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Courant Events Right: He will tell us about the review of probabilistic and street models for JT Granth.

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Courant Events Right: Okay, so thank you very much for the invitation. I'm very honored and glad to give a talk here. And so I was asked to give a review, more like talk, so at least a large fraction of it will be a review.

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Courant Events Right: And, and then I will explain also new things.

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Courant Events Right: some published, but also a large part that is not published yet. So here you have two papers of 2024.

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Courant Events Right: on which I will base part of my presentation, but as I said, the talk is mainly review and papers to appear. So that's a plan. I'm not going to go through in detail, I'll let you read the plan. We'll see if we have time to cover all these subjects. And

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Courant Events Right: Let's start now here with… Bomb Quantum Gravity. So…

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Courant Events Right: The defining feature of the Jackie title of quantum gravity is that it's a quantum gravity theory for which the bulk curvature is fixed.

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Courant Events Right: Okay, so it's, like, even in the quantum theory, you don't have any bulk curvature fluctuation. Okay, so you completely rigidify the metric in the bulk, but things can happen because you will have boundaries.

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Courant Events Right: So you have 3 models, 3 types of models, depending on which curvature you choose in the bulk, positive, zero, or negative.

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Courant Events Right: They're all very interesting for physics. The negative curvature theory is relevant for holography, etc, etc. It's been the most studied one, with some particular boundary conditions that we shall review.

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Courant Events Right: But the zero curvature or positive curvature are also very interesting for physics. For example, the positive curvature is relevant for cosmology, you know, the city of space.

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Courant Events Right: etc.

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Courant Events Right: The Schwarzan description, that probably you know quite well, only applies to the negative curvature model.

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Courant Events Right: And… The study of the other models.

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Courant Events Right: requires to go beyond the Schwarzan description, and so this will be one of the main,

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Courant Events Right: point of my talk will be to go beyond the Schwarzan description, and it's required when you have models that cannot be defined on asymptotically hyperbolic geometries or on infinite geometries. Nearly, for positive curvature models, you don't have this notion of asymptotic geometry.

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Courant Events Right: So I start with presenting, you know, the usual action for JT gravity. The field phi, capital phi, is called the gelaton, it's really like a Lagrange multiplier field that enforces in the bulk

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Courant Events Right: the constraint r equal to eta. So R is the Ricci scalar, okay? I didn't use the Gauss curvature, even though I'm in a math institute. So, okay, 5, you can think of it just as a Lagrange multiplier field.

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Courant Events Right: and you have a term that is proportional to the bulk area, proportional to capital lambda. You may also add

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Courant Events Right: boundary terms, and you may also couple this model to some matter quantum field theory, possibly a matter CFT.

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Courant Events Right: Of course, this action, written down this way, will not make much sense on infinite geometries.

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Courant Events Right: It needs to be regularized. On finite geometries, the boundary terms will not make sense, because we shall see that the typical matrix on the boundary are very rough, very irregular objects, so you cannot really write it this way, but that's, okay, okay.

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Courant Events Right: That's a starting point that is okay at the classical level.

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Courant Events Right: when you study JT, so you fix the curvature of the bulk, but you also have to pick boundary conditions, because you will always define the model on many folds with boundaries.

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Courant Events Right: And, I want to emphasize that there are really three qualitatively distinct type of boundary conditions that you may want to impose. You have others, but the other boundary conditions are just obtained by doing some Laplace transform, if you like, so it's sort of trivial. But you have really three different… qualitatively different starting points.

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Courant Events Right: The first is something that was known as topological gravity that people actually have considered since the 80s, maybe, or the 90s.

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Courant Events Right: In JT, it would be like taking three boundary conditions for both the delater turn and the metric.

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Courant Events Right: That imposes not only that the bulk curvature is fixed, but also that the boundary extrinsic curvature is fixed.

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Courant Events Right: Because if the boundary conditions for the D-laton is 3, you have this coupling on the boundary with little k, which is the extrinsic curvature. So if you integrate out also the boundary value of phi, it plays also the role of a Lagrange multiplier for the extrinsic curvature on the boundary. So at the end of the day, you're getting many folds which have

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Courant Events Right: constant curvature in the bulk, and geodesic boundaries, or constant expressing curvature on the boundaries.

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Courant Events Right: This is an old story. The moduli space of metrics with such properties is just finite-dimensional. It's a very interesting math story. That is not going to be my main point of emphasis today.

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Courant Events Right: The second type of boundary conditions?

