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Jorge Pullin: Okay, so our speakers today is Lauren for those who will speak about corners, symmetry, and new pyramid for bombarded.

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Laurent Freidel: Well, thank you. Welcome everyone. I'm happy to be here. I want to thank you for you know, helping you organizing this great event, and for being so patient with me

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Laurent Freidel: today, I want to talk about corner symmetry and quantum geometry, and and

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Laurent Freidel: it's more of an overview. I'm not going to go into in details into any any aspect, and it's motivated, and I i'm going to try to convince you how it fits with a a broader program of providing a a form of paradigm for common gravity, or, as i'm going to explain, maybe a bottom up approach instead of a top down approach.

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Laurent Freidel: and it's motivated by the the recent chapter, we had to write with the Mark Geller and both gangland, where we summarize some of the recent developments for for quantum gravity TV audience, and which okay in the in the endbook of quantum gravity that

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Laurent Freidel: about it to organize.

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Laurent Freidel: I also want to. Thanks. You know, the many amazing collaborators, I I add, and I still have that help me develop my understanding, and then

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Laurent Freidel: I me already

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Laurent Freidel: get to to where we are today, in the picture, which is still a work in progress.

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Laurent Freidel: maybe a special thanks to the I've done a lot of work. So let me focus on the question. So you know the I. It's important when we we think about quantum gravity to ways kind of reset back of what? What do we think are the important questions, and that we, making progress

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Laurent Freidel: to wild them. So for me some of the important questions I I I am really deeply interested by our what are the fundamental quantum cavity, degrees of freedom. You know, if there's an atomic structure of space time, if there are constituents.

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Laurent Freidel: what are their property, how do we understand and and reach that another more general, broader question, which is like, what is the geometric or anthropic counting? You do. The area

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Laurent Freidel: well, for the horizon, you know, is is a count of entropy. Then it's giving us, of course, certain crew. But what what is it that it's counting

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Laurent Freidel: and related to? That is that we are talking about gauge theories. And we're talking about gravity where the observable, our default invariant, and we really use to understanding the nature of observable in usually in the context of usual kind of theory which are not Defo environment. So it's a you know. There's a challenge of understanding. What are the the fundamental observable.

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Laurent Freidel: and also some for me, something that really started this. This this quest is is we can we provide? It's a more broader question. Can we provide a model of common gravity that respects the presence of a planking kind of.

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Laurent Freidel: and the principle of general covariance, and the ability to that is, that is there an optimal way to discretize covering theory that break symmetry in a controllable manner. So let me emphasize that because I think

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Laurent Freidel: it's specially important for this audience that works for the people that works on speed Forum, or on low gravity, especially for the more junior audio, but also, you know, for some of my more senior colleagues, so it's it it. The following point is very important to the

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Laurent Freidel: in the form of a graph, or in the form of a triangulation.

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Laurent Freidel: You are by nature breaking the fundamental symmetry you know of of the theory which is the general covariance so general the form ofism under all all possible translation. You just have to accept that. So you know, if you study your spin for model.

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Laurent Freidel: it is very important to appreciate that that's been for model. It's a model. It's an approximation. But until we know

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Laurent Freidel: to what extent that approximation breaks the symmetry we care about, we we cannot really promote that model to the level of a theory. You know I've seen many people

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Laurent Freidel: trying to do that. I think we have to be more modest in our or I precise in our in our scale. So it's important that we have these amazing models that that

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Laurent Freidel: so? A lot of of the problem. But it's also important to appreciate that. You know we we need to. If we want to improve our models, we need to understand, you know. Do they respect the symmetry? How can we make them respect the symmetry of this year.

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Laurent Freidel: So this is, I mean especially because in gravity the expression of the symmetry principle is the dynamics of the theory. This is what we don't for match that. So I I just want to make that point very clear.

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Laurent Freidel: because sometimes it is forgotten in many discussions. So that's the the basis of of what we're trying to do. It's a quest to try to see. How could we improve? Lqg: Improve the You know speed for theory, and beyond improve our understanding of fee theory, so that we have a deeper understanding of what what it means to be

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Laurent Freidel: a generally Covariance. So the the approach I'm talking about is local. It's a perspective that comes from asking you questions.

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Laurent Freidel: One of these question is which was one of the most fruitful questions for me it was, how do we understand the decomposition of a gravitational system

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Laurent Freidel: into subsystems? It's a very simple question, but it's a very.

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Laurent Freidel: I don't know, deep and and and powerful a question, and then a a question that many people ask is, what is the nature of entanglement across sub region. If you are decomposing the sub regions, you need to understand the nature of that entanglement. A more broader question is, how do we understand quantization of one of in finite region. If I talk to my friends in fi theory, I they seem to believe that there's only one observable in nature. It's so the observable which is attached to a a. Some to take infinity, so it's size of the S. Metrics and fields in flood space.

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Laurent Freidel: or the as cft partition function.

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Laurent Freidel: and you know it. It seems strange to me, because we do physics in local region, and we should be able to have a framework that allow us to do and understand physics in in finite region, a little bit along the line of algebraic quantum theory. But as we're going to see, or maybe you know, as as it's becoming up right now, there's certain challenges that needs to be overcome.

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Laurent Freidel: And then these questions led: let me let us in it a lot of us to the questions, what are the symmetry of gravity? And then along the way, there's many interesting tools that I've been give up, you know. Covariant face space question to a big representation theory of I, you symmetry, group and calorie and geometry.

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Laurent Freidel: Okay, just so. If you fall asleep during the talk. Let me just give you the the main message there, and and hopefully that will become clear at the end of the talk. So the main message is that if you study the decomposition of gravity into subsystems, we can prove that the entanglement

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Laurent Freidel: is controlled by a symmetry group. not to gauge

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Laurent Freidel: by symmetry group I mean a physical symmetry which has non-violent charges. We checks not trivially on the phase, space and and around the physical little bit space so it's controlled by a symmetry group called the Corner Symmetry Group, which follows from the the the gauge redundancy of the total space so gauge

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Laurent Freidel: this symmetry. Group. The beauty is that it's it's corner symmetry group. It is universal in the sense doesn't depend on really the gravitational theory. Only you know, derivative you add, and you know what the shape of the region is

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Laurent Freidel: as we're going to see, and it gives us what's beautiful is that give us a semi classical phase space tool to understand quantum geometry in the continuum. So we don't have to discretize

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Laurent Freidel: classical geometry to arrive at a control on on some element of quantum geometry. In that sense it's kind of a bottom up approach, in the sense that if you have any theory of common verity could be looked, could be. String it has to submit to this a general principle that it has to carry a representation of this color symmetry group

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Laurent Freidel: at some level. In some you know semi-classical limit. So at the quantum level now we can, we can study the gop representation of this symmetric group. Think about the symmetry group a little bit as the the conformal symmetry group, for there was a invented by discovered by De Lavin polyac of the monarchy, of the study of of phase transition here. We claim that we want to understand gauge theories and and gravity. Then you know, one thing to do is to understand the gop or presentation.

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What what are they classified? The unitary presentations and their fusion works.

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Laurent Freidel: and what we find is that the symmetry generators, and they are Casimir's, act as geometrical operators, but because now they represented as as a symmetry generators, we know what to pondize on them in a controllable matter.

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Laurent Freidel: Okay.

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Laurent Freidel: Now, there's 2 kind of recent development. One recent development is that if you take the limit of infinity, large regions, you can connect the the analysis you can do at finite regions connect seamlessly with

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Laurent Freidel: people. Do with the S matrix, you know, and the study of a synthetic symmetries of theorem. And so that's a good check that you're on the right track. And then more recent development, which I'm. Very excited, about which

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Laurent Freidel: I I might not have time to really explain is that we are showing now the the area operator.

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Laurent Freidel: a a and and elements of the symmetry group are profoundly connected to the, to me that the modular action that is interesting to.

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Laurent Freidel: So what is the we? We're interested in your understanding, the composition in subsystem and space and tanglement. So the setup is as I like you. You, you take a a full Sigma is a full slice.

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Laurent Freidel: Okay, which has a boundary condition defined, and then we want to choose which is a 2 dimensional surface that divide this.

