, because. 00:30:09.000 --> 00:30:15.000 From the point of view of topical field theory, this comes singular backgrounds appear non-natural. 00:30:15.000 --> 00:30:28.000 Anyway, otherwise it's the same… same… thing you would like to compute the partition function of a string theory with limiting value of this one. 00:30:28.000 --> 00:30:36.000 Um, so let's try to understand this sort of background in a… in a way that's useful for applications to gravity. 00:30:36.000 --> 00:30:40.000 Um, so if you choose this manifold where you're putting. 00:30:40.000 --> 00:30:50.000 Then the symmetry narrator, Sigma. To be the boundary of some D-dimensional manifold, so the boundary of some ball, for example. 00:30:50.000 --> 00:31:00.000 Then the background field, which is a delta function, you can understand as acting on the trivial background with a gauge transformation. 00:31:00.000 --> 00:31:12.000 Which is AC equals to a0 plus D lambda, as you saw. With the gauge parameter being constant in the ball and zero outside. 00:31:12.000 --> 00:31:21.000 I just want this essentially like a heavy side function. I'm going to take the derivative, you're going to get the delta function precisely where you want it. 00:31:21.000 --> 00:31:28.000 Um, so we're acting on the boundary conditions for A with the generator of gauge transformations. 00:31:28.000 --> 00:31:36.000 is the Gaul's law generator wrapped on. So this point of view is going to become useful in a minute. 00:31:36.000 --> 00:31:43.000 And here, when I'm talking about Gauss' law, again, I'm thinking about this Hamiltonian picture I was mentioning before. I'm thinking already or quantization. 00:31:43.000 --> 00:31:53.000 And there's a Hebert Space, which states are the boundary conditions. 00:31:53.000 --> 00:32:00.000 Alright, so to understand this in a way that's useful to my analysis of gravity in a minute. 00:32:00.000 --> 00:32:05.000 I'm going to represent Maxwell theory in the following way, which is perhaps. 00:32:05.000 --> 00:32:10.000 Not the… it's definitely not the standard Maxwell way, but it's equivalent. 00:32:10.000 --> 00:32:19.000 So I claim that in this theory, with two form field B and a one connection A, is equivalent to a standard Maxwell unit. 00:32:19.000 --> 00:32:22.000 And the argument is probably familiar to most of you. 00:32:22.000 --> 00:32:27.000 Um, here you notice that B appears quadratically, so you can integrate it out. 00:32:27.000 --> 00:32:37.000 And up to a one-loop term that discussed in detail in this paper, for example, you get Maxwell's action, the standard one for eight. 00:32:37.000 --> 00:32:40.000 Another thing that you could do, you don't have to, but you could. 00:32:40.000 --> 00:32:52.000 is integrate out A, which appears as Lagrange multiplier here. This imposes that B is pure gauge. Let me call that B is DC, where C is a one-form connection. 00:32:52.000 --> 00:33:10.000 And then what you get after integrated the Lagrange multiplier, so this term is gone, you get VC, which starts DC. So this is Maxwell theory with coupling 1 over e squared. So this is the standard argument for duality going back to… 00:33:10.000 --> 00:33:19.000 Okay, so I'm going to think of this representation of Maxwell here. 00:33:19.000 --> 00:33:28.000 In this formulation, the symmetry generator, the one that acts by shifting my background A0. 00:33:28.000 --> 00:33:32.000 Well, how do you shift the field by acting with the conjugate momentum? 00:33:32.000 --> 00:33:37.000 And with the conjugate momentum for A here, well, that's B. 00:33:37.000 --> 00:33:42.000 So… The momentum generator, in other words. 00:33:42.000 --> 00:33:53.000 the better that, you know, shifts of A. Is simply the integral of B on the manifold sigma, or you're shifting a. 00:33:53.000 --> 00:33:59.000 Is it okay? This is just a statement that P generates a shift of X. 00:33:59.000 --> 00:34:15.000 quantum mechanics. Um, here, assuming as before that sigma is the boundary of some ball, so you could also write this as the integral over D of dB. 00:34:15.000 --> 00:34:24.000 And this is useful to think in this way, because here you go back to the statement about the case transformations. In this representation of Maxwell. 00:34:24.000 --> 00:34:29.000 B is the filler strength for the dual of H. 00:34:29.000 --> 00:34:42.000 I'm writing that as FD. And in standard coordinates, so in terms of A, we will say something slightly impressive that fd is something like a star f. 00:34:42.000 --> 00:34:54.000 So you could read this operator if you want as the action of integral over d of D of star f. And D of star f is precisely thing as gauge transformations. 00:34:54.000 --> 00:35:00.000 Doesn't the A equation of motion say that db is 0? 00:35:00.000 --> 00:35:06.000 Yeah, yeah, but that's a… I mean, yes, but at the United States, you are imposing the cash flow, so Hilbert. 00:35:06.000 --> 00:35:17.000 The stage in the Hibbert space are annihilated by the… by the Gauss law generic. 00:35:17.000 --> 00:35:22.000 But it does still united gauge transformation. Many gauge representative you you do work. 00:35:22.000 --> 00:35:27.000 But in your second equation, you have the integral of db, which is 0. 00:35:27.000 --> 00:35:36.000 Oh, sure, yeah. Um… 00:35:36.000 --> 00:35:46.000 Yeah. I mean, you can do that. You can do that exactly if you integrate out A, but then I don't think I'm allowed to represent the state in terms of A wave functions, right? 00:35:46.000 --> 00:35:52.000 So I can choose to… parameterize the state in terms of A. 00:35:52.000 --> 00:35:59.000 In which case, my wave function has to be gauge invariant. 00:35:59.000 --> 00:36:04.000 But A, itself can shift. So A itself communicates variant. 00:36:04.000 --> 00:36:13.000 But if I integrate out A to impose BB, then… I'm sorry. 00:36:13.000 --> 00:36:17.000 Alpha is going from identified with alpha plus 2 pi i, right? 00:36:17.000 --> 00:36:24.000 That's true. Yeah. 00:36:24.000 --> 00:36:25.000 Okay. 00:36:25.000 --> 00:36:32.000 So if I replace B by star F. Then, if I change the metric, I change the normalization of the integral. 00:36:32.000 --> 00:36:33.000 That's true. Um, that's why… Yeah. 00:36:33.000 --> 00:36:37.000 So that and alpha can't be identified by 2 pi i 2 pi. 00:36:37.000 --> 00:36:49.000 You're correct, yeah. So this wiggle here, Drew, is imprecise, right? It's the usual thing that you can quantize and question properly. You cannot quantize f and start F at the same time. 00:36:49.000 --> 00:36:53.000 Uh, so this was a little heuristic. 00:36:53.000 --> 00:36:58.000 I can choose to work in this language as I did here. 00:36:58.000 --> 00:37:05.000 And then B is not… quantized. 00:37:05.000 --> 00:37:22.000 Yeah, and then you'll say… 00:37:22.000 --> 00:37:24.000 Okay, just go on. 00:37:24.000 --> 00:37:35.000 Yeah, yeah. You're right, because if I work in this representation, it would be… B is not necessarily quantized in any way. 00:37:35.000 --> 00:37:44.000 I think I agree. Can we discuss after the talk? I think I know how to answer that. 00:37:44.000 --> 00:37:46.000 I don't know, just go on, don't worry about it. 00:37:46.000 --> 00:37:57.000 Thank you. Thank you. Alright, so… In fact, I think the answer is related to what I'm going to say. 00:37:57.000 --> 00:38:03.000 So, in this description of Maxwell theory, the beach HSRB term is not invariant. 