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Courant Events Right: are the ones that people have studied a lot, in particular in the physics literature, and for us, I will be precise and call them conformally compact boundary conditions.

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Courant Events Right: So those boundary conditions are only valid for the negative curvature model, and these are boundary conditions for which the boundary is actually pushed to infinity.

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Courant Events Right: So you constrain your space of metrics, the space of metrics that you consider to be such that any points in the bulk is actually at infinite distance from the boundary. Okay, so these are very specific sort of boundary conditions.

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Courant Events Right: And the third example Which is new.

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Courant Events Right: is when you take direction boundary conditions for the D-laton.

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Courant Events Right: So, like, that's a little bit like 4.2, but 3 boundary conditions for the metric.

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Courant Events Right: Which means that the boundary is going to be freely fluctuating.

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Courant Events Right: And that will yield models on finite-sized geometries for which, you know, a lot of new things will emerge compared to the second type of boundary conditions. So this…

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Courant Events Right: Third type of boundary conditions, you can think of it as the correct one to go to finite cutoff, or to consider the quantum gravity, quantum JT gravity, on finite side geometrics. Okay, so you have really this free…

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Courant Events Right: Three different types of… Things.

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Courant Events Right: Let me start, I wanted to give you, since I know this is an audience, many of mathematicians, I wanted to give you, maybe for 5 minutes.

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Courant Events Right: or something like that, an idea of why we're interested in JT gravity. There are actually many reasons, so I could have a long list and talk for an hour just about the motivations, but I picked just one, okay?

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Courant Events Right: So, the example I want to briefly treat is the case of Raschler Nordstrom's black hole in real-world four-dimensional, you know.

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Courant Events Right: space. We could also do care, so Raschner Nordstrom means a black hole that has some electron charge. Kr would be a black hole that is rotating, so Kaer is very realistic, and you have these objects in the universe.

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Courant Events Right: Pretty much everywhere. And they have similar properties, rational, and kale.

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Courant Events Right: So here I wrote a metric in the Euclidean, but the Minkowskian signature metric just has minus signs. The important aspects of these black holes is that they have generically two horizons, an outer horizon and an inner horizon, okay? Positions R plus and R minus.

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Courant Events Right: And there's a hooking temperature.

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Courant Events Right: And an entropy. S, which is pi r plus squared. So the area of the outer horizon is the entropy.

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Courant Events Right: Now, the nice thing about these, guys, is that you can consider a limit where the inner and the outer horizons are very, very close. So this is when the temperature goes to zero, the rocking temperature goes to zero.

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Courant Events Right: In this case, a very elementary analysis shows that the near horizon geometry always contains a hyperbolic two-dimensional float.

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Courant Events Right: Okay? So that's a completely universal feature of near… so-called near extremal black holes, so these black holes that are very low hogging temperature. So for Rachelor-Nostrom in four dimensions, you have H2 cross S2, okay? The S2 will remain finite size.

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Courant Events Right: It won't play a big role in what I want to say, but there is this hyperbolic force that develops.

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Courant Events Right: No!

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Courant Events Right: when you look at Einstein theory.

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Courant Events Right: in that limit, you realize that the dominating gravitational fluctuations are actually governed by the JT gravity.

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Courant Events Right: Action.

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Courant Events Right: Okay, so that object just emerges as some effective description in these near-horizon limits of near-extremal black holes. That's very nice. So we can use, actually, JT gravity to study the very low temperature limit of black hole physics.

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Courant Events Right: Now, this has yielded some very… Sorry, yeah? Can you mind where does the gelaton come from? Okay, so the gelaton will be the mode that governs the size of the two spheres, if you like.

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Courant Events Right: In the form… the four-dimensional picture. Why is it allowed to portray, really?

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Courant Events Right: Almost pretty.

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Courant Events Right: on the boundary? Oh, well…

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Courant Events Right: In the mode, well, it's… it is… it is a gravitational mode, okay? So the size, I mean, this is… so the D later term, the leading term is the size.

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Courant Events Right: So that will be some constant. We will use that, if you like, as some boundary conditions.

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Courant Events Right: And then there is, you know, plus some fluctuations. What I call phi is the fluctuations. Why in this 14 description, this happens?

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Courant Events Right: So I think, you know, when you do the dimensional reduction from 4D to 2D, you have, indeed, subleading terms. You know, so if you like, the D later turn will have a potential. The linear term is what governs JT. That will be the leading term in the really t goes to zero limit.

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Courant Events Right: But immediately after, indeed.