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Laurent Freidel: Now the surface is the entangling surface. It, and it defines what we call a could dimension to corner. It's a corner because you can think of that surface as being the corner of the cosine of the dependence

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Laurent Freidel: of the the slicing my L, which is the interior of of this from this corner. And if you think about

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Laurent Freidel: because that domain of dependence, or let's say i'm going to call it the cause of diamond, then it's entirely determined by this, the presence of this 2 2 dimensional corner. So this 2 dimensional corner represents, if you want some kind of a regions of space time. Now we know that if you want to understand an entanglement, we can assign to these regions of space time

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Laurent Freidel: a number of observable, and then we call a Sigma. I'm guessing you're seeing my pointer right if I move it.

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Abhay Vasant Ashtekar: Yes.

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Laurent Freidel: okay. And then we can also assign a Hilbert space right, which is a attached to this original Sigma, and then, you know, we, the herpid space is usually constructed by action of a see mount from Vacuum States.

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Laurent Freidel: So now we have 3 different cases of of entangling or factor visibility of the theory, because we have these 2 structure so in quantum mechanics, which is the simplest, and this is the really of of information. What we have is that the algebra

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Laurent Freidel: of a prior to kind of split, you know, if you want as a here. I could even have written the tensor product of algebra

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Laurent Freidel: and and Gilbert Space also split between the space inside and outside. So we have double factor visibility of the algebra and the but space. Okay. Now, if we move on to relativistic quantity theory.

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Laurent Freidel: one of the most beautiful result in relativistic quantity theory related to this kindler. A property of vacuum space is the bizon nano-vigman theorem

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Laurent Freidel: That's essentially proved that you know it's not possible to factorize hilbert space across a sub-region in quantity theory. So we don't have the the factorizability of it but spaces sheets to do Co. Total cohesi slice into Tens of products associated to individual region. And the reason we don't have that is because we have infinite vacuum entanglement at the cuts.

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Laurent Freidel: Okay, Another way to see that is that maybe as vector species we could, we could try to decompose them. But what happens is that

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Laurent Freidel: the the Hilbert space on the left is not October, not to the hipaa space on the right. And, in fact, this matrix here is almost invertible, which is the expression of the duration that you know. If I act here on any States here I can represent, I will be as close as represent any States on Cmr: okay, so infinite vegetables.

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Laurent Freidel: which is the another way to say, to say the same thing is that this universality of an of of the future like you, you give me any state in quantity theory. You look at at high energy. It's going to look like the vacuum state, because it has to be addemar.

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Laurent Freidel: and because it's at the marit, as in Fields document, and it's satisfied with the very beautiful. Now the beauty is that we still have that the of observable still factorize.

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Laurent Freidel: you know, as tens of product on subbridge.

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Abhay Vasant Ashtekar: So are you all saying law on that this entanglement that you are just like so beautifully explained to us.

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Abhay Vasant Ashtekar: It has something to do with.

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Abhay Vasant Ashtekar: That's where I'm going next. Yes, but even even for just

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Laurent Freidel: no, i'm not I. Didn't say corner symmetry yet.

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Abhay Vasant Ashtekar: Yeah, but i'm just asking your viewpoint, and just to understand where you are

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Laurent Freidel: at these stage.

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Laurent Freidel: Is this something that is a related only for gravity? Or is it also for it's going to be related only for gravity, for for quantum fee theory. What we have is is in some sense the portal version of the corner symmetry group in quantum theory is called the modular group.

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Laurent Freidel: Okay. but

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Laurent Freidel: it it's really only for gravity. That's suddenly something a a new weapons that that makes sense. That yeah, that that's fine. That's okay.

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Laurent Freidel: So in gravity and gauge 3, what? What's happening is that now? No. Okay. We You know we it's so. We don't have to

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Laurent Freidel: of the space, but we lose now factorizability of the the observable. Okay, it's essentially the statement of the No locality of the deserable, and we shed more time to kind of explain you that you know more nicely. But it's actually think about. You know you have a. C miles in my right. So you're here, and imagine that you're imaging a a with some line. So if i'm engaged, you're in with some mind, so the recent line is gauge in and the total space. But when I start to cut that with some line intersection.

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Laurent Freidel: my boundary surface, and therefore it's no longer a a gauge in variant with respect to as a as an observable restricted to signal, and therefore it's not on my a/C mile, because my job of observable is to see my left to be, you know, observable in Sigma, supported on signal, which are gauge in violence. Okay, so it means that we we are finding an extra level of of an entanglement or or connections engage theory and gravity, which is really the purpose of our study.

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Laurent Freidel: Okay? And this is what what I was saying is that now something new happens, and and which is where this corner seem to go, and and the proper way to understand and the resolve, this new, a a condition. The fact that the gauge environmental observable is is essentially to introduce, to understand that the zoom.

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Laurent Freidel: So this is the 3 case here, and that the factor is ability, property of the other above the in quantum gravity is really governed by a symmetry group. And now

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Laurent Freidel: there's some kind of

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Abhay Vasant Ashtekar: you are explaining it so well, so just to I mean even the last line that you're writing down, strictly speaking, is only for gravity.

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Abhay Vasant Ashtekar: because, for example, for Maxwell. I can just consider. You know it.

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Abhay Vasant Ashtekar: The electromagnetic field, electric and magnetic field, as we are then locally

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Abhay Vasant Ashtekar: in Sigma. I and Sigma are. and I would get an algebra of observers. Right? I mean.

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Laurent Freidel: Yeah. But we you you. We know that because of this, with so what's happening in you? You you have to different. So this what's going to happen is that in the union of Sigma of C Ma. Right? They're going to be. They're going to exist observable which are gauge invariant in for the Union

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Laurent Freidel: which are not gauge environment when restricted to a a subset.

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Laurent Freidel: and and and in order to understand what what's happening is that the observable which are gauge and for the union of signal we see my right now. They are kind of charged observable when you restrict to the subsets right. And and in fact.

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Laurent Freidel: understanding what are the nature of these charges, and and is is already. And you understand that these charges are already discharge, upering which the presentation of this Colonel Symmetry group.

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Laurent Freidel: So there is a a simpler but also interesting applications of this idea of Connor symmetry group to to gauge theory, especially in my 1 billiongauge theories.

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Abhay Vasant Ashtekar: It wills up later. Thank you.

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Laurent Freidel: Thanks.

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Laurent Freidel: Okay. So this is the the general ground. And and I think this perspective, you know it

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Laurent Freidel: for some information. I think it's very important in general, and and we need to be developed so now.

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Laurent Freidel: what what is the the issue? I'm trying to just trying to to say the same thing. So the the in gravity. Let's go to my gravity, because, as as as I mentioned the Gish theory, there's some element which are there. But there's something rightly code about gravity, so it's focused now purely on gravity. So you know, when essentially, but it's it's about gauge symmetry. So when no boundary exist and you have a gauge symmetry, the time evolution operator. So in

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Laurent Freidel: okay, now, see you, know this is called the Problem of Time.

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Laurent Freidel: Now the situation changes in the presence of a spacetime boundary or a space-time corner, and here I am going to be so. Space-time Corner is when I just have a surface like that. I just looking at a causal domain of dependence and I Don't move the corner.

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Laurent Freidel: In that case, what's very important is that this is the statement that if i'm studying corners, I do not need a specification of boundary condition.

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Laurent Freidel: Okay, I do not need to talk about boundary condition. It's absolutely generic. The theory is kind of the phase space of here is well defined. Now. Often people, when they introduce a

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Laurent Freidel: they introduced batteries and countries that when you have a line there that goes up and down like you need. Yes, so. And if you have batteries, then you need to introduce battery condition, and it becomes obviously. And the theory depends on the boundary condition. I don't want to do that. Okay. So just to specify the language, because often

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Laurent Freidel: people think that corners are boundary, they're they're fundamentally different. Okay, Boundaries are what people study before, and callers are are what we are studying, and it's simply that I don't need to specify boundary condition, because I can really access the full information we're in the back.

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Laurent Freidel: Okay? No.

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Lee Smolin: This is because corners are where boosts dominate. Where

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Lee Smolin: is this all really touching the the effect of the boost?

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Laurent Freidel: This is where it's. Yeah, this is. I hope I have time to explain more of that. But this is really where it's going. This is why suddenly the symmetry of gravity is good. Yeah, you you kind of a

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Laurent Freidel: I agree. We'll see if we have time to go there. But of course, yeah, a deform of that fix a surface is called a boost, and we we're going to look at that. Really, how important this this subgroup is. Is there?

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Laurent Freidel: Okay? Oh, hey, can I ask you?

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Laurent Freidel: And yes.