00:38:03.000 --> 00:38:14.000 under gauge transformations of B. So this theory just does not have these gauge invariance. 00:38:14.000 --> 00:38:19.000 And this operator, e to the i alpha integral of B. 00:38:19.000 --> 00:38:27.000 There's no need to be gauging variant under this transformation. It has to be gauging variant under the rest of the digital transformations. 00:38:27.000 --> 00:38:37.000 Okay, clearly it is. But it's no gauge invariant and shift away, but it doesn't have to be because the action isn't equal. 00:38:37.000 --> 00:38:42.000 Um, so… This operator, it's a well-defined operator in this theory. 00:38:42.000 --> 00:38:51.000 Without having to attach an open shortage. And it's well defined for any alpha, for any value. 00:38:51.000 --> 00:38:56.000 When we're studying just the BF theory, so omitting this term here completely. 00:38:56.000 --> 00:39:11.000 Uh, we often discard that operator because the gauge theory has… in which B going to B plus D beta is a gate invariance. 00:39:11.000 --> 00:39:21.000 But in the Maxwell theory, we don't need to… So the operator needs to be included, because we want to discuss Maxwell theory. 00:39:21.000 --> 00:39:26.000 But otherwise… 00:39:26.000 --> 00:39:32.000 The analysis… Apart from the fact that we want to include visuals. 00:39:32.000 --> 00:39:43.000 If I need an integer. Otherwise, everything we need from this action is what's being given to us from the last term, the 2 pi i. 00:39:43.000 --> 00:39:53.000 B128. Okay, the limit of this action when e equals to 0 gives us the right action on the boundary conditions of A. 00:39:53.000 --> 00:39:57.000 A different way of saying that is all I want from this action is to shift. 00:39:57.000 --> 00:40:02.000 A, and for that, I just need the B field, which is the momentum. 00:40:02.000 --> 00:40:06.000 whether I have the EA squared term or not. 00:40:06.000 --> 00:40:13.000 So I could take the limit of Maxwell theory and the limit in which e goes to 0, and obtain this topological field theory here. 00:40:13.000 --> 00:40:21.000 With the provision that if I really am talking about Maxwell in this limit, I probably want to talk about this opinion. 00:40:21.000 --> 00:40:28.000 Um, so at this point of view was emphasized in this paper idea, Wang, and Chang. 00:40:28.000 --> 00:40:42.000 And it's a bit of a certainty. It might seem like, um… Uh, you know, been a bit too careful here. But the problem has to do with the fact that the reason I want to be careful. 00:40:42.000 --> 00:40:47.000 Is that when you think of this theory, the gauge invariance of the theory changes. 00:40:47.000 --> 00:40:54.000 And quite a bit whether E is a small but non-zero or equal to 0. 00:40:54.000 --> 00:41:02.000 The theory with E identically 0 has more gauge invariance than the theory with E is small but larger than 0. 00:41:02.000 --> 00:41:07.000 And you're trying to do the quantization carefully, as this group of people did, for example. 00:41:07.000 --> 00:41:15.000 Um, then you run into subjects. 00:41:15.000 --> 00:41:26.000 purposes of my talk, I… I'm interview… I'm not going to dwell on this more, but it's an important thing. 00:41:26.000 --> 00:41:33.000 Right. So… That's been understood, the simplicity for continuous symmetries. 00:41:33.000 --> 00:41:44.000 Then should be some version of this. BF theory that allows you to write the operators e to the i alpha integral. 00:41:44.000 --> 00:41:50.000 And this can be reached by taking the limit on which e goes to 0 in the faculty. 00:41:50.000 --> 00:41:59.000 So this was argued in a series of papers, and many of the things that you would expect the same TFT to do in the case of continuous symmetries, in this case you want to do work, and it also works. 00:41:59.000 --> 00:42:08.000 Bono Navilion International Centers. 00:42:08.000 --> 00:42:15.000 So, that was enough, I think Max was here. Let me try to understand what happens with the space-time symmetry. 00:42:15.000 --> 00:42:18.000 Let's say that we have a d-dimensional conformity theory. 00:42:18.000 --> 00:42:29.000 In principle, my conjecture is broader. I was claiming that it should be BF or spacetime integral, but I'm going to test this in the case of conformity. 00:42:29.000 --> 00:42:32.000 I think I'm going to better control and also. 00:42:32.000 --> 00:42:37.000 or graphic dualities are going to be easier when you have conformative. 00:42:37.000 --> 00:42:46.000 So, omitting discrete factors, the space-time symmetry group is the conformity, so d plus 1 component. 00:42:46.000 --> 00:42:53.000 And the conformal group of the boundary arises from boundary behavior for a gravitational receptor in the case of. 00:42:53.000 --> 00:43:04.000 Please leave a favor. Thank you. So, in general, can we write gravity as a BF theory for… The conformal group. 00:43:04.000 --> 00:43:13.000 Plus some corrections, perhaps. These things like this term here, which is not a BF theory. 00:43:13.000 --> 00:43:22.000 By something that becomes an important ad. 00:43:22.000 --> 00:43:34.000 Okay. Um… So, see, we can… For even D bigger than 0, when the boundary dimension is even. 00:43:34.000 --> 00:43:44.000 Positive. There's a conformal anomaly. Um, because as a conformal anomaly, you will have to write something more complicated than a PF theory. As I mentioned before. 00:43:44.000 --> 00:43:50.000 So you know, you have a BF theory assumes that you don't have anomalies with this. 00:43:50.000 --> 00:43:56.000 Um, so certainly what we have to require more. 00:43:56.000 --> 00:44:03.000 For example, something like James Simons. So, in the paper, we say more about what happens for boundary. 00:44:03.000 --> 00:44:12.000 when there is of even dimension. But we don't understand the situation fully. So the summary of what we understand is that we have, I think, a decent handle. 00:44:12.000 --> 00:44:13.000 On the A-type anomalies. Um, but not of receipt type of need. 00:44:13.000 --> 00:44:19.000 Okay. 00:44:19.000 --> 00:44:22.000 I have a quick question. 00:44:22.000 --> 00:44:23.000 Yeah. 00:44:23.000 --> 00:44:29.000 Do you want something other than just the first-order formulation of Euclidean conformal gravity with… 00:44:29.000 --> 00:44:33.000 Some topological terms in it? 00:44:33.000 --> 00:44:34.000 Because you always have that in any dimension in principle, right? 00:44:34.000 --> 00:44:39.000 Um… To clear and confirm account. 00:44:39.000 --> 00:44:43.000 I mean, you're working with the Euclidean conformal group, so… 00:44:43.000 --> 00:44:44.000 Yes, but the… 00:44:44.000 --> 00:44:50.000 the Euclidean aspect is not that important, but I'm just curious. That seems to be the way you're setting it up. 00:44:50.000 --> 00:44:53.000 But that's not presumably what you were… 00:44:53.000 --> 00:44:54.000 I have in mind. 00:44:54.000 --> 00:44:58.000 You're right. There was a possibility, yes. I think all I can say, yeah. 00:44:58.000 --> 00:45:03.000 Because that will… that has a Simon's term in it, uh… 00:45:03.000 --> 00:45:04.000 So… 00:45:04.000 --> 00:45:13.000 Yeah, you're right, but… The bulk we want to keep as the bulk, so as a standard gravity in the bulk. So when we are doing holography. 