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Courant Events Right: you have, you know, phi squared, you have a V of 5, like, so you will have a delay on gravity with an arbitrary potential. So this JT is really the leading term that you have, you know, when you really go near the horizon. It's a leading approximation. But it does capture the essential

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Courant Events Right: Gravity fluctuation, I want to emphasize now.

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Courant Events Right: So it's a rather, yeah, rough sort of approximation, it's really a leading term, but it captures the qualitative feature that is fundamental.

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Courant Events Right: So, the formulas that I show here, the mass and the entropy, are just, you know, can derive that very elementary using known techniques, so the mass or the energy of the black hole.

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Courant Events Right: you know, if the temperature is not strictly zero, it starts as a T squared, the entropy grows linearly with the temperature.

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Courant Events Right: These two formulas have been a big puzzle for a very long time in physics, for, you know, two obvious reasons. First of all.

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Courant Events Right: you learn that the energy at small temperature grows like T squared, which means that if the temperature is very small.

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Courant Events Right: the total, if you like, sum of energy of the black hole is much smaller than something that would be linear in T, because T squared would be much smaller than t. However, a single Hawking quantum that is emitted

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Courant Events Right: has an energy that is of order T.

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Courant Events Right: That means that the whole semi-classical picture is completely breaking down here, because the emission of the single quantum

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Courant Events Right: Carries more energy than the full, you know, sum of energy that you have in your system.

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Courant Events Right: Clearly, this approximation cannot work. And that is surprising in some sense, because those black holes can be enormous.

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Courant Events Right: You know, they can be, like, astrophysical. So, the problem is not about having some strong curvature.

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Courant Events Right: It's something else, okay, it's related to the fact that we are at very low temperature, but okay, that's, that's very strange.

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Courant Events Right: Another puzzle is that, of course, the entropy at zero temperature is also a huge number, non-zero, because, you know, for the extra… even at the external limit, you have this huge

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Courant Events Right: horizon.

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Courant Events Right: So, we are not used… I mean, it's very strange to have, in quantum mechanics, a system that is sort of generic. By generic, I mean there's no supersymmetry in those systems.

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Courant Events Right: And that would have a macroscopic degeneracy for the ground state.

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Courant Events Right: That is extremely bizarre. How do you build that? We know how to build that in supersymmetric examples, because supersymmetry allows you to have that… this sort of degeneryses, but in a non-supersymmetric setup, this is really strange, okay? So these were, you know, two basic problems in this,

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Courant Events Right: Two basic puzzles for these near-examole black holes.

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Courant Events Right: Now, genetic gravity gave the clue how to solve that.

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Courant Events Right: And the remarkable, truly remarkable thing is to realize that even though those black holes, again, are very large, there is no problem with the curvature. The curvature can be arbitrarily small.

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Courant Events Right: In spite of this, there is one mode in the gravitational fluctuations that is not tamed.

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Courant Events Right: That is not semi-classical. That strongly fluctuates.

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Courant Events Right: And this large mode is this famous, by now, JT Gravity reparametrization.

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Courant Events Right: That needs to be Treated fully quantum mechanical.

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Courant Events Right: In spite, again, of the fact that the black hole is very large.

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Courant Events Right: When you do this, So it's really quantum gravity, to my…

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Courant Events Right: Knowledge, it's the first application of quantum gravity to something that looks like the real world.

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Courant Events Right: We found this particular instance

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Courant Events Right: where a strong quantum gravity fluctuation is actually fundamental to understand the physics, and not related to Planck scale, or height curvature, okay? So when you do this, okay, we know how to deal with JT, and the buzzers are solved.

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Courant Events Right: like, the energy now has, at very low temperature, a term that is proportional to T, so that really comes from the quantum fluctuations of this JT gravity mode. And the entropy gets a log T term.

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Courant Events Right: log T correction, which, of course, when T goes to zero, no longer makes any sense, but it's associated with the density of state, and we now understand that this is, if you like, some signal of a discrete spectrum at very low energy.

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Courant Events Right: And this approximation only gives the continuous limit of the spectrum, but there will really be a discrete spectrum, which means that the huge degeneracy that you believe the…

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Courant Events Right: Joaquin-Bekenstein formula computes is not really a degeneracy of the ground state, it's just the density of state that is very large, very near the ground state. So you have a huge number of states near the ground states, but they are discrete, actually, okay? And the ground state probably is unique.

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Courant Events Right: So if you go to very, very strictly zero temperature, the entropy will go to zero, actually, in those systems.

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Courant Events Right: Alright.