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Deepak Vaid: so I mean. I was also under the impression that the account owner, and then boundary that

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Deepak Vaid: you make this. you know. explain that a little bit

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Laurent Freidel: again. Why? Why? Why? Why? Boundaries and and corners are very different. So if I have a boundary, so your imaging you have a line that goes up there. Okay? And now you you make a little perturbation here. Then, you know, whatever the perturbation is going to cook to cross

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Laurent Freidel: that boundary, and therefore my system, in some sense a is kind of open, and then I will have to specify what what's happening there, and usually that doesn't work. So you don't have you don't have a so in the presence of boundary. The simple kick structure is not conserved. This is system is open, and therefore I need to close the system by putting it's a dirichlet back Recognition.

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Laurent Freidel: Okay, if I have a quarter, and you know, I look at what's what's happening at the corner, and I here Boom! You know it. It it doesn't really change the boundary condition there. Of course it's going to change the the aviation status here. Okay. But but

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Laurent Freidel: in terms of what type of conditions I put on S. There is no need for any restriction, because the system

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Deepak Vaid: is it? Is it like a space? I can. I can think of it as a space like boundary. Yeah, yeah, yeah, you can. You can think of it. No, no, it's not a space like boundary. A boundary is called. They mentioned one, and a corner is called they mentioned to. So they're completely different. That's that's what I mean. That's what I mean by when I

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Laurent Freidel: when I said boundary. What the picture in my mind is that you have a 3 dimensional manipose.

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Laurent Freidel: Yeah, that's why the corner is the boundary of a of a 3 dimensional slice. Yes.

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Deepak Vaid: okay. But that's why you distinguish the 2, because they have such a different If I start to call them boundaries, everybody will get confused. So

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Laurent Freidel: yeah, yeah, it is. A corner is a space like boundary. If if you think that's 100% correct. Okay. So here as Lee's mentioning, you know, because these corner is fixed, what is the time? Evolution, which is the time evolution generating by I mean that look like translation in the bulk. But I really kind of like boosts near the corner. Okay? And then we we can compute. In that case we go back to a theorem that was proven by cone, and I mean it's like we have to kind of the same theorem, you know, Con and the Rovelli have this beautiful paper that says

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Laurent Freidel: you give me a some region there's a notion of time. Well, here you go. So in gravity there's no time.

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Laurent Freidel: If you give me a submission. There's a notion of time in the sense that at least there's a notion of the conjugate variable to time, which is the Emiltonian variable that generate the boost. The translation here, and what we would call time is, in my language what I would call the edge mode. I don't have time to explain the edge mode, but in some sense, you know, we find at in the in the level of classic, or we are exactly the the analog of this.

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Laurent Freidel: Okay, so that thank you for your question. Maybe I sell us, but it's important people understand the thing. So now this property, that the what's very potential, so that this energy is a pure surface syntax, it has no bulk contribution, because

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Laurent Freidel: it's a constraint, and this is something that people in quantity theory repeatedly cannot

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Laurent Freidel: absorb or understand, because we use to study the energy which is a bulk expression. So this is anywhere. Gravity is really special, and I would I want to mentioned on my heart because it was the first one to say that autography, if you want to call it to gravitational autography, it was in the context of boundary, but it doesn't matter

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Laurent Freidel: is in aerence

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Laurent Freidel: to the principle of equivalence. Okay, so this is essentially why gravity is from the motor. You look at. Okay, the energy localized on the battery

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Laurent Freidel: Erez agmoni. In this case he was at infinity. So essentially what we did was really am in 2,006, and it's not. It sounds like self advertising, but I I can assure you if you read that paper, you 2.

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Laurent Freidel: The paper is very enlightening on my front. You know we we show that this this properties, what is the foundation of what we call local

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Abhay Vasant Ashtekar: Laura. But since we're talking about it, I mean this idea about infinity and

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Abhay Vasant Ashtekar: the boundary I mean, we. We have been discussing this since the early eighties, and I have a lot of discussions with Carol to C, for example, who did not First, I appreciate this, but in the in the space of in the case of infinity. The idea that the Hamiltonian is really the boundary to something that I can, emphasizing

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Abhay Vasant Ashtekar: very, very strongly, since

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Laurent Freidel: totally agree. You cannot agree more. And and I yes, that's somehow No, no, you're you're right. The the you mentioned only that matter of as being the first one. No, no, no, it's okay. So it's true. No, you you you're right. You're right. I was. I was just saying that yeah, it was more explaining. Okay, because I was you all right. You're right. I should. I should change, upgrade my slides.

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Laurent Freidel: What I was. I was. It was because more of this quest of oligarchy, because when I use the word of a log Ii. People say what you do, i'm not doing it If you i'm not doing it. If you look at the about having a a boundary and specific boundary condition and a lot of other I put. This is so when I

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Laurent Freidel: It's just that I don't have a better time, for when I talk about the I i'm just saying gravity is a little graphic in the sense that that done first the explain. But as you as you're in, and you're totally right. I should have mentioned that I think. Phoenix. In fact, what we're doing

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Laurent Freidel: a a lot of the checks of what the we're doing, or or or the the knowing that we're we're doing is right, is that he connects seamlessly and perfectly with all this beautiful work that you started, and then people rediscover, and they're under redevelop right now at the infinity. Okay. So if you're in comfortable with having a corner symmetry group.

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Laurent Freidel: Be my guess. And just think that this corner symmetric group is the Asymptotic Symmetry group, and then you you know, and then you're the same. I I know, for an audience in quantum gravity. This is okay for people in quantum theory and string. Often this is where they relax. They like You're not allowed to talk about finite region and say, okay, fine. Just you think it's it's infinity.

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Laurent Freidel: but totally true. I I should. I should upgrade and mentioned that, and I wish I had more time

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Laurent Freidel: to give credit back to and and for people who know, as you can see, I i'll.

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Laurent Freidel: Oh, this is the same structure. So there could be another motivation. My motivation was the the ambiguity of discretization. Another motivation is, you know, would be well. Can you do in finite region? Can you do what what you can do at infinity?

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Laurent Freidel: I hope that gives back the the right credit.

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Laurent Freidel: Okay? And tangling corners. So what I you know the main point is that then you call those carried the presentation. Therefore the fundamental group of symmetry that that you know, acts on this region, and how do we distinguish gauge from symmetry is very simple. A gauge transformation as 0 charge.

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Laurent Freidel: Repeat the gauge information as 0 charge. So it's on physical. It's pure redundancy, the symmetry

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Laurent Freidel: transformation as non-zero charge so there's a clear cut between what you call gauge and what you call symmetry.

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Laurent Freidel: So you look at the way you understand your symmetry is that you look at this expression for the so. This is an integral on the sphere so clearly shows that if you do a little blob in the middle, where it will have 0 charge unless your theory is fundamentally completely non-local, like string theory. But in field theory, you know.

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Laurent Freidel: then the charge will be supported by transformation. That act on the boundary. And then you look at the charge aspect, and if it's 0 or not, then the determines whether it's symmetry or pitch. Okay, so this 0 would be with you. But what is symmetry and what is, c. And as much as you can gauge.

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Laurent Freidel: gauge, and then and see you cannot gauge symmetry. And that explains a lot of the back end force in the literature and the confusion you you You can feel, if you read the the past literature, that there there was a bit of a confusion about what is gauge and what is image? I think now it's completely clear. The the confusion is being completely resolved.

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Laurent Freidel: No, I again, I want to say this. This confusion, even in the case of spatial

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Abhay Vasant Ashtekar: space, like surfaces, was completely resolved in the eighties. When I, you know, when we were first doing new variables and such thing that th this was when the discussions took place that there is a distinction between gauge and symmetry. It was something that was that that has been a highlighted in many papers, and I

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Abhay Vasant Ashtekar: but I I mean the key point. Here is, however, this requires an extra ingredient which is Hamiltonian framework.

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Abhay Vasant Ashtekar: Right? Yes. Only then. Yeah. So that is it. Yes, I wish. I wish, you know I will have the version, which is the spin from version and the path integration I have to rely on entirely on monetization in the yeah, 100%.

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Laurent Freidel: Okay. So the the RAM tells us that not only we have this no 0 charge. But there's these shows represent elements of the spacetime. So essentially, you know, I know exactly how they did the quantum commutator of these charges

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Laurent Freidel: A. On the quantum, you know the the quota algebra, or on the classical phase space. Then you know, I know exactly what all the structure a constant, and if that object is only algebra, these structure constants are going to be constant, not not algebraic or functions, as we encountered in in the quantization of the we know the the equation.