00:45:13.000 --> 00:45:15.000 Yeah. 00:45:15.000 --> 00:45:18.000 In August 5, for example, you want the standard supergravity. 00:45:18.000 --> 00:45:21.000 And you can write it in a first-order formalism, which is what you're after, presumed. 00:45:21.000 --> 00:45:30.000 And I don't know. Yeah, I'm definitely going to go for the first order formation, but it's not going to be conformant. 00:45:30.000 --> 00:45:38.000 It's going to be a standard double number. 00:45:38.000 --> 00:45:47.000 So let me go through the other… Uh, cases, so for D equal to 1, so one-dimensional boundaries. 00:45:47.000 --> 00:46:01.000 This has been a topic of a little research in the last few years. I could think of citations, but… there anyway. You have a bug described by JT gravity, which is already BF theory, used to the Lagrangian level. 00:46:01.000 --> 00:46:07.000 Um, so… Well, what I'm saying seems reasonable. 00:46:07.000 --> 00:46:12.000 But let me try to describe the case that I think is closer to. 00:46:12.000 --> 00:46:18.000 What we would like to do, which is, uh… high arbitrary image. 00:46:18.000 --> 00:46:25.000 And in particular, I would like to understand. Gravity in cases where it's not just a polarity. 00:46:25.000 --> 00:46:30.000 So the first thing that happens is when the boundary is three-dimensional. 00:46:30.000 --> 00:46:39.000 And then the bulk is four-dimensional. So the bulk will be described by gravity in ADS4. 00:46:39.000 --> 00:46:49.000 And as Vivek was saying, would like to formulate things in a first order way and a formulation of areas for gravity in this space exists. 00:46:49.000 --> 00:46:58.000 I should say that everything here, I need to say for the… financial life is classical, and it's a random claims of quantization here. But classically, at least. 00:46:58.000 --> 00:47:02.000 You can write Einstein gravity in the following way. 00:47:02.000 --> 00:47:06.000 or is the gravity plus a couple of times each other. 00:47:06.000 --> 00:47:14.000 In the following way. It's a Bf wave. So roughly, it's a BF plus b squared, as we had before. 00:47:14.000 --> 00:47:17.000 But let me explain the details of what happened. 00:47:17.000 --> 00:47:22.000 So B here is a 2 form in my 4 manifold could be. 00:47:22.000 --> 00:47:33.000 foreign for manifold, taking values in the adjoint. of SO41, which is the relevant conformal group for 3 inch. 00:47:33.000 --> 00:47:38.000 So that's be the… you know, doing value field as before for this gate. 00:47:38.000 --> 00:47:43.000 And F is the fill strength for a connection in the same. 00:47:43.000 --> 00:47:50.000 This group. So, SO4, 1. The indices? 00:47:50.000 --> 00:47:59.000 Here ijfen. The whole vector indices of SO41, so for example, these two will be antisymmetric, given me by joints. 00:47:59.000 --> 00:48:09.000 And, in addition, I have this VM here. Which is a dimensionless fixed SO4, 1 vector. 00:48:09.000 --> 00:48:17.000 So, in terms of a space-time indices, B is a true form. F is a 2 form, and B is a scalar. 00:48:17.000 --> 00:48:22.000 And in terms of SO41 indices, which you should think as internal gauge indices. 00:48:22.000 --> 00:48:27.000 This is a joint, this is a joint, and this. 00:48:27.000 --> 00:48:37.000 Now if I put an epsilon here, you can construct something which is, uh… The top form, which is also a gauge inlet, and that is a formula. 00:48:37.000 --> 00:48:43.000 Okay, so this theory that was written down to my knowledge in these papers here. 00:48:43.000 --> 00:48:51.000 reforming some other ideas of McDoug Mansory. The claim shows that is. 00:48:51.000 --> 00:48:57.000 Classically equivalent to Einstein gravity. So how does this go? 00:48:57.000 --> 00:49:02.000 Well, first of all, you have to choose V in a specific way. 00:49:02.000 --> 00:49:11.000 So by Gauge invariance, it doesn't matter. very much in which direction you pick it up, and just doing a case transformation, put anything in. 00:49:11.000 --> 00:49:22.000 This direction. And if you choose the value of V to be of the non-zero component to be lambda Gn over 6. 00:49:22.000 --> 00:49:30.000 This is a specific value. Then you're going to get a B that appears quadratically, so you can integrate it out. 00:49:30.000 --> 00:49:35.000 And you get this action in here. So let me explain what happened. 00:49:35.000 --> 00:49:43.000 He has only a non-zero diff component, so the epsilon here will pick i, j, and k. 00:49:43.000 --> 00:49:53.000 Um, to address that to the IJKM. So, this Bijkl. 00:49:53.000 --> 00:49:58.000 Um, to have indices in the first four indices. 00:49:58.000 --> 00:50:04.000 So this is what this expression is doing. We integrate out, they still satisfy the problem. 00:50:04.000 --> 00:50:09.000 And now, because everything takes values in the first four indices, which are in SO4. 00:50:09.000 --> 00:50:15.000 Then I can reduce my index structure to SO4. 00:50:15.000 --> 00:50:21.000 VGS notation, what I mean by putting the hat is that. 00:50:21.000 --> 00:50:30.000 F has MN, which is a parentheses, is f and n were now the same effect as special for comma 1. 00:50:30.000 --> 00:50:39.000 Is that clear? So, I'm just taking the… The indecision is also for Kuma want to be restricted to the foreign. 00:50:39.000 --> 00:50:49.000 Alright, so these are basically doing the… the Gaussian integral, because in classically, the… integrating out of B. 00:50:49.000 --> 00:50:56.000 And it's important here that I'm choosing a B, which is non-zero, and in this particular direction. 00:50:56.000 --> 00:51:05.000 One very interesting question in this game is what happens if V happens to be null or zero, and I don't know. 00:51:05.000 --> 00:51:10.000 I'm going to pick this one and claim it was the same as… 00:51:10.000 --> 00:51:16.000 So to see that, what you do is you expand A in a way that's very similar to within Transimus in three dimensions. 00:51:16.000 --> 00:51:25.000 So, your A is, um… connection in that joint, and you're going to separate them to separate. 00:51:25.000 --> 00:51:29.000 the indices that involve 5, which is this one's here, from the. 00:51:29.000 --> 00:51:36.000 The ones that involve five, you're going to call BL vines or VR vines in this case. 00:51:36.000 --> 00:51:40.000 And the ones that don't, you're going to think of as a spin connection. 00:51:40.000 --> 00:51:46.000 So if you expand A in this way and you work out what f hat is, and you plug it in this expression. 00:51:46.000 --> 00:51:51.000 bit of algebra, you end up with. This Lagrangian here, or this action here. 00:51:51.000 --> 00:51:56.000 Um, which is familiar, is the first-order formulation of, uh. 00:51:56.000 --> 00:52:05.000 and then gravity, so this is the… term and this is the cosmological constant. 00:52:05.000 --> 00:52:12.000 And a moment in a term, which is something like R, which is proportional to the Eular density, so it's a topological matter. 00:52:12.000 --> 00:52:24.000 So, at least classically in four dimensions. I can write Einstein gravity if I introduce the presence of a cosmological constant, and if I allow myself to put the Euler term. 00:52:24.000 --> 00:52:27.000 I can write it in terms of a PDF here. 00:52:27.000 --> 00:52:33.000 So, quick, quick comment about the previous slide. I think you cannot choose a B2B… 00:52:33.000 --> 00:52:38.000 Uh, space-like or null, because then the stabilizer group is not going to be the Lorentz group, right? 