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Courant Events Right: Any question about this? So that's just to give you, you know, one of the reasons why we've been excited about JT.

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Courant Events Right: Agreed.

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Courant Events Right: Now, let me review the Schwarzene Very scary.

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Courant Events Right: Okay, so the Georgian field theory

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Courant Events Right: is, can be completely well-defined probabilistically, I mean, and there's been beautiful work by Belo Kurov and Jacob Liz, and that made much more precise and elegant by Boher Schmidgetal, where we understand completely what this Russian figure is, you know, even at the level of

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Courant Events Right: probability, and mathematical rigor. The path integral is, as I indicated, so you integrate over circle dipomorphisms F,

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Courant Events Right: DLF is this formal left invariant measure on the group of circuit diffiomorphisms that, of course, does not exist, but we write it anyway. And SSCH is disruption action.

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Courant Events Right: And the dots represent possible insertions that you may want to include in your path integral. For instance, insertions of these operators OH, which are PSL2R invariant bilocal operators that geometrically really compute the normalized length of geodesics in your geometry.

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Courant Events Right: So this is the objects that defines the Schwarzan field theory. Yes?

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Courant Events Right: Well, I mean, so the measure is infinite.

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Courant Events Right: Because you have this PSL2R invariance, so this is a formal, you know, division. You have to fix the gauge, okay? So you have to do a slice, it's the usual, like.

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Courant Events Right: physics, we call that the Fadyev-Popov procedure, so that's a trivial thing. Otherwise, you have to think that this measure factorizes into an infinite piece, and

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Courant Events Right: Yep.

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Courant Events Right: Alright, so the precise definition of the measure is actually very simple.

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Courant Events Right: You just write down that the derivative of your differ morphisms is proportional to exponential of Q, where Q is a completely standard Bronian process.

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Courant Events Right: Okay, or Euclidean quantum particle, as you would say in physics. And the measure, so no, the well-defined measure which combines the formal DLF…

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Courant Events Right: with some piece of the Schrodzen action is just the Wiener measure.

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Courant Events Right: So at the fundamental level, the Schwrogen field theory is just the linear measure plus some additional terms that modify the measure, but it's like a Radon Euclidean derivative, if you like, what you add. At the fundamental level, everything is defined just by the Vienna process. Okay, so it's a very simple thing.

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Courant Events Right: Yes. For a mathematician, it's a bit sharp because of the equation, so the right-hand side is totally fine, but as you say it's equal to the left, so how do we know that the measure is the right one?

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Courant Events Right: So, how do we know? So, first of all, it has all the required properties, so in particular, the PSL2R invariance, and it transforms in some particular way under left multiplications, which is what we want.

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Courant Events Right: So you could say, okay, it satisfies a list of axions that is what physicists would like, and then it's a precise mathematical object. On the other hand, maybe you would like to have a derivation from JT, and I will give a hint of how this is done.

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Courant Events Right: In the next few minutes. I won't have time to do a full thing, but yeah. So, you know, the important thing here is to realize that F is a smooth, completely smooth object, okay? F prime is continuous, almost surely.

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Courant Events Right: So F is like C1, you know, so it's… this Russian field theory really deals with the smooth guys.

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Courant Events Right: Nice, smooth guys.

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Courant Events Right: Okay.

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Courant Events Right: So maybe to answer your question in more detail, so because when you look at the Schwartland field theory, why is this theory of quantum gravity? You know, you could say, what is the relation with JT, with JT gravity?

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Courant Events Right: So let me give you some idea of the relation. It might be completely mysterious.

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Courant Events Right: So, when you have a quantum gravity model, in all cases, if you want to have, let's say, a first principle approach, you have to choose two things. The first thing is the space of metrics that you will know, if you like, the space over which you will integrate with the path integral.

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Courant Events Right: So this is sometimes… this choice is sometimes referred to as the choice of boundary conditions, but more generally, if you want to be more precise, it's really a choice of the space of metrics that is supposed to be relevant or a load for the model you want to consider. And then you also have to choose a gauge group.

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Courant Events Right: Which is, and which are the metrics that you will identify?

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Courant Events Right: Okay? And so this is a group of diffiomorphisms, but when you have boundaries, you have many group of diffomorphisms that you may want to consider, so you also have to be careful here.

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Courant Events Right: Okay, and then after you've done that, appropriate choices.

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Courant Events Right: We yield the Trojan description, so how does… how does that work?

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Courant Events Right: So this is the precise setup.