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Laurent Freidel: And so but the subject. These, these charges, you know, are are also a element of the metrics. So you give me a matrix around the the surface? S. And I can say, oh, here is the component of the metric which i'm no MoD. They generate this and to are the one which are tangential. They generate something, the mixed index that you like something else. So I can look at the metric and tell you exactly after I impose the constraints which component commute with what and how and and what's exactly the the detail of the commutation relation.

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Laurent Freidel: and that's Why, once you have this information, it's much easier to discretize the theory, because then you know that you have to discretize with you in terms of this. No commuting variable.

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Laurent Freidel: and it's not commuting variable. So representing the geometry directly gives us a direct picture that the geometry is not commutative. It gives us the no communicativity of the metric component at the location in the corner, and therefore finding the polynomial presentation of this symmetry group is equivalent to quantizing geometry.

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Laurent Freidel: That's the message. Okay, if there is no

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Laurent Freidel: question on that, I'm: Just okay. This is just a a slide to move on where you know the lots of development and covariant phase space. So if you know a little bit, you know we have a simplic potential, You have an action of a symmetry group. There's a part which contains a you know the symmetry charge, and there's the notion of Ch. Of flux. Okay. And and so, when we have this notion of symmetry at the corner. We have to distinguish the symmetry that

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Laurent Freidel: do not. It's not enough to not have 0 charge. We can distinguish the symmetry which are kinematical. Let's say that have 0 flocks and the sync, which are like the the boost that that to Lee was talking about from this image free like translation. If I translate the the surface, it is an on it it it carries an on your charge, but it will carry flux.

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Laurent Freidel: and the beauty is that we can know. Not only we can understand the child, but imagine you give me 2 corner charges at 2 different corners. Then you can understand the evolution, a a

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Laurent Freidel: A, a, as as kind of some kind of transition between these 2 charge. And so we can recast the the entirety of a locally conservation equations of motions of govity as local conservation, and that's gonna become very useful because I do quantum level. I don't know what is the content I mentioned the question. But I know what is the what is the the the quantum analog of for conservation, though?

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Laurent Freidel: Okay. So as i'm saying, there's a it's going to be important that you know. It's a question of lingo, the when there is no flux. We have what we call the Corner Symmetry group gs.

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Laurent Freidel: and we also have a. We have non-zero charges like translation charges, which, from what we call the extended corner of symmetry group, and they they're both quantizable, and they have a very different role. Okay. So when I mentioned the corner symmetry group, I really mean the one with no flux at this stage.

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Laurent Freidel: Okay.

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Laurent Freidel: Yes.

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Lee Smolin: it's just quickly. A. If you're in Lorentzian signature the way you can

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Lee Smolin: trap this ambiguity is in a corner

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Lee Smolin: to to approve. If you're in the or you're getting the signature. Is it right that the only way you could?

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Lee Smolin: It is in a topological.

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Laurent Freidel: Yeah, I think what i'm what i'm saying. It's profoundly Lorentzian. I cannot do. Euclidean quote on gravity. I'm: sorry for those who wants me to do that. Yeah, no, it's profoundly. Once you know it. Yeah, okay, No, for me. It's clear at this rate. You the question of bye bye, it's. I mean it's Lorentz. Yeah.

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Laurent Freidel: And

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Jerzy Lewandowski: what is the meaning of this symbol. Capital! I

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Laurent Freidel: it's a it's a field space contraction. So here, you know, on fee space. We have a capital calculus, and we have a differential, and we have a notion of a custom contraction.

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Laurent Freidel: So it's a contraction of a 2 form, a longer a fee space. Okay, it's just that.

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Laurent Freidel: Okay, we have to develop a technology which is capital calculus and free space. Once you have it, it simplifies your your life and your calculation by effect of 10. But

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Laurent Freidel: So we which for people who worked on that, I I promise you. Just so. It's just the techniques. But but sometimes techniques is useful because it allow you to do stuff that you will not be

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Laurent Freidel: able to do before.

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Abhay Vasant Ashtekar: Yeah, just just to sort of make sure that I understand that. But so the corner Symmetry group is the one who just tangential. The call number? No, no, no, it's more complicated than that where we get it's the one so symmetry group. Extended corner is Don't do charges which are non 0 at the corner.

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Laurent Freidel: and the corner symmetry group is also the group which which have a non-zero flux. So now let's look at what what it is in. This is flux with the null surfaces. Where is the flux being calculated? The flux is calculated on the corner. So there's a notion of that's the corner. Only. Yeah, it's flux at the corner. It's it for you. You would know it's the pull back of the the contraction of the simple technique for anyway it there's a this precise formula for it.

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Laurent Freidel: Okay, it's at the corner. Everything is at the corner. Yeah. Now, if you know, if you did this, this is where right now I live at the

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Laurent Freidel: So okay. So I I repeat, here we have this on tanglings here. So what is the group we're talking about? And and so in gravity.

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Laurent Freidel: Let's say you take the Einstein gravity. You do the calculation you compute the charges and the you. You look at at what what you get, and you find these groups. So this group has a very

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Laurent Freidel: simple I mean a relatively simple structure. It is a semi-direct product. So this symbol is submitted by products. So the the core of the group is Df. S. So this is what I was mentioning. This is the space of transformation along the corner.

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Laurent Freidel: Okay, then, the next part of the group, which is extremely interesting and super super important is the boost transformation. Sl to R S. So what's happening is that at the corner? You know I have 2 normal direction. Think of it as 2 in our directions. And I can do kind of transformation in that null plane. Okay.

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Laurent Freidel: by by any arbitrary confirmation. As long as that transformation doesn't, you know, fixes a point on the sphere. So at every point on the sphere I have an asset to our, you know, family of transformation, and that's what it means you. This is our the this power s means functions from S to S to our okay.

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Laurent Freidel: and this this. This part, which is in blue is what I was calling the Corners image group. It's the keynote, Kinematical group, a part of the group. So it contains defomorphism and local boosts.

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Laurent Freidel: Okay? And then there's an a next track open, on which is, I have 2, no directions, and I can translate into these 2 in all directions. So this is the super translation part

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Laurent Freidel: which is then dynamic, or this one has flux. The black pot has flogged the blue as no. So what's very important to appreciate that it

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Laurent Freidel: honestly kind of surprising is that this group is doubly universal. So first, it's very important to appreciate that this group is the same group of symmetry. If you take a you know, a small sphere, a big sphere, or a horizon, or a isolated, or rather it is, you know

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Laurent Freidel: it doesn't depend on the shape. Okay, and you know, small diamond and large diamond. And, in fact, you know we have. We have a proof that if you you take the limit of this here to infinity, you can over recover the asymptotic

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Laurent Freidel: generalize. Bms group. Okay. Now the other thing, which is that's maybe not so surprising. But it's very important. So what's changed? What? Yeah, what changes when you change the size of this year for me in terms of our presentation theory is the is the value of the Casimir. Okay, but not the group.

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Laurent Freidel: Now it's the same group. It's more surprising for Einstein gravity, or you could say, Well, I want to look at the most general theory of gravity of a metric which is def for environment, but contains I a derivative.

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Laurent Freidel: And I Don't, I don't care how many I your derivative you will find you would think. Oh, but that group is going to be extended by the number of derivative that appears, and that's not the case.

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Laurent Freidel: It's the same group for ensuring gravity of any higher order derivative formulation, no matter how many derivative you have, what changes when you do. That is kind of do not a charge get modified. But the group is the same.

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Laurent Freidel: Okay, very important. Okay. So that's the the 2 things. You know the 2 point about about this, and you are the references where some of this is proven.

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Laurent Freidel: So Just a a word about symmetry on not surfaces. I mean, right right now. There's a lot of study trying to understand how your radiation works infinity, and then so it's. It's kind of interesting to try to rest. Take this group of symmetry to to large surfaces. What happens is that if you take this symmetry group here, the Universal symmetry group, and I I just look at the the subgroup of the group that preserve a particular, not direction. I'm going to find

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Laurent Freidel: which we we call now a thermal current structure, but the the name is not important. Then what you find is that

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Laurent Freidel: the group of symmetry that preserved this not structure and Telmon means that you also preserve a notion of a temperature. It's called the Bmsw Group. So the defense stays. The reset to our of course, becomes just one direction, because now you can rotate, you can just rescan the null direction, and the 2 superposition become one and locally. If I have coordinates where my sphere is at U, equal to 0 along a north of faces.