00:52:38.000 --> 00:52:44.000 I think you want to get to four-dimensional gravity. 00:52:44.000 --> 00:52:45.000 Yeah. 00:52:45.000 --> 00:52:51.000 Yep. Sorry, yeah, so that's what I mean. You're right that if I pick V2B now, I don't get gravity. I just don't know what I get is. 00:52:51.000 --> 00:52:59.000 Because the stabilizer group of SO41 for V… stabilizer group of V is SO3-1, right? 00:52:59.000 --> 00:53:07.000 Got it. Yeah, so the… depends on how I choose B, I get the stabilization could be so poor or so 3 comma 1. 00:53:07.000 --> 00:53:15.000 And the interpolating thing goes through a null vector. I don't know what you mean. 00:53:15.000 --> 00:53:24.000 I was just throwing the question there. This is a question of this theory that I haven't seen analysed in the paper. I'm not clear how. 00:53:24.000 --> 00:53:29.000 In the case that V is not. 00:53:29.000 --> 00:53:33.000 Well, in the case that V is of this form, then you get the Einstein gravity. 00:53:33.000 --> 00:53:38.000 20 minutes before my project. 00:53:38.000 --> 00:53:46.000 All right. So this is… quite close to the kind of thing I wanted. I have a BF theory. 00:53:46.000 --> 00:53:59.000 coupled to some perturbation, if you want. I haven't justified visual perturbation, but coupled to some extra term, which is B squared times one parameter, which is lambda times t. 00:53:59.000 --> 00:54:03.000 Maybe I should mention that one of the motivations for these papers. 00:54:03.000 --> 00:54:17.000 Were the realization that lambda times z in our universe is extremely tiny. So we wanted to think of this as a small perturbation, but… Because of the same reasons that I was mentioning before, it's not clear how to do the gauge fiction and so on, it's not clear how good there. 00:54:17.000 --> 00:54:23.000 trouble that works. I think it's interesting. 00:54:23.000 --> 00:54:32.000 So… Right, so we have a BF here. Let's try to see those corrections. Let's try to see how this BF theory couples to. 00:54:32.000 --> 00:54:37.000 to a boundary, how this gravity be a form couples to the boundary. 00:54:37.000 --> 00:54:45.000 And here I'm doing something similar to the video I was mentioning before, I think. If I take the boundary value of A. 00:54:45.000 --> 00:54:50.000 Uh, to be some… sorry, if I take laboratory conditions of A. 00:54:50.000 --> 00:54:55.000 That means that my CFT is coupled to some specific background. 00:54:55.000 --> 00:54:59.000 Which has to do with the… precise form of A. 00:54:59.000 --> 00:55:09.000 So, if I split A in this way. That is I split the generators of SO4, 1. 00:55:09.000 --> 00:55:16.000 into the algebra of S03ism, SO3s of algebra, which is the relevant one in the three-dimensional boundary. 00:55:16.000 --> 00:55:29.000 Then, the different components of A can be arranged in this way. You have something which is essentially the spring connection multiplying the generator of rotations times something which is a real drive. 00:55:29.000 --> 00:55:38.000 Uh, multiplying the generative momentum and F and B, which have to do with the component. 00:55:38.000 --> 00:55:44.000 So, this is how a digitally boundary value for A. 00:55:44.000 --> 00:55:53.000 Would the couple to the CFT. It would be a CFT coupled to these conformal data. 00:55:53.000 --> 00:55:58.000 This is very different to the way that you will couple Northern CFT to. 00:55:58.000 --> 00:56:04.000 to an internal symmetry. But that doesn't really matter. We have a background. 00:56:04.000 --> 00:56:13.000 for the CFC, that we know how to manipulate using topographic operators, because we know how to do topographic operators in the. 00:56:13.000 --> 00:56:19.000 I'll just do that. We'll see how acting with the topological operator in the bulk. 00:56:19.000 --> 00:56:26.000 BF theory shifts this data in K. 00:56:26.000 --> 00:56:34.000 So I'm going to… because I can write my action as a BF3 plus a small correction. 00:56:34.000 --> 00:56:45.000 And in the paper, we have an argument that this small correction could be not close to the boundary. I'm going to… focus on the BF edits you get if you ignore that less than the B squared. 00:56:45.000 --> 00:56:52.000 And let's see if I get something that acts as the CMTFT for spacetime symmetries. 00:56:52.000 --> 00:56:59.000 Oh, yeah. I think I have like 3 minutes, right? 00:56:59.000 --> 00:57:07.000 you do have 5 minutes, but I guess we can continue for a few minutes unless people have to leave, so… 00:57:07.000 --> 00:57:08.000 Gotcha, yeah. 00:57:08.000 --> 00:57:17.000 Okay, I'm not at least give the punchline and then I will try to be self-contained in 5 minutes, and if anybody has other questions, I'm happy. 00:57:17.000 --> 00:57:24.000 All right, so… I'll give evidence in a minute for why this BF theory behaves as you would expect. 00:57:24.000 --> 00:57:33.000 generators of symmetry to behave. Um, but there's a couple of remarks I think you've learned from now, especially because I might be a little bit fast when I'll discuss them later. 00:57:33.000 --> 00:57:46.000 One is that we're working in a first-order formulation of gravity, and the famous issue with first order information of gravity is not clear why the turbines must be non-degenerated. 00:57:46.000 --> 00:57:49.000 In the first order formation, there's no obvious reason for that. 00:57:49.000 --> 00:57:55.000 And that gives problems, and those problems will also come for us. 00:57:55.000 --> 00:58:04.000 Later. And everyone should have taken on your test. 00:58:04.000 --> 00:58:11.000 So let me try to give the… The punchline, how this behaves, the track definition time. 00:58:11.000 --> 00:58:19.000 So… We have a BA function for the conformal group, trace of B which F. 00:58:19.000 --> 00:58:25.000 And there's an equation of motion, which is that DA of B is equal to 0. 00:58:25.000 --> 00:58:33.000 As I mentioned, we postulate that the way this couples to the vendor is as it couples in the gravitational case. 00:58:33.000 --> 00:58:39.000 the composition turbines has been connected. 00:58:39.000 --> 00:58:45.000 Now, B can also be expanded in a similar way, but it lives on the dual algebra, so I'm putting the stars. 00:58:45.000 --> 00:58:48.000 Their star is the dual with respect to the object. 00:58:48.000 --> 00:58:52.000 But otherwise, you can expand B. 00:58:52.000 --> 00:58:57.000 No, because you have an equation of motion, which is DA of B equal to 0. 00:58:57.000 --> 00:59:01.000 If you express it in components, you get this system of equations, which. 00:59:01.000 --> 00:59:07.000 No particular transparenta, it was flashing to. 00:59:07.000 --> 00:59:14.000 But if you go to Flood Background. Choose your boundary to have a flat metric to start with. 00:59:14.000 --> 00:59:20.000 Uh, then the question simplify quite a lot, and they all become D or something equal to C. 00:59:20.000 --> 00:59:27.000 That is the equations of motion for… The BF theory implies that you have conserved covers. 00:59:27.000 --> 00:59:33.000 Not only that, with some… if you integrate this conserved currents. 00:59:33.000 --> 00:59:37.000 You get some generators that are the familiar generators of the conformal algorithm. 