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Courant Events Right: you consider… so let's say you have a manifold M with one boundary. You could have many boundaries, you focus on one, that's where you're going to impose the boundary conditions that will hit the shorten.

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Courant Events Right: You consider metrics, by definition, that must be conformally compact. So you take this form G bar A over A squared, where A is the defining function. You have a similar formula for the D lateron.

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Courant Events Right: So the G bar A extends smoothly to the boundary, as usual, so HA would be the induced metric on the boundary of G-bar A.

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Courant Events Right: And since you also have this D-laton, you have a ratio, phi A, divided by the square root of the induced metric on the boundary, which can be fixed, because this is an object that is independent of the choice of your defining function.

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Courant Events Right: And you also impose that the curvature, R, will be minus 2, so negative curvature, plus corrections that must be offered a squared. For us, this last condition, I will not deal with it, much longer, because we will integrate out the D latons and impose r equals minus 2 as an exact equation, eventually.

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Courant Events Right: So that's the first…

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Courant Events Right: set of conditions I put on my matrix, okay? So that's really conformity compact metrics. These are the boundary conditions

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Courant Events Right: That we always use in oligraphic setups, even in higher dimensions. Okay, so there is no thing too special here. In higher dimensions, you have similar, very similar, sort of conditions.

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Courant Events Right: Then the choice of the game.

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Courant Events Right: The choice of the gauge group, and that's crucial.

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Courant Events Right: it needs to be only the small diffus, what I call here diff of M, S1, which means the diffiomorphisms that act trivially on the boundary.

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Courant Events Right: You do not gauge the boundary re-parameterizations.

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Courant Events Right: This is also a crucial feature of any holographic theory.

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Courant Events Right: When you have an asymptotic boundary.

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Courant Events Right: you pretend that the boundary coordinates are sort of fundamental. So there is a boundary observer, okay? And this is why, on the boundary in olography, the theory does not gravitate. The boundary theory is not gravity.

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Courant Events Right: So the coordinates are fixed on the boundary, and you will not gauge boundary reparametrizations.

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Courant Events Right: That's good.

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Courant Events Right: So that's the second point. Third point that you also need in order to make sense of your model is you have to add data.

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Courant Events Right: So I said that you… we were having this boundary coordinate for S theta, that is fundamental. Actually, you undo, you know, your manifold with some color structure, so you have some privilege r theta coordinates.

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Courant Events Right: You have the boundary, R equal to 0 is, let's say, the location of the boundary, and you have a boundary

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Courant Events Right: a metric that you pick. That's the data in the definition of your model.

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Courant Events Right: Okay, so you need this data to define a holographic model. A colored structure, and a particular metric on the boundary that you fix. So data is your problem.

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Courant Events Right: With all this, we're going to be able to define very, very grossly what we mean by dischargeant theory.

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Courant Events Right: First of all, the space of metrics are not takes the form that I have indicated.

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Courant Events Right: So it's the quotient of the space of conformally compact hyperbolic matrix. No, hyperbolic because, okay, I know that eventually in JT, R equals minus 2, so I impose that they are conformally compact and R equal minus 2,

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Courant Events Right: Quotiented by the group of small differos, you can compute that modularized space, this modulized space, and it's the well-known quadjoint orbit, if you like, of the Razoro, the quotient of the diffiomorphism of the circle by PSL2R.

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Courant Events Right: This really comes from the fact that if you had gauged the large DFAOs.

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Courant Events Right: You could go to conformal gauge, but there's only one conformal, complete metric, if you like, or conformally compact metric.

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Courant Events Right: So, any elements in this modulized space

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Courant Events Right: can be obtained by applying a large DFO on this unique guide, which is, of course, the point-carry metric on the disk.

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Courant Events Right: which has some PSL2R isometry, so the little group is PSL2R, and that explains why you have this formula, okay? So there is no mystery here about this computation of the modulized space.

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Courant Events Right: Another element, another piece of information that is important to know, is that in any orbit in this Gaussian space.

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Courant Events Right: You can always find a unique element that has the so-called Fifferman Gramfall.

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Courant Events Right: Okay, so you can write it in this way in the arc theta coordinates we've picked.

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Courant Events Right: So, you know, it's a very simple formula, EB is the data we have fixed, and E2 is actually the function that encodes all the… all the freedom in your matrix. Okay, so these metrics are super constrained, and everything is in a unique function, E2FUTA.

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Courant Events Right: That will take some particular form.

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Courant Events Right: Alright.

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Courant Events Right: No, you can check!