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Laurent Freidel: This is what it simply looks like. You know the super translations are just the time derivative we don't do, and there are functions on the sphere. Defense is just a vector and the vial transformation is this boost local boost transformation. Here we depend on here and then at infinity. There's a next high level of the generation. That is, that the boost.

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Laurent Freidel: a a, a, a a component of the group, because you fix the area, form at infinity, becomes it's called the Gbms group. It's essentially that these vial factors simply the divergence of the of the vector fee. Okay, so that summarizes. And so

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Laurent Freidel: at the end of the day, what i'm saying here, this is just one group, whether you're in infinity. Fine. I know.

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Laurent Freidel: Okay, Gravity I a derivative of gravity. This group is universal, and that's why i'm confident in saying that any theory of common gravity needs to connect the presentation theory of that group. So that's

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Laurent Freidel: in that sense that I You know we. I think of it as a bottom up approach.

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Laurent Freidel: Now.

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Laurent Freidel: you know if you want to say, okay, Well, how does he connect to loop gravity and who gravity? Somehow we have different group, and you know we have the essence of the emails, the parameters. So there's this work with we did with with Mark, and and then delay where we explain a little bit what happens when you kind of.

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Laurent Freidel: and I think you know also, Simone, as some some work earlier than that where we look at the connection between, you know first order, and so on, all the other formulation. So I I recommend you're interesting that you look at this works of Simone, and then

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Laurent Freidel: and then us here, and essentially what what you do when you adding is boundary term, is that you might be adding extra edge modes or degrees of freedom that are kind of really located only on the boundary.

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Laurent Freidel: and in that case you you you might. You might get different groups, and it's possible that at the quantum level. This means that we're talking about in the equivalent quantum theory, right? And so what's happening is that in the presence of first order? So you still have this general structure of defense semi direct product, with what was before the Booth group, and then super translation.

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Laurent Freidel: But they say, if you introduce Einstein, Captain Old's formulation of gravity. You're going to find that this group here not only contains the boost, but also contains, you know, 150.

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Laurent Freidel: The electric flux generator, the internal. Listen to see generator, and this this is where the loop gravity variable survive. Right? So do the gravity variable survive there they don't appear in in the in the metric formulation, but they appear here now.

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Laurent Freidel: Yes.

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Deepak Vaid: Can I ask you for him?

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Laurent Freidel: Yep.

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Deepak Vaid: So what happens? Like, for instance, if you

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Deepak Vaid: include the you know, gauge fields or scalar fields.

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Deepak Vaid: does it? Does the connect to group change?

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Laurent Freidel: Yeah. So if you include scalar field, No, it doesn't change. If you go gauge symmetry. Yeah, there will be a extra.

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Laurent Freidel: a gauge gauge transformation on which to this this object acts as a natural, You know, symmetry like product. I am not talking about that. But yes.

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Laurent Freidel: okay. Yeah.

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Laurent Freidel: Okay. So so in general, we might have this group here that contains the boost, but also contains like extra factors. So it's the one with the 2 when we know is the boost factor I described, but also the electric flux vector which is a To Cs right? Okay.

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Laurent Freidel: which is functions from this year to a. So this one we organize, and the charge of the symmetry of the the our famous electric charge right, but covariantize for us to see.

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Laurent Freidel: So now to to to remark first, what's beautiful is that even if that factor and they have several example is is the product of the booze times, the local time. Something else.

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Laurent Freidel: There is a general notion of that I call simplicity constraints is that if I look at the Casim years of these groups.

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Laurent Freidel: so I can have several factor. The casualties of this group. This internal group are always proportional to the area form. Okay, so that's really what matters? I'm: I I have completely little group, but somehow they are all sharing the same information that the the in this group is always the area for

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Laurent Freidel: okay, and and somehow it is important because DC. Area form and this universality of the era from is related, I think, and it's it's it's coming. A paper is coming about that to the universality of the modular group action on sub-

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Laurent Freidel: but I hope it. It explains a little bit. Do, and look. Gravity is when somehow you you restrict this case, you fix a normal gauge, and you're in this case to be just a function of this fear on this, you 2. Okay. So you you can see that look. Gravity fits into that. But that's just a a

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Laurent Freidel: 1 point subsets okay of of of the whole presentation theory of that curve.

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Laurent Freidel: Okay. And then yeah.

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Laurent Freidel: And this is explaining that in several papers there, that, and I recommend people to look at the review with the reference would be there. So okay. So now I have this group, and I hope I've convinced you that they are universal, and they're worth your time and study. So you know

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Laurent Freidel: Well, if we are there, then, and we, you know we need to study their presentations, and then understand, You know, what are the representations of this group? What are the Casim is because this is already the elements of quantum geometry.

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Laurent Freidel: So, as I say, the little group of the Boost group here is the is the area element, and so when we have a semi-direct product like that. It's clear that to study the opposition through that group you need to study the representation. You're very little group.

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Laurent Freidel: and of course, the group that preserve any area form. It's called the it is the little group, and it it is well it is you can identify. Does the local modular boost group the the booth do so. You have a cas in the area square of queue. This is a generator of transformation.

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Laurent Freidel: The claim is that that generator of transmission to generate local books at at the corner. The one

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Laurent Freidel: is this S. UN. And unlimited goes to infinity. Well, so this is another. I'm. I'm. Going to mention that later. So he's referring to the fact that well, I could do something. And this is something we have done, and it's really fascinating is to just to say that the little group is the group of area preserving the form of zoom, and there is a canonical way to introduce a fundamental cut off into that group.

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Laurent Freidel: Okay, so here it it would be like going beyond just quantizing the group. If you, If you have a group of symmetry, you classified the representation theory. What it is inviting us to do is to do something even more radical to a a proposed, a deformation of that group. I'll just mention that on past. But yeah, yes.

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Laurent Freidel: okay. Now, the key object. So we have this quote of queue that generates the the boost, and then we have a generator of the phomorphism which is the generator of of dephomorphism, right? And and you know, if

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Laurent Freidel: yeah. okay, and then okay. And if I denote and a do, a set to 1 one. Okay, essentially, you know, we showed in one word that we can specify the the generator of this little group, and it it

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Laurent Freidel: it. It is related to the outer virtual geometrically, and it's a generalization of of vaulticity. If I think of the subject as the freed, and then we can classify some of the okay. I'm: not going to go into the detail of that. What's important is that when we start to do that work.

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Laurent Freidel: We realized that some all part of the work we need. It was done by another mathematical physics community, which is the community that inherited the idea of Long Dow that reads our our fundamental quantum mechanical system because they are Milton, you. And there's a beautiful series of where by Kezin and Anna and Marsden and Rescue, where people have developed this theory of representation of this group, because the group of symmetry of gravity is isomorphic to the symmetry group of 2 dimensional, idle, dynamic.

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Laurent Freidel: Okay. And the analogy is profound. It's just means that the area density score of queue plays the role of the free. Density, the the the the the generator of the from office in place, the role of the of the freed momenta and the outer coverage replace the role of the free devices, and we know that in

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Laurent Freidel: the classifications of quantum free this is a characterized by the problem or presentation theory of this symmetry group, and it's really a a characterized but testified by choice of area and vorticity. Measures

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Laurent Freidel: or and and W. Are densities on this.

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Laurent Freidel: and then depending on the nature of the density you have. You have 3 types of read. Whether these density are both absolutely continuous or row is discrete. This one is continuous. That's type 2, you know that's a molecular feed, or both role and and W and the are discrete. In that case you have a quantum freed of a type 3 like alum, 3.

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Laurent Freidel: Okay. So now we can understand the role and W. To be labels of the courage and the classic or level label of the courage. It's a bit, and then the presentation of the free group.

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Laurent Freidel: And then, as I was saying, classical freed grew, correspond to a choice of density measure which is absolutely continuous and quantum for its correspond, to which we're both role and the vorticity are now counting measures. Okay, discrete measures. And in that case when Row and W are both scanting measures, we have a constituent picture for the free.

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Laurent Freidel: Okay, some some kind of atomic representation with the 3D is made up. So if we do that for for for gravity, let me give you some elements of the representation theory. So.

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Laurent Freidel: because of this analogy, you know the area is the free density. The auto curvature is the then essentially, you know, we can do the same. A picture of of discovering what is the atom of geometry by quantizing this group, and we get a constituent picture where

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Laurent Freidel: essentially we have what's the what happens in free is that we have a free molecularization or atomization. And this is essentially the fact that we have area constituents. So this is the fact that we say, okay, the area measure can be written as a sum of their tough functions.