00:59:37.000 --> 00:59:42.000 So expanding this BF on particular components of B. 00:59:42.000 --> 00:59:49.000 In the dual algebra, and integrating with the standard expressions that you find in any review of conformatics. 00:59:49.000 --> 01:00:02.000 Here, the only thing I'm doing, which is in trivia, is instead writing the start of T, which is what you will find typically, and write in TA, which is… 01:00:02.000 --> 01:00:16.000 So, in contraction with Gerda. So in this way, you can identify with the generators of the conformal algebra should be, if this whole picture works is p, j, k, and b. 01:00:16.000 --> 01:00:25.000 And what we did, and with this, I will finish the following sort of experiment. Imagine that you have some operator here. 01:00:25.000 --> 01:00:37.000 And you act with a symmetry generator. Which is the integral of T, which… Through my previous identification, it's essentially the energy-momentum tensor, so the technology is translations. 01:00:37.000 --> 01:00:45.000 So if I do this operator, which, remember, you say hologrami or some B field in the back, which I'm pushing to the boundary. 01:00:45.000 --> 01:00:54.000 Does it generate translations of this object? And indeed, it does, and the way it does is kind of interesting. 01:00:54.000 --> 01:01:02.000 So, if you act with this. B, which is in the dual of translations. 01:01:02.000 --> 01:01:11.000 Remember that B and A are canonically conjugate, in the BF theory, so it will act by shifting the canonically conjugate field, which is the VL1. 01:01:11.000 --> 01:01:22.000 So by acting with the generator translations. You will shift the build line precisely at the position where you have inserted this generative translations. 01:01:22.000 --> 01:01:24.000 So the pictures that you put with generative translations. 01:01:24.000 --> 01:01:31.000 And between two insurgents of operators. And when you put the generator of translations, what has happened. 01:01:31.000 --> 01:01:36.000 is that the Virbine, which was constant here, suddenly gets a delta function. 01:01:36.000 --> 01:01:45.000 And then it goes back to the present. So… 01:01:45.000 --> 01:01:53.000 There is a way of… You have a beer vine, which is a delta function. 01:01:53.000 --> 01:02:00.000 Localize here. You can undo this modification of the verbine by acting with the different morphism. 01:02:00.000 --> 01:02:09.000 we can construct. And the deformorphism is such that it's non-zero on the right of the line and zero on the left of the line. 01:02:09.000 --> 01:02:14.000 And so we have a proof that you can always construct these diplomorphisms for regular scan. 01:02:14.000 --> 01:02:20.000 these transformations and so on. Um, sorry. 01:02:20.000 --> 01:02:26.000 We have a partial proof, and there's some details of the proof that we have quite worked out. But in this case, at least. 01:02:26.000 --> 01:02:32.000 Uh, you can do, you can act with a deformorphism that. 01:02:32.000 --> 01:02:38.000 Brings back the metric for the Virvine to the flat one, but separates. 01:02:38.000 --> 01:02:43.000 The two points in particular, I suppose this point amount proportional to the momentum that you have at it. 01:02:43.000 --> 01:02:53.000 So in this way, by undoing the action of this, uh… 2 metre narrator that we identified abstractly with P. 01:02:53.000 --> 01:02:56.000 We obtain what the translation should do, which is to move this point. 01:02:56.000 --> 01:03:00.000 It generates a deformorphism which is of. 01:03:00.000 --> 01:03:06.000 Do I understand correctly that you're moving all the points with X positive? 01:03:06.000 --> 01:03:15.000 Yes, correct. Yeah. So, I'm putting the… Symmetry in it, or on this slide. 01:03:15.000 --> 01:03:25.000 So X equal to 0. And the interpretation, you could write either the diffamorphism marks by shifting the X positive once a fixed amount. 01:03:25.000 --> 01:03:36.000 all the narrative ones, 15 minus X. It separates your points and amount X. 01:03:36.000 --> 01:03:48.000 Okay. So I think that the… sort of main lesson, there's subtlety is that… And they're important, so I'm happy to stay and discuss. 01:03:48.000 --> 01:03:55.000 But let me conclude. 01:03:55.000 --> 01:04:00.000 So, we wanted to construct a simplistic description of continuous space-time symmetries. 01:04:00.000 --> 01:04:07.000 The necessary condition for that was to have the same TFT description of continuous internal symmetries and non-avilion ones. 01:04:07.000 --> 01:04:13.000 And luckily, there has been lots of progress on that in the last couple of years, and now we have technology. 01:04:13.000 --> 01:04:23.000 space-time centers bring their own conceptual complications. I think more than technical, these are conceptual points that we have to solve. 01:04:23.000 --> 01:04:34.000 But we have made some progress on those. In particular, I would use the connection to gravity and the holographic duo. I think we have reasonably satisfactory answer in four dimensions, bulk of four dimensions. 01:04:34.000 --> 01:04:39.000 And also, what is the connection of operators in SMTF2 to deformorphisms? Again, we have, uh. 01:04:39.000 --> 01:04:45.000 or some beginning of an understanding. 01:04:45.000 --> 01:04:55.000 Um, and there are lots of open problems. Here I list a few. Let me just highlight the two main ones that I think are… much confusion? 01:04:55.000 --> 01:04:59.000 Um, maybe 3-1. So, we should strengthen the connection to gravity. 01:04:59.000 --> 01:05:07.000 In all dimensions, so everything I told you was about four-dimensional gravity, because there's this formulation by Antonio Mansouri. 01:05:07.000 --> 01:05:17.000 Um… I'm not… I'm not sure how to do this in D bigger than 4, for example. 01:05:17.000 --> 01:05:26.000 Also, in even dimensions, we have the issue of the conformed anomaly that we haven't fully captured, and… I think it's important to control. 01:05:26.000 --> 01:05:35.000 Something I had to skip was that when you act with localized operators, this kind of gauge transformations with a delta function profile. 01:05:35.000 --> 01:05:43.000 This background fields with a delta function profile, which are written naturally in gravity. You do get unnatural. 01:05:43.000 --> 01:05:53.000 boundary metrics. In particular, with some sort of the generate or even… It's not clear what. 01:05:53.000 --> 01:06:01.000 You should do in those cases. Probably the most confusing bit of the whole story. 01:06:01.000 --> 01:06:08.000 And there are other things you could do. And the other thing that's to me very much mysterious is how to incorporate finite space-time symmetries in this story. 01:06:08.000 --> 01:06:16.000 Um, which is, I think, an entirely different. Right. Thank you. Napoleon. 01:06:16.000 --> 01:06:29.000 Alright, let's thank Inyaki for any nice talk, and if there are questions, please feel free to unmute and ask. 01:06:29.000 --> 01:06:32.000 Can you ask a question? 01:06:32.000 --> 01:06:38.000 So, uh, the SIM TFT theory is a topological theory in the conventional case. 01:06:38.000 --> 01:06:40.000 In case of, uh, gravity, is it topological gravity, or that's 01:06:40.000 --> 01:06:44.000 Yes. 01:06:44.000 --> 01:06:48.000 conventional gravity with a dynamical graviton. 