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Courant Events Right: that large GPOs

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Courant Events Right: will act on E2. If you… if you act with a large liferal jihad, it will act on E2,

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Courant Events Right: And the action is exactly the coadjoint action of the result.

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Courant Events Right: Okay, so that's a nice calculation to make.

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Courant Events Right: Okay, so the large CFOs act on U2 as the quadjoint quadjoint action of the reserve.

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Courant Events Right: And, essentially, with this, we have everything that we need.

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Courant Events Right: So, yes? So, this corresponds, like, what's the… what's the minimal regularity for this to make sense?

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Courant Events Right: For what? So all this is completely regular, so all these metrics are, like, completely smooth. Yeah, but when I say, like, say, the regularity of the Schrozier measure, so the right-hand side is kind of, like, C1.5 minus epsilon, does this still…

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Courant Events Right: It's okay. As I said, at the end of the day, it's a veneer process.

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Courant Events Right: So, you know, it's exactly like, of course, when we write the Schrodzen action, you have second derivatives.

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Courant Events Right: In the same way as when you write, I don't know, the free particle action, you have Q dot squared.

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Courant Events Right: We know that this, you know, Almost surely does not exist.

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Courant Events Right: But that defines the measure, and that's what I mean with that is not.

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Courant Events Right: Yeah, yeah, I mean, kind of the… can you actually write down the metric? Like, if I give you… if I give you the measure, which was sensitive, can you write down this… So, the entries of this form. Maybe you want the relation between B2 and F.

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Courant Events Right: on the circle deformorphism. That I will give now, in one minute.

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Courant Events Right: then you'll see the relation with the Schrozen. Here, I'm not at the Schrozen field theory yet, almost, but not completely.

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Courant Events Right: All right, so with all this data, the important thing is… thing is that you can define a renormalized action, which is, you know, like the renormalized volumes that you know very well in hyperbolic space. Here is some sort of renormalized volume.

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Courant Events Right: But the volume is replaced by the JT action. You have a renormalized JT action, which is defined

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Courant Events Right: By regularizing your manifold, you replace your manifold by some M epsilon, which is cut off at this particular coordinate R equal epsilon. Okay, so you remove

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Courant Events Right: what is very near the boundary, and then you take the epsilon goes to limit, to zero limit of the GT gravity action evaluated on the G-hat.

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Courant Events Right: You know, on this unique element in the orbit that had the fulfillment ground form.

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Courant Events Right: And that defines the model. That defines the action.

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Courant Events Right: Very little calculation shows this formula, just the integral of the boundary value of the D-laton, E2 thista.

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Courant Events Right: And to get the Schwarzene theory, you just have to now realize that E2 is going to be the Schwarzene derivative of F, but that's not difficult to see.

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Courant Events Right: If you're on a disk, Okay, let's assume we are on the disk topology.

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Courant Events Right: then E2 has to satisfy constraints, because the holonomy of this matrix must be

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Courant Events Right: Trivial. That tells you that the relevant orbit of the… of the Virazoro, of the quadrant orbit of Virrazero, is the one that contains the constant one-quarter. Okay, that's a non-trivial exercise.

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Courant Events Right: Very interesting to make, that, you know, you realize that for metrics that are trivial lonomies, there's only one coadjoint orbit that can do that, and that's this constant one-quarter orbit.

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Courant Events Right: you add, with the Razoro, with, you know, you do the quadrant action of the Razoro on this constant minus one quarter, and you get E2 equal minus 1 half of the Schrozen derivative of the tangent of f over 2. And that's exactly the Schrozen action.

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Courant Events Right: Okay, so this is how you derive from JT the Schwarzene field theory in these particular boundary conditions, which restrict enormously the set of metrics that you're allowed to consume, of course, entirely parameterized by these circle differmorphisms.

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Courant Events Right: You can make the link with what is known in the physics literature as the smoothie-Wrigling boundary configurations by realizing that if you look at the finite geometries that you obtain by cutting off at r equal epsilon.

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Courant Events Right: Then you have, indeed, because it's cut off, note the geometry as finite size.

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Courant Events Right: Any constant curvature metric of final size can be represented as an immersion in hyperbolic space.

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Courant Events Right: And this immersion will be this smoothly wielding disk, and you can check that the F that I have been talking about here is exactly the F that you find in the physics literature introduced by hand-waving and probably incomprehensible arguments, even by a lot of physicists, but that's exactly the same object.

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Courant Events Right: So, so, so, so, so we are matching everything, but here I gave you the, I think, the height, the rigorous presentation of it.