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Laurent Freidel: And then Vtex quotation is really a deeper form of quantization that we have moment tag monetization. And this is where, for instance, spin is going to appear right

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Laurent Freidel: now. Here I give you the like a fundamental, discrete representation, where role is a sum of that function. It's point like, and then the momentum. So what's important is the opposition of the deform of his own group, and I think it's important because that's one thing that

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Laurent Freidel: that in Lqg. There is sometimes of a presentation of the different office in group, because we just look at functions, and we look at the action on pull back, but I think it meets an important quantum number. The claim that we're making here a a a, a, you know, in in our studies that the opposition of Deformorphism group is characterized by 2 quantum number. One is the dimension of the representation.

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Laurent Freidel: the call for more dimension of the presentation or the weight, and then the other one is the spin, and i'll do the answer Well, PA, so the 9 representation is when you put Delta and S to 0. This is when people doing loop. Okay, and we just say opa act just by translation. Okay, that's the first term here.

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Laurent Freidel: But in general, you know there could be a divergence to PA, and it's like it's related to a notion of multiple moments. If you, if you study multiple moments or not horizon. And it's this call from a dimension. And it could be also an issue of spin here which is carrying demo. The

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Laurent Freidel: Okay. So it's a lot of work, but words here. But just keep in mind that i'm saying, yeah, we need to represent the action of different move of zoom group, and by looking in a in a proper manner to the representation theory, we don't necessarily assume that the representation carries 0 charge and 0 spin as

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Laurent Freidel: as it is implicitly assumed in that Fuji. Okay? And so again, we get what I call a notion of area constituents in the continued. So it's in the consume. I mean, I didn't. I never introduced a graph or anything like that. I'm just looking at classifications of our presentation theory of continuum group, and they have this kind of focal presentation, if you want.

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Laurent Freidel: So it's similar. Some aspects are similar to refugee. So here the molecularization has nothing to do with the fact that the area a row I is quantized. Of course.

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Laurent Freidel: in Lqg: we have a next line to that. We said that this object row I as discrete spectrum we're under, and the and the image the parameters enter into this quickness. Of course, if we have a local area, G, we are force

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Laurent Freidel: to have this molecularization of the free. Okay, and this, in fact, Volt gang will, and was the first one to really kind of realize that it's possible to have a area discreetness into continue a from from kind of a focal presentation. So that that's a very landmark kind of a shift in our understanding.

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Abhay Vasant Ashtekar: Okay, I think your clarifications, Laura, and paper. So you you, You'll be talking about this. What is it is? Is that the same as just the you know in the context of the isolated or the various horizons where the rotational one form is that the same as what is city or

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Laurent Freidel: exactly exactly I a is the same as the and the is exactly the magnetic multiple generator.

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Abhay Vasant Ashtekar: Okay? And the second thing is that I mean when when John buys and I. We we're doing the entropy thing, I mean. We really had to go to this mapping class group precisely because they're punchers in this here, and therefore it was not just the you of the group. You might, or something which is extension. So it looks.

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Abhay Vasant Ashtekar: That's some elements you are in common. Do you know anything about relation. I I I I agree. In fact, I mean, you know there's there's a good reason. It's so. Me that paper which goes back to John and and created. That was very influential, because I remember that at the moment I like it. Looks like it's the good of, you know, coming from this

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Laurent Freidel: discrete object that seems to be leaving the continuum. But what are they right? So? And I think it's that question stayed with me since the day I was at at at Penn State when it happened. So yeah, that for me it's the same. And I think there's a reason that it's been pushing, because for him he worked a lot on the

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Laurent Freidel: the black hole horizons. And then, you know, by the more you try to understand the nature of these degrees of freedom and what they mean classically. So. Yes, there is a and it would be nice to kind of now, redo back the connection. It's. It's entirely. Of course it's it's kind of connected to these those other line of thoughts. Yes.

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Abhay Vasant Ashtekar: Yep.

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Laurent Freidel: Yeah, that's another motivation where you will end up in the same place. I agree.

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Laurent Freidel: Okay, as we mentioned earlier, you know, there's something really kind of fascinating to me that somehow

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Laurent Freidel: it it seems that if you cannot for a moment forget that the sphere is a usual sphere, and you say so, Fuzzy sphere. Okay, the coordinate Sigma don't commute. Then you you can do exactly. You do everything.

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Laurent Freidel: It just turns out that now this row and this PA they are going to be generators of S. U. N. Elements or generalization of sun, we sure that the the boost generator generate Sqn. But now you have, like a you know, a locally compact group that appears after this regularization. So that's something

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Laurent Freidel: to think about, and we get some kind of natural matrix model coming from gravity. But it's.

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Laurent Freidel: you know we're going beyond studying the symmetry. So I think it's it's kind of interesting. I think I need to wrap up.

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Laurent Freidel: Let me. Okay, let me just say in one word, and I I don't have to time to explain. So I just talked about the corner symmetric go. Maybe we want to understand what happens into 2 Corner. In general this is super complicated. So let's understand the relationship of 2 corners which are related by in our translation.

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Laurent Freidel: and it's a. It turns out that there, okay, we can do a lot of progress. And this is where a lot of developments arise, and we can understand the dynamics so clearly. Maybe, instead of being space like a time like going not is is really the key.

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Laurent Freidel: And and there's a beautiful. The beautiful new understanding is that we can just understand the conservation of charges as as some kind of calorie and conservations over energy momentum tensor. Yeah, which is written. And we have a canonic or simplectic form on any north surfaces with no restriction on the level of symmetry.

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Abhay Vasant Ashtekar: And you just say, what about what is Tau and what it started?

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Laurent Freidel: In some sense Tower is like the edges check one form and Tau a. B contains the trace of Tai B contains the the expansion and the traceless spot is really the sheer. But but yeah, it would.

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Laurent Freidel: I. I got to give you the element, and then we find that this Igj connection is is conjugated to the variation of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of of of of, and Of course the shear is is conjugated to the metric component.

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Laurent Freidel: Yes, yeah.

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Laurent Freidel: Okay. So sorry. I'm: I'm: I'm jumping away just just to tell you if you want in the in the review there's a there's a kind of more in-depth analysis of this and what I what I like is that we put together? Maybe I don't know 30 40 years of study, not surfaces. And now it kind of gel down to something

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Laurent Freidel: very manageable and universal, and and you know, connected to this corner symmetry that you can, you know, take on your own and and study if you want, and and it it. It connects a lot of different. So literature that comes from different groups and country.

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Laurent Freidel: Okay, so maybe one Id. This is a an idea, I really i'm really found it's something we discovered with the need in one of our study about something infinity. I think it's something that Barrett tried to do, and then, I think, working at some point that something really is that? Okay? Well, once we have this

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Laurent Freidel: this element of representation. So now I have for every corner I have a representation, and it's it's a discrete object. Then then I can think you know of a more non perturbative definition of what I mean by had the issue. I mean if you think of the concept of validation, it's a self hard concept of study or understanding

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Laurent Freidel: of of gravity, but it's very perturbative. So it is very classical in some sense. So so here I think it does the possibility to define something where we define what is quantum radiation. So you know you need to call, or they have 2 different representation: R, one and R 2. And of course.

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Laurent Freidel: if there is no radiation between well, R, one is going to be equal to R. 2 right. That's the definition of an evaluation, the the presentation doesn't change. But if there is some amount of radiation, then my your presentation. I One is going to change. It's going to change into a now 2. So the Casim year is going to change.

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Laurent Freidel: and and classically there's this beautiful work of Penrose that that shows that it's possible to recallstruct space time. By by somehow growing together I mean you just did it for one. But what we propose is that maybe it's supposed to be like a post right to an arbitrary signal. It's infinity by doing in both ways, and he perceive ways are beautiful solutions of

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Laurent Freidel: Einstein equation, the exact solution of answering equations that carries non-violent radiation? They don't carry any energy momentum tensor.

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Laurent Freidel: But because you glue what you do that you do a cut and paste technique along non slices, and it's possible to do that without generating a energy momentum Tense right. If you glue 2 different spacetime region on the time, like a space like you, will You'll create a non 0 energy momentum time. So so.