01:06:48.000 --> 01:06:59.000 The way I like to think of it, so… 01:06:59.000 --> 01:07:02.000 So, this action here, this is the standard gravity. 01:07:02.000 --> 01:07:09.000 At least on the classical level. And the same TFT that we are proposing is the first term, yeah. 01:07:09.000 --> 01:07:22.000 So, standard gravity is not topical in… They're saying that it doesn't have, um… 01:07:22.000 --> 01:07:27.000 And it only becomes topological, in the sense that you don't have propagating. 01:07:27.000 --> 01:07:33.000 The limit in which you take Vn to 0. So this is the analog. 01:07:33.000 --> 01:07:38.000 of taking E equal to zero with Maxwell. 01:07:38.000 --> 01:07:46.000 So I went a little bit fast, but… The standard gravity is this expression here, or at least classically this expression here. 01:07:46.000 --> 01:07:50.000 And the sympia extract out of that is, like, in the Maxwell case. 01:07:50.000 --> 01:07:54.000 The limit in which this theory becomes topological, in particular topological limit of gravity. 01:07:54.000 --> 01:08:01.000 But you might want to call those imprecise, that lambda times unit goes to 0. 01:08:01.000 --> 01:08:06.000 You might want to think of that as unit and go to zero, which is imperative. 01:08:06.000 --> 01:08:16.000 Unitone has dimensions, but sort of heuristically, it's… Um, so I'd like to think of this is the behavior of gravity close to the… boundary of issues. 01:08:16.000 --> 01:08:19.000 You with respect close to the boundary of ideas. 01:08:19.000 --> 01:08:28.000 gravity to become non-dynamical. Something to freeze out, and… This BFA usually should be what remains of gravity. 01:08:28.000 --> 01:08:35.000 Very, very close to a synthetic power. 01:08:35.000 --> 01:08:44.000 Do you expect something like that to happen in the conventional holographic settings? 01:08:44.000 --> 01:08:50.000 Uh, yeah, I don't think there's anything… Yes, I would expect that to happen. 01:08:50.000 --> 01:08:58.000 For Radius 4, at least. So, of course, the histories are different, because the bug is already topological. 01:08:58.000 --> 01:09:03.000 Can we discuss alternatives? But in four and higher dimensions. 01:09:03.000 --> 01:09:10.000 Gravity is not too political. There's a propulating graviton. 01:09:10.000 --> 01:09:16.000 I would still respect in every dimension that close enough to the boundary at infinity. 01:09:16.000 --> 01:09:21.000 give us some sort of BF theory, but we have only been able to argue for it. It's on care in major states. 01:09:21.000 --> 01:09:27.000 We have this formation on gravity. But we sort of know what happens. 01:09:27.000 --> 01:09:31.000 Um… It's very important, maybe… 01:09:31.000 --> 01:09:37.000 What about the derivation of the conformal anomaly of a boundary, of the boundary CFT? 01:09:37.000 --> 01:09:41.000 Doesn't that depend on the coupling of the graviton to the. 01:09:41.000 --> 01:09:43.000 Team, you know. what you need? 01:09:43.000 --> 01:09:53.000 Uh, that's… Yeah, so the… the sea… Yes, on the derivation service. 01:09:53.000 --> 01:09:56.000 Addington, Skanderas, and all that. 01:09:56.000 --> 01:10:06.000 Yeah, yeah, exactly. So yeah, indeed the. The more careful way of doing what I'm doing, but unfortunately I have some data to run that. 01:10:06.000 --> 01:10:12.000 is… will be to do the holographic randomization program. 01:10:12.000 --> 01:10:17.000 And in a first-order formation of gravity, it's very important. 01:10:17.000 --> 01:10:26.000 They will work in first order for us, because we want to have a BF structure. I think we know how to deal with CFTs otherwise. 01:10:26.000 --> 01:10:32.000 And unfortunately, I found very little about holographic normalization to first order. 01:10:32.000 --> 01:10:38.000 So, in first order formulation. Maybe a couple of papers. 01:10:38.000 --> 01:10:44.000 So yeah, the careful way of proving what I was trying to argue. 01:10:44.000 --> 01:10:59.000 So, what I'm doing here is a conjecture, but it… One word to do more carefully, you will have to… It started writing first-order versions of gravity, which are of BF type. I only know how to do that in. 01:10:59.000 --> 01:11:05.000 Core dimensions, this kind of… version generalized to D bigger than 4. 01:11:05.000 --> 01:11:14.000 But assuming that you have that, then you'll want to do the secondary holographic randomization. 01:11:14.000 --> 01:11:20.000 And then try to argue that, uh. The holographic counter terms somehow. 01:11:20.000 --> 01:11:22.000 Don't care about this last term. 01:11:22.000 --> 01:11:32.000 Let's see, if I'm… this is the SIM TFT, right? So that you're describing. 01:11:32.000 --> 01:11:33.000 That's it. 01:11:33.000 --> 01:11:42.000 translations of three-dimensional field theories. But… So going back to an earlier part of your talk, what are in the. 01:11:42.000 --> 01:11:48.000 You mentioned this. VF theory for B. 2 and C. 2. 01:11:48.000 --> 01:11:49.000 Mm-hmm. 01:11:49.000 --> 01:11:57.000 And type 2B supergravity. you know. So what are the boundary conditions that? Yeah, that one. 01:11:57.000 --> 01:12:02.000 One of the boundary conditions that killed the singletons? 01:12:02.000 --> 01:12:11.000 Um… They're typically written, but there's not very… yeah, so the short answer is I know, because it's a connoral system. 01:12:11.000 --> 01:12:17.000 We typically say that we take something like BSLEF or B2, and take Neumann for C2. 01:12:17.000 --> 01:12:22.000 You can't do both. 01:12:22.000 --> 01:12:29.000 Okay. So what I meant is, if you take for this theory, if you take divisile for B2. 01:12:29.000 --> 01:12:37.000 When you are taking Neumann for C2. They're not independent. 01:12:37.000 --> 01:12:46.000 Well. In the quantizations of this theory that I'm familiar with, you have a singleton mode. 01:12:46.000 --> 01:12:47.000 I would say… 01:12:47.000 --> 01:12:55.000 and actually the singleton. The singleton mode is important because type 2B string theory is s duality invariant. 01:12:55.000 --> 01:12:59.000 and the Langlands dual of SUN is PSUN. And so the singleton mode restores the U1. 01:12:59.000 --> 01:13:04.000 That's true, um… 01:13:04.000 --> 01:13:06.000 Yes. Um… 01:13:06.000 --> 01:13:13.000 So you really have the dual. My picture has always been that the dual type 2B supergravity. 01:13:13.000 --> 01:13:21.000 his UN Super Yang mills. and that nicely restores the s duality of type 2B supergravity. 01:13:21.000 --> 01:13:30.000 That's true, but I think one other thing you could say is that if you quantize this theory without including a singleton. 01:13:30.000 --> 01:13:39.000 And look at the boundary modes to the… The boundary conditions for this B2 with DC2 theory. 01:13:39.000 --> 01:13:43.000 The boundary conditions don't have to respect edge duality. 01:13:43.000 --> 01:13:50.000 They have to transform covariantly, and there is value. 01:13:50.000 --> 01:13:51.000 Well, that's why I'm asking. I'm not sure how you get rid of the singleton mode. 01:13:51.000 --> 01:13:55.000 They have to be Norwich. And the way the set of… 01:13:55.000 --> 01:14:00.000 Yeah, the short answer is I don't know either. Nobody has understand this carefully, that's what I know. 01:14:00.000 --> 01:14:04.000 But I offer a smoke? 