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Courant Events Right: Okay, the beta S is completely free, or is it fixed by a matter of… The what? Better S. So the beta S is completely fixed, so that's about right. It's an arbitrary parameter. It's not fixed by the interaction with matter or something? No.

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Courant Events Right: It's an arbitrary parameter that you can have in your model.

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Courant Events Right: And, you know, beta S… the important thing is that, because it is arbitrary, a number of order 1,

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Courant Events Right: This action is not to be treated semi-classical.

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Courant Events Right: The semi-classical limit of digitize goes to zero.

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Courant Events Right: Some sort of very high temperature limit.

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Courant Events Right: But we don't want to do that. The mode is not going to be…

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Courant Events Right: weakly coupled or semi-classical. Vita is also the one, and you have to treat that model

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Courant Events Right: Non-culturally. How do you fix it in physics?

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Courant Events Right: So, it will be related to the temperature.

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Courant Events Right: And we'll interpret it as a temperature.

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Courant Events Right: All right.

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Courant Events Right: Now, if I had more time.

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Courant Events Right: I would explain how you derive the measure.

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Courant Events Right: the, you know, this formal left invariant measure, that would require to use some fairly effective procedure in a setup where you only fix… you only gauge the small diffiomorphisms. There's a lot of…

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Courant Events Right: interesting technical issues associated with that, but I don't have time to talk about it. In private, if some of you are interested, I can give you some updates, but I won't have time here. So, if you, in JT, you couple this better, how does it affect this,

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Courant Events Right: No, no, it does not affect at all. So, to do this, I have to add the matter action.

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Courant Events Right: That will be coupled to the metric, which is parameterized by F, exactly the way I explain.

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Courant Events Right: Okay, so the… the… there will be, indeed, the matter will be called to gravity.

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Courant Events Right: Biabetrics that are very, very specific.

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Courant Events Right: Bahami Khan by death.

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Courant Events Right: So you have a coupling, if you like, of your matter action to this circle DFO, which is very explicit right now. But it doesn't affect the law. It doesn't affect everything I said. Because what I said, if you like, is the JT action piece, and the measure also to be treated.

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Courant Events Right: If you add a term in the action, it will not be affected. It will not affect the discussion.

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Courant Events Right: I'm in the beta, I assume.

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Courant Events Right: Beta can still be chosen now.

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Courant Events Right: I'll mature.

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Courant Events Right: No.

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Courant Events Right: This one is not related to, to the magic.

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Courant Events Right: What time is it? Because I don't have time here, and I don't see any clock.

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Courant Events Right: 05? Yeah.

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Courant Events Right: Okay.

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Courant Events Right: Now I want to go to the story of finite geometries.

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Courant Events Right: The first thing to realize is that the space of metrics for finite-sized geometries in JT will be much, much larger than the space of conformed compact matrix that we've just discussed, and that are relevant for the Schwarzan model.

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Courant Events Right: And that will imply that the physics would be completely different.

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Courant Events Right: So, something I've already said quickly, it's quite nice to represent the matrix, to realize that at least if you have metrics on the disk.

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Courant Events Right: of constant curvature, you can always represent them as, in an isometric way, as immersed disks in some canonical target space. And the canonical target space, you choose according to the curvature of the model you study. So if you want to do flat JT, you just immerse your disk

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Courant Events Right: into a Euclidean space, or hyperbolic, or the two-sphere, if you have curved models.

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Courant Events Right: So this is why I can present you these sort of pictures.

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Courant Events Right: So all these pictures represent finite size constant curvature matrix.

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Courant Events Right: that have to be included in the path integral if you want to consider JT on finite-sized geometries. On top of the smoothly wiggling guys, you have the, let's say, self-avoiding guys with a lot of zigzags, but you also have overlapping configurations.

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Courant Events Right: These are perfectly good, because they are immersed disks.

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Courant Events Right: Then you also have boundaries that can take the shapes of the inset, you know, the lower insets in the figure. That looks extremely complicated.

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Courant Events Right: So all these configurations are included.

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Courant Events Right: In parentheses, they're included even when these configurations, let's say, are very large.

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Courant Events Right: The boundary of these guys can be arbitrarily large.

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Courant Events Right: So that's a parenthesis, because in physics, sometimes they present the Schwrogen model, not in terms of conformally compact metrics, but in terms just of metrics that have very large, very long boundaries.

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Courant Events Right: That's completely wrong, I mean, these guys, super complicated guys, can have arbitrarily large boundaries, but they won't be conformally compact in the limit where they become infinitely large. Okay, so… but they have to be included in the finite geometry models.