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Laurent Freidel: anyway, so what it means is that these representations, you know here at the classical level, the able, the coordinates. So they label the status of of of your of your vacuum here, and then the radiation is a transition. So at the bottom level we expect these transitions to

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Laurent Freidel: to to. So this is how we can. I'm thinking about translation translation, something that intertwines different corner representation, and I think there's a lot more to be said about that.

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Laurent Freidel: But I think you know. Let me conclude. So. What I've tried to convey to it was very general, and I didn't give, you know, many detail, but hopefully, the references are clear enough is that we we see that the profound consequences of not the theorem for gravitational theory leads to a new picture of quantum geometry.

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Laurent Freidel: We call them geometry is a state of representation of the corner symmetry. and and this kind of symmetry group characterized capture the essence of sub-region and tanker.

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Laurent Freidel: Now that that representation theory includes the non-committee of geometric or observable after you impose the constraints associated with sub regions, and therefore you know, I would argue that it represents a quantization of the equalization of geometry, or at least a universal part of the quantization of geometry.

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Laurent Freidel: And

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Laurent Freidel: what's beautiful to me is that it? It? It realizes what was the the most fascinating part of the Lqg project, which is to to

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Laurent Freidel: to get discretization of space, not from putting extra discrete structure, but purely

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Laurent Freidel: from the representations of continuum, a symmetry algebra. And this is what you know. We're investigating and the discretization that we're saying is 2 phone. It allows the does this optimization right? And this is maybe the possibility of discussing whether each atoms that will have itself. You know this could spectr in their in their representation.

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Laurent Freidel: and then I just briefly mentioned the dynamics of a null surface which is now and understood as guardian conservation laws, for the charges are activated at the quantum level by the presentation theory, or by if you want fusion, because transition between 2 or presentation is also another way for what we call tensor product of fusion.

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Laurent Freidel: And so it becomes. You know, in that sense you you seamlessly go from pure geometry. It's an impulseive wave to something very natural from the algebraic perspective.

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Laurent Freidel: And then, as as was mentioned, and I no time to give a credit to that. But for me it's a proof of concept. The fact that we're doing is not completely crazy. It's just I was seamlessly these these ids

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Laurent Freidel: can be extended to asymptotic infinity. And then, when we do that in that context we recover

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Laurent Freidel: all the slew of all the results. You know the people established in the in the eighties and and the and before, and this picture of a by over, you know, the non perturbative, synthetic phase space, but also the more modern results where you know people connect to these, to self symmetry, and and in fact, you know, with we we were able to, in fact, show that this this canonical perspective allow you to understand something kind of remarkable is that at infinity there's the possibility to

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Laurent Freidel: to a a unrivaled, at least, maybe in some sub sector of the theory, a new tower of Ios p symmetry.

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Laurent Freidel: That's responsible for all the okay. I thank you

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Laurent Freidel: for your attention.

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Jorge Pullin: So any questions

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Abhay Vasant Ashtekar: good. I I I have a couple of questions lot of I mean. I wanted to have your sort of reaction to the

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Abhay Vasant Ashtekar: couple of ideas a long time ago.

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Abhay Vasant Ashtekar: People people like

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Abhay Vasant Ashtekar: we're quite concerned about different, mobile and various issues.

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Abhay Vasant Ashtekar: And so their idea was that well, what one needs to do is to find. say. for

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Abhay Vasant Ashtekar: invariance, form of curvature. which are generically

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Abhay Vasant Ashtekar: in linear independent right in the sense that their gradients are generically

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Abhay Vasant Ashtekar: they that they can serve as coordinates

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Abhay Vasant Ashtekar: and then express everything else in terms of this coordinates, and that will be automatically be from office

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Abhay Vasant Ashtekar: any observable express in terms in this expressed in this call there should be.

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Abhay Vasant Ashtekar: and at the time my kind of big reservation was that? Well, I mean it's very nearly impossible to find this coordinates in the last sector of the face space.

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Abhay Vasant Ashtekar: and that is why, you know I

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Abhay Vasant Ashtekar: I I was looking at these things like just in the context of

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Abhay Vasant Ashtekar: drawing a distinction between gauge and symmetries and symmetries, being the symmetries at infinity, and then spy, group and all that, and all that. But now, coming back, what is your viewpoint? I mean, supposing somebody tomorrow where to find such a

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Abhay Vasant Ashtekar: such coordinates, you know, such a in in values. And then then I got a local representation

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Abhay Vasant Ashtekar: of observables and everything. And then I don't need this this whole framework.

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Laurent Freidel: Yeah, very, very good, very good. I that very good question. So it it's interesting, because i'm going to. There's there's not 2 element to that answer. So the first one is okay. Let's say, if you have such a so it it goes back to. So if you have such a a sense of coordinate, and you use it to gauge, fix, and localize.

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Laurent Freidel: Then, if they were so, if they were, do so, if you do that in the bulk No problem. I don't have any problem. Okay. But the the thing i'm saying is that if there were a notion of coral symmetry attached, then you, you might miss it so you will not see in this gigantic formalism you will you will not? You may not see it

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Abhay Vasant Ashtekar: the the other. We are also. There are many observables that you will not see in the gauge in your corner Symmetry Algebra.

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Abhay Vasant Ashtekar: because I got sure sure. But this one is the key to distinguish this. So I in fact, I I did what what i'm saying, because I did a very interesting exercise. I wrote with a feel of a paper that goes back to this debate between Einstein and Krachman.

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Laurent Freidel: which which I understand lost, which was, oh, the notion of the form of a zoom environment is is empty. It's useless because I can take any theory and make it by adding these coordinates I can make it Defo environment, right. I can introduce you know, of ghost to on fields, and then it's defined. What we can prove now is that if I take the theory of catchman.

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Laurent Freidel: or if I take any theory which is customized, and I compute the kernel charge. I'm. Going to have a just environment theory. I compute the kernel charge, and I find there all 0.

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Laurent Freidel: Okay. So I can distinguish between fake and through gauge and violence. Now, which is, which is very good. So now second point is that, in fact, okay, I didn't talk about that. But to some extent part of what Kushar is talking about makes its appearance. Okay, let me.

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Laurent Freidel: So. The the I Haven't talked about is that when I really isolate the feeling, I really want to understand the presentations of the of the translation generator, and and I really want to understand how I can isolate that system.

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Laurent Freidel: In fact, it's not enough to just have the group. I need to add the variable conjugate to the group. Think about the If I think about the the group. If I go back to the gauge, it's a link. So I have a group there, but also have the electric field, which is the charge, but also have the group elements.

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Laurent Freidel: your lunar me, which is the variable conjugated, and these these conjugate variable. That's what we call the edge mode, but it is, if you you can think of it as an ex e e instantiation of these coordinates. So it's it's it's a field that tells me. Where is that surface placed in space time and at the corner it becomes physical.

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Laurent Freidel: And and somehow.

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Laurent Freidel: anyway, maybe I should stop there because this is yeah, this is. And the technical thing is the corresponding. So I think, thought of it, this Id is is gone, and part of this Id is going to is resurfacing.

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Laurent Freidel: And maybe to understand. The last question is that you know there's a physical difference between having corner symmetry, 0 or corona symmetry non 0.

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Laurent Freidel: So I would be able to say the difference.

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Abhay Vasant Ashtekar: And but I I think that there's this other view other approach that somebody else tomorrow may come up and find such code, you know, as you say you can call it gase fixing, if you like, but you know it's really choosing

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Abhay Vasant Ashtekar: these 4 core nets made out of just invariance, and then expressing everything in terms of them.

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Laurent Freidel: And they will have lots of observers that may not be will not be captured in the I will say so, then do the exercise. We zoom your theory, understanding the decomposition into some region, and you know, if if you find that the corner symmetry group is is trivial, then we have different pro quantum gravity proposal

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Laurent Freidel: Right? Exactly

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Abhay Vasant Ashtekar: ready. Thank you.

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Jorge Pullin: One

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Jorge Pullin: good morning.

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Simone SPEZIALE: Thanks.

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Simone SPEZIALE: Concerning this question that was just discussed. Of course I agree with Lauren's answer, but I would like also to see that

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Simone SPEZIALE: in the idea of using the curvature scale as as coordinates, Typically, one problem is that you end up having maybe, horrible Poisson brackets for the construction you do in terms of those observables.

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whereas one of the nice things of using these

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Simone SPEZIALE: corners, c. Meeting and covariant free space construction is that it gives you a nice way of this, describing the the brackets and the algebra.

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Simone SPEZIALE: So that's something that is quite advantages in this formalism. On the other hand, maybe also a word of caution.