01:14:04.000 --> 01:14:05.000 This… excuse me. 01:14:05.000 --> 01:14:11.000 So the singleton mode appears. 01:14:11.000 --> 01:14:17.000 I mean, I understand that if you. If you take the sandwich. 01:14:17.000 --> 01:14:29.000 you have B2 and C. 2, which are in differential colomology, then that's equivalent to a ZN gauge theory. 01:14:29.000 --> 01:14:34.000 But. I think it's different in holography. 01:14:34.000 --> 01:14:35.000 of her… 01:14:35.000 --> 01:14:39.000 Yeah, I think… I think the status is in your paper with Belo. 01:14:39.000 --> 01:14:51.000 Yes, in that case, if you took a 5-dimensional, as you said yourself, if you take the 5-dimensional Zm gauge theory, then you can explain the different global forms of. 01:14:51.000 --> 01:14:57.000 any young mil. So it doesn't have to be an equals force to be young mouse. 01:14:57.000 --> 01:15:06.000 any young mills there. You can explain the different global forms of the gauge group by different choices of row boundary, topological theories for the. 01:15:06.000 --> 01:15:08.000 the VF theory. take the conversation on ideas. 4 is id 5 is different. 01:15:08.000 --> 01:15:16.000 Yeah, you're completely correct. 01:15:16.000 --> 01:15:27.000 You're completely right. I don't know the boundary conditions that will kill the singleton in the UN case. I know that if you do the analysis quantization this theory, you reproduce the… which gives me a bit of confidence. 01:15:27.000 --> 01:15:32.000 And I also know that in the case that you put orienty force that kill the singleton mode for you. 01:15:32.000 --> 01:15:41.000 Then, by this same sort of analysis. you look at the whole set of global problems. 01:15:41.000 --> 01:15:42.000 So the whole picture is… 01:15:42.000 --> 01:15:46.000 I mean, you know, I could put in the kinetic terms for B. 2 and 8 and C. 01:15:46.000 --> 01:15:48.000 Yeah, yeah, correct. Um… 01:15:48.000 --> 01:15:54.000 And then and then then it's a much less subtle quantization. 01:15:54.000 --> 01:16:01.000 And the only the only one that I understand gives you a singleton mug. 01:16:01.000 --> 01:16:02.000 Could I offer… could I offer a small comment? 01:16:02.000 --> 01:16:03.000 So I'm not understanding this claim here. 01:16:03.000 --> 01:16:04.000 I don't disagree, but… Yeah. 01:16:04.000 --> 01:16:10.000 If we do what Greg's suggested at connections for beta NC2, 01:16:10.000 --> 01:16:16.000 Then, of course, taking Beto the boundary will not impose the boundary condition 01:16:16.000 --> 01:16:22.000 Which… which we want, which will give SUN or PSUN. 01:16:22.000 --> 01:16:29.000 In fact, the wave function will be some holomorphic function of a combination of B2 and C2. 01:16:29.000 --> 01:16:37.000 But the space of these functions will be complete in the sense that this will be conformal blocks, which 01:16:37.000 --> 01:16:41.000 form a basis in the Hilbert space of this topological theory. 01:16:41.000 --> 01:16:49.000 Which means that there is a particular way to impose boundary conditions by integrating over 01:16:49.000 --> 01:16:54.000 this columnorphic combination of B2 and C2 at the boundary. 01:16:54.000 --> 01:17:03.000 It's not just taking B2 to 0, but if you take this holomorphic combination and you integrate over this holomorphic combination with a particular weight, 01:17:03.000 --> 01:17:11.000 That should be sufficient because of the completeness. To project on a particular conformal block you want. 01:17:11.000 --> 01:17:12.000 So that would give you… yeah. 01:17:12.000 --> 01:17:20.000 And that will be a way… and there will be a way to cook a particular SUN or PSUN theory in the bulk. 01:17:20.000 --> 01:17:24.000 That's very interesting. I don't think I've seen that. Is that written down somewhere, or something? 01:17:24.000 --> 01:17:27.000 Yeah, I'll send the reference. 01:17:27.000 --> 01:17:32.000 Please, thank you. 01:17:32.000 --> 01:17:41.000 Yeah, um, if I want to make these statements a bit more carefully, they are more completely technically, but I could put an identity fold, then there's no issue with the. 01:17:41.000 --> 01:17:55.000 With a single-tone mode. Then everything else is still closed. So you get the set of boundary conditions matching the set of lower for the… So I'm speeds, and so on things. 01:17:55.000 --> 01:18:04.000 We do more computing. 01:18:04.000 --> 01:18:08.000 Thank you for the communication. You can send me a paper. 01:18:08.000 --> 01:18:09.000 Thank you. 01:18:09.000 --> 01:18:10.000 Okay, well, thanks for the talk, and Yoki. 01:18:10.000 --> 01:18:23.000 Thank you. Thanks for the questions. Thank you. If anyone has more questions. 01:18:23.000 --> 01:18:26.000 So, your only motivation for the… 01:18:26.000 --> 01:18:33.000 McDowell Mansouri was to get this BF termed cleanly, as of in the first order? 01:18:33.000 --> 01:18:43.000 Yeah, because the way I'm thinking of these things is the way I can think of these things is by. 01:18:43.000 --> 01:18:48.000 By asking myself what's the limit? or the dynamic of you in the park. 01:18:48.000 --> 01:18:54.000 In the limiting which the coupling goes to zero. That's how I tend to think of these questions. 01:18:54.000 --> 01:18:57.000 And I only know how to answer this question for gravity. 01:18:57.000 --> 01:19:02.000 If it has any sort of magic reform, so B squared. 01:19:02.000 --> 01:19:03.000 plus BF. 01:19:03.000 --> 01:19:12.000 Right, so my only point was that if you were to write first-order conformal gravity in that form, then the expression you wrote for A as a linear combination of the generators and the conformal algebra, 01:19:12.000 --> 01:19:17.000 is kind of, uh, fairly natural, right? Because those things are coming from the… 01:19:17.000 --> 01:19:19.000 Oh, yeah, yeah. No, those things are coming from exactly from coupling component gravity to CFT. 01:19:19.000 --> 01:19:25.000 Yeah. And in fact, the point is that Euclidean gravity… sorry, 01:19:25.000 --> 01:19:31.000 ordinary gravity or other theories of gravity can, in principle, be obtained from, uh… 01:19:31.000 --> 01:19:35.000 conformal gravity is more or less the same way as 01:19:35.000 --> 01:19:45.000 forms of supergravity, as you know very well, are… can be obtained from conformal supergravity, so by using compensators or something, the equivalent of the compensators here would be auxiliary 01:19:45.000 --> 01:19:49.000 fields that you're adding, just like your B field. 01:19:49.000 --> 01:19:59.000 Because what's happening in, as I understood, was that for a fixed value of the vector V, F is basically proportional to Hodge star of B, 01:19:59.000 --> 01:20:02.000 Where the hot star is determined by a choice of V. 01:20:02.000 --> 01:20:16.000 Right? So, effectively, that's kind of the flavor of what you're doing. So, that kind of first-order structure can be replicated by conformant gravity. I'm not sure how useful that is, but it seems like a more natural way to think about 01:20:16.000 --> 01:20:19.000 uh… adding extra terms that 01:20:19.000 --> 01:20:28.000 already exist. Like, here, adding the Euler curvature or other topological terms seems like something one has to do by hand. 01:20:28.000 --> 01:20:29.000 And in conformal gravity, one has a… 01:20:29.000 --> 01:20:30.000 Mm-hmm. 01:20:30.000 --> 01:20:35.000 More or less systematic way of constructing those curvature invariants. 01:20:35.000 --> 01:20:37.000 that's one point. And the other thing that I didn't understand was, uh, it's probably not… 01:20:37.000 --> 01:20:41.000 Mm-hmm. 01:20:41.000 --> 01:20:44.000 Probably related to Anatoly's question, 01:20:44.000 --> 01:20:49.000 So you, in any dimension, in principle, you can write down topological gravity the way Witten did. 01:20:49.000 --> 01:20:56.000 And the minimal, uh, version of it is… I mean, the famous one is in two dimensions, which was in the… 01:20:56.000 --> 01:20:58.000 Consevich, uh, Witten Koncevic, 01:20:58.000 --> 01:21:00.000 story, but in four dimensions, it would just be, like, a metric, a gravitino, and an odd vector field. 01:21:00.000 --> 01:21:05.000 Okay. Mm-hmm. 01:21:05.000 --> 01:21:11.000 Like, uh, sorry, and even, uh, bosonic vector field. 01:21:11.000 --> 01:21:15.000 And they satisfy a simple algebra. Basically, it says that Q on G is the gravitino, 01:21:15.000 --> 01:21:17.000 Q on the gravitino is, uh… 01:21:17.000 --> 01:21:28.000 del mu phi nu plus del nu phi mu, the symmetrized derivative, and Q on phi is 0. So that's just a simple model for topological gravity, and one can write down an explicit 01:21:28.000 --> 01:21:32.000 action, which is fairly simple. 01:21:32.000 --> 01:21:33.000 Mm-hmm. 01:21:33.000 --> 01:21:39.000 I imagine you want to do that in the first order, so why would that… could that not be a candidate for the BF? 01:21:39.000 --> 01:21:49.000 that you have. I'm saying gravity, but I really just mean the conformal, the full group that you want. And I know how to get that out of conformative gravity. 01:21:49.000 --> 01:21:55.000 So that's also another… I'm positing this not as an alternative, but more like 01:21:55.000 --> 01:22:00.000 to say that the issues you have with the first-order formulation are a little more transparent in that way of thinking about gravity. 01:22:00.000 --> 01:22:05.000 Mm-hmm. 01:22:05.000 --> 01:22:13.000 I'm not saying that the conformal anomaly can necessarily be solved at all. I'm not making any deep statement. I'm simply saying that 01:22:13.000 --> 01:22:14.000 Mm-hmm. 01:22:14.000 --> 01:22:20.000 Uh, those formulations of conforming gravity seem somehow more natural to get the topological bulk theory. 01:22:20.000 --> 01:22:25.000 I'm curious if there is some connection there. 01:22:25.000 --> 01:22:30.000 So I was confused, I don't know much about topological gravity in higher dimensions. What's the relation of that to. 01:22:30.000 --> 01:22:32.000 Ah, well, uh, so there is a paper, uh, by Witten on just 01:22:32.000 --> 01:22:37.000 Standard gravity. 01:22:37.000 --> 01:22:42.000 topological gravity. That's the one where he writes down the model. I think his model, uh… 01:22:42.000 --> 01:22:49.000 is inspired by the 2D model, but it's actually supposed to be a four-dimensional model. 01:22:49.000 --> 01:22:56.000 And what's the relation to standard gravity? Is this a limit or is it? 01:22:56.000 --> 01:22:57.000 Restriction? 01:22:57.000 --> 01:23:06.000 Well, uh, that's a good question. So, the relation to standard gravity that I understand is that you can get both out of limit of conformal supergravity, or conformal gravity, let's say. 01:23:06.000 --> 01:23:07.000 Mm-hmm. 01:23:07.000 --> 01:23:11.000 The written model can be obtained by doing BRST on conformal supergravity. 01:23:11.000 --> 01:23:13.000 The super part is not that important, I'm just using super because there are extra fields that I can play with. 01:23:13.000 --> 01:23:17.000 Mm-hmm. 01:23:17.000 --> 01:23:20.000 The final theory is just essentially a model for topological gravity. 01:23:20.000 --> 01:23:24.000 Mm-hmm. 01:23:24.000 --> 01:23:31.000 So, that's the relation. I don't know if it's useful, but it's definitely a topological theory. 01:23:31.000 --> 01:23:34.000 So, uh, to the extent that 01:23:34.000 --> 01:23:41.000 you want to study a theory that's naturally coupled to gravity in the bulk, uh, and must be a bulk. 01:23:41.000 --> 01:23:42.000 Mm-hmm. 01:23:42.000 --> 01:23:49.000 Topological theory just occurs to me that that's also an interesting, uh… 01:23:49.000 --> 01:23:52.000 structure that comes out of… 01:23:52.000 --> 01:23:57.000 Basically, what I'm trying to say is that I think the McDowell-Mansuri, as I understood it, 01:23:57.000 --> 01:24:05.000 you get, in a particular way, as a first-order formulation, which is very natural, and it replicates 4D gravity, but that's… 01:24:05.000 --> 01:24:13.000 that's the only, uh, motivation for you studying it, right? 01:24:13.000 --> 01:24:14.000 Right. 01:24:14.000 --> 01:24:16.000 Yeah, I wanted to stay with the standard gravity ideas for just written in a way that made the limit more transparent. 01:24:16.000 --> 01:24:21.000 Yeah, but because the conformal gravity would… 01:24:21.000 --> 01:24:32.000 reproduce the symmetries of ADS4 somewhat more systematically. That's why I was curious why you weren't starting from conformal gravity and… 01:24:32.000 --> 01:24:33.000 and writing it in the first order formation. 01:24:33.000 --> 01:24:35.000 Well, that's… 01:24:35.000 --> 01:24:46.000 This quick push was just my ignorance. I looked into conformal gravity and… I didn't get much out of it, but I'm not very familiar with it either. So, could you get, um… 01:24:46.000 --> 01:24:55.000 Yeah, I can… I can probably send you something offline. I'm not sure how helpful it is, but I'm sure you already know those papers by Fredkin and Zeitlin and so, so… 01:24:55.000 --> 01:25:06.000 I did. Yeah, I did look at it even. I think I said them at some point, right? But… I was always thinking of those as a prescription for coupling CFTs to gravity in a nice way. 01:25:06.000 --> 01:25:10.000 Right. 01:25:10.000 --> 01:25:11.000 Yeah, this one here. 01:25:11.000 --> 01:25:18.000 Anyway, this is just, uh, this expression that you have on the slide right now somehow seems suggestive that if I were to work with conformal supergravity, this is a very natural 01:25:18.000 --> 01:25:26.000 way of getting, uh, expansion like this. Instead of, uh, saying that this is… 01:25:26.000 --> 01:25:32.000 a coupling of the boundary CFT to this decomposition. That looks, like, more like a choice that I have to make, but somehow 01:25:32.000 --> 01:25:43.000 if I already had a gauge theory with generators of corresponding to these symmetries as having been gauged, which is… which conformal gravity would do for me, then 01:25:43.000 --> 01:25:48.000 these couplings would be very natural. 01:25:48.000 --> 01:25:49.000 Mm-hmm. 01:25:49.000 --> 01:25:56.000 And so with the couplings in the action, and so would the limits of the couplings in the action. It's just that remark. I'm not saying anything more deeper than that. 01:25:56.000 --> 01:26:01.000 No, no, that was only how these guys derive these couplings. 01:26:01.000 --> 01:26:06.000 The thing always confuses me, I'm happy to have this discussion if you have time. 01:26:06.000 --> 01:26:07.000 Sure. Maybe we can stop the recording, otherwise the file will probably get too long. Not that we're… 01:26:07.000 --> 01:26:11.000 The thing… Uh, is this… recorded. 01:26:11.000 --> 01:26:19.000 Yeah. Yeah. Christina, can we stop the recording, probably?