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Simone SPEZIALE: I will not go as far as saying that the issue of gauges versus symmetry is fully resolved, because, especially in the radiative case these charges are ambiguous, and one always needs a prescription in order to define them.

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Simone SPEZIALE: And so.

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Simone SPEZIALE: if they were a unique prescription by for all cases, then maybe I would agree. But since this is possibly not the case, then one is to a discovery. One could choose some

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Simone SPEZIALE: purely mathematical prescription with no physical ground, and then one ends up having a charges which are no 0, which, with the other prescription, would, on the other hand, not be there. For instance.

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Laurent Freidel: Yeah, yeah, sorry. So yeah, You I mean, that's a word of question for the young people. If you hear any any older people telling you that everything has been done before, like I mean. I was in a tok where Carol was saying, we understand matter, and we understand quantum gravity team model. Then, as as Simone. A correctly quote me, just don't listen to that like any any other people that tell you I have done it before. There's nothing to be done you. I wish there was a button where you I don't know.

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Laurent Freidel: Anyway, that's Florida, and how I started my quantum gravity lectures in Paris earlier in the year I am 100% agree with you for fair enough.

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Abhay Vasant Ashtekar: Yeah, no. I think everything is incomplete and everything can be can be done. Well, let me just ask one more quick, quick question. Unless there is somebody else's hand is up.

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Jorge Pullin: Somebody else hands it up.

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Abhay Vasant Ashtekar: Okay, please.

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Jorge Pullin: How?

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Hal Haggard: Hi, Lauren? Thanks for the nice talk.

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Hal Haggard: I wanted to understand one aspect of the corner symmetry group a little bit better, and I thought maybe to to make things concrete and simple. For myself, I would discretize. So instead of having a corner sphere, if we just had a corner tetrahedron.

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Hal Haggard: and we asked what these local boost transformations did to that.

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Hal Haggard: So I understand. In the in one of the facets of the tetrahedron I could take the say

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Hal Haggard: time like normal to the whole tetrahedron and the space like normal to that face and about the boost in that plane.

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Laurent Freidel: But then, what do I have to do with the neighboring planes? Do I have to do the boost in some coordinated way, so that I keep closure. How should I think about this? Thank you for your question. I forgot to mention that, and I think you wrote a paper that answers that question.

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Laurent Freidel: So let me quote that paper. So you you wrote a paper a long time ago. Right You're You're part of that people. They're trying to understand the nature of twisted geometry

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Laurent Freidel: in the geometry. You have a touch right. You have to try and go left. Triangle right match. and and you much exactly what you show in that paper is that in 2 weeks the geometry you need an sn 2 our transformation to change their shape.

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Laurent Freidel: That's what the boost is so twisted. Geometry is a much better description than than than the regular geometry, because it contains an action of the boost, and of course it looks like the they are now mismatch because we are in the wrong reference frame, and the boost maybe, is not I I again. Here I think you are. The answer would be nice to kind of. I wish I had more time, and i'm happy to discuss that that somehow twisted geometry is a discretization. At least this sent to our component that that would be

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Laurent Freidel: hey great! And you can come in because you are the one that introduced this.

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Hal Haggard: I'm: I'm happy with the answer. I wanted to understand. Yeah.

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Jorge Pullin: am I?

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Abhay Vasant Ashtekar: Yeah. So a lot of it. This is very.

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Abhay Vasant Ashtekar: extremely interesting. Thank you, for this is a great talk.

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Abhay Vasant Ashtekar: Oh.

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Abhay Vasant Ashtekar: now it look kind of gravity. Per. Say. First of all, I would say that you know I I like very much your

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Abhay Vasant Ashtekar: caution to the younger people about the way that if you mobilize them are often treated.

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Abhay Vasant Ashtekar: But in the canonical picture at least right. I mean, I I think that the one never one doesn't does not want to stick one up to one graph, and that is why this full projected construction was there, so that one can go from one to another, and so on. So, as I had the full and a few hours of action on okay.

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Abhay Vasant Ashtekar: But

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Abhay Vasant Ashtekar: nonetheless, it is kind of still a kinematical frame. But and the whole point there was really, you know, how do we implement specific dynamics, such as the one coming from the Hamiltonian, and there has been a lot of progress like another one, and recent work, and so on. But

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Abhay Vasant Ashtekar: I think the problem is not so out there.

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Abhay Vasant Ashtekar: Now, what you have done is in some, in a way, and so I would like your reaction to that is really a very nice.

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Abhay Vasant Ashtekar: powerful way of generalizing that can those kind of magical considerations, by bringing in this the the coral symmetry, and by

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Abhay Vasant Ashtekar: and and the representations of that group and observables constructed the auto, symmetry, etc. And as you emphasize, in fact.

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Abhay Vasant Ashtekar: this is very very universal. It doesn't really distinguish, but in various theories.

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Abhay Vasant Ashtekar: and you know it could rise and be higher curvature, etc., etc. And so in that sense it really is. It's kind of a sense of

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Abhay Vasant Ashtekar: of commonality, of the kinematical structures there. The dynamics of this thing is very different.

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Abhay Vasant Ashtekar: and so isn't. I don't we in some ways at the grander level back to the same issue about, you know.

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Abhay Vasant Ashtekar: But now this is all fine. But now I would like to know what is kind of drive. This theory. I don't want all of them to be the only one of that, so to say and

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Abhay Vasant Ashtekar: isn't that the next big question.

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Laurent Freidel: Yeah, yeah. So to command, yeah, at some level, you you're right, like like what I was talking about is this universal structure of of the kinematics of

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Laurent Freidel: it's super important that we accept it, we because it's universal, so it has to be there, and it no matter what your mother is, and and then you're right. So so.

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Laurent Freidel: But what do we? What that we gain from from Fuji? I think what we have gained? And this is the next step. And this is this type in process. So this is where we have gain is a direct link to the continue

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Laurent Freidel: in the in. The, for instance, is that now I can take this very atomic or presentation. But I can take the usual of a presentation.

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Laurent Freidel: and and so and you're totally right that. Now, what matters is the study of the dynamics? So

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Laurent Freidel: which is the study of the constraints, if you want. But for me it's clear that. And this is what i'm doing now the study of the dynamics is really the study of dynamics around null surfaces.

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Laurent Freidel: That's what's happening right now, this is what has been cranking up. You know the work for the past 3, 4 years. So, and it's what we're doing. It right now is understanding much better. What is this dynamics of long surfaces connecting to other work? But what is the face, space, and really kind of rephrasing everything now in terms of of of symmetry, because I I gave this idea, which which is that you know, if if I think of a cut of a north surface and look out of another, and our surface

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Laurent Freidel: as as a representation of the Corners image group. Then the dynamics there tells me how how these these cuts are needs to be fused. So somehow.

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Laurent Freidel: in that long goes the dynamics is should be. It is ability to the fusion world. So this is the next step. But what I the new things. I think this was not available before in that that that elements of understanding, the super translation and the fusion role of that bigger group.

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Abhay Vasant Ashtekar: No, but but the the Meta question is told you one of those future rooms will be correct.

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Laurent Freidel: Yeah, yeah, yeah, I know I some guidance for that I will. Just. Yeah. Yeah. I mean the paper I'm writing now, I some title. It's not exist. Comply to quantum ratio 3. And I think once we understand what is going on, I should agree in no ways. It it gives us a basis

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Laurent Freidel: to be fair that I was kind of a yeah.

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Abhay Vasant Ashtekar: No. But I mean right on the question is valid in every theory of privacy.

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Laurent Freidel: It's nothing more dynamic of privacy. Yeah, the phase space is very different. The commutation is there. I mean, there's the details matter there. Yeah, because matter the do. The the nature of the representation is going to be different. So

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Laurent Freidel: there may be at the end that you know my first go would be well. Can we have a quantum analog of Einstein or it? And then, you know. Is that a a universal object? Okay.

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Abhay Vasant Ashtekar: So I think one would do it.

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Laurent Freidel: I mean, you know that

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Laurent Freidel: I think if anybody gives me that first one. I'm. I think i'm happy to retire and give give the the next step to the next generation.

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Laurent Freidel: Alright. Unfortunately, I have another commitment here, so i'm i'm kind of short on time.

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Laurent Freidel: Thank you very much. Any other questions.

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Jorge Pullin: It's not. It's like Moran. Can we reconvene in the fall?

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Laurent Freidel: Thank you.

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Lee Smolin: Thanks to you.