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Courant Events Right: Alright?

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Courant Events Right: There is, on top of these complications, an absolutely remarkable phenomenon. If you've never seen that, you will be surprised. It's called the Milner phenomenon because, I mean, the legend says that it's Minor that first wrote this curve. So the Minor curve is this one.

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Courant Events Right: This one?

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Courant Events Right: I represent it, like, in its sort of Darth Bador, sort of…

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Courant Events Right: figure. And the disk curve has the amazing property that it actually is the boundary of two distinct disks.

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Courant Events Right: So you can fill it up inside with two distinct disks. By distinct, I mean that the metric that corresponds to the filling will be deformorphism in equivalent.

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Courant Events Right: So a given boundary can bound many disks, and here I tried, you know, to represent it by separating the disk into pieces that are bounded by simple curves, so that you clearly see how it works, and then you merge, you glue those different pieces, and you reproduce

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Courant Events Right: Two distinct disks, but the boundary curve will be the same in both examples.

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Courant Events Right: Okay, so that's really remarkable. The example here, the one on the lower right side of the slide, a more complicated one, there are 3 disks in… 3 distinct disks in that guy.

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Courant Events Right: you can try to work it out, okay? And you can have configurations with an arbitrary large number of disks inside.

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Courant Events Right: So that's a remarkable thing that tells you that one of the law of GT gravity, you know, that the dynamics is governed by fluctuating boundaries is wrong, actually, on finite geometries. It's correct in this… for this conformally compact metrics, but for finite-sized metrics, it's not correct.

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Courant Events Right: The boundary does not encode the degrees of freedom. You have to associate with the boundaries a multiplicity that counts the number of disks inside.

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Courant Events Right: And it will turn out that this will be the typical configurations.

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Courant Events Right: The typical configurations have a non-zero… a non-trivial multiplicity.

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Courant Events Right: Okay.

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Courant Events Right: So this looks horribly complicated, and, you know, you can… there's even another layer of complication, because you could say, okay, but maybe, okay, there's this multiplicity, but maybe at least any closed loop bounds on this.

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Courant Events Right: That's not true. So some closed loops have multiplicity zero, if you like. They're not a load, okay? So the set of closed loops that bound disks are called self-overlapping.

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Courant Events Right: And those self-overlapping guys come with some multiplicity.

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Courant Events Right: And that's what you have to deal with. So here, for example, these two simple examples on the lower center and lower right-hand side are forbidden. There is no disk in those simple things.

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Courant Events Right: There you go.

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Courant Events Right: Alright, so how are we going to deal with this? At least.

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Courant Events Right: Maybe it looks complicated, but at least there is a way to go to define regardless of the model, because clearly this can be discretized.

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Courant Events Right: So let's say we're looking at the flat GD gravity, discretize?

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Courant Events Right: Take some square lattice, or any regular tessellation of the plane, but let's do the square lattice, and consider a model of random self-overlapping polygons.

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Courant Events Right: that you count with the multiplicity I was talking about.

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Courant Events Right: That is the same as counting the flat disk matrix, Uniform.

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Courant Events Right: That's a combinatorial problem.

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Courant Events Right: Very similar to…

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Courant Events Right: the way UV gravity is formulated by considering triangularizations. Here, these are triangularizations, or quadrangulations, with the constraint that the bulk curvature is zero.

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Courant Events Right: So I have a constraint both on the degree of the vertices and the length of the faces.

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Courant Events Right: That makes the model more complicated, because it's constrained. The boundary has no constraint.

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Courant Events Right: But inside, you have this constraint.

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Courant Events Right: You can formulate this in terms of a matrix model. I like DeFrancesco and Isickson, called, like, these dually weighted graphs. You know, DeFrancesco and Isickson considered that a long time ago. You can, you know, weight, consider matrix models where you wait arbitrarily

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Courant Events Right: the degrees of the vertices and the length of the faces, and this is an instance, of course, where you will froze, you will freeze the degrees to be 4 and the length of the faces to be 4 as well. So it's a complicated matrix model.

294
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Courant Events Right: techniques have been developed to deal with those matrix models by Kazakov and collaborators in the late 90s, but they didn't push it enough to immediately solve what we needed. But it exists. This formulation exists, that's an open problem, but it exists, in terms of the matrix model.

295
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Courant Events Right: All right.

296
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Courant Events Right: So, you, you know, we have started to work on this.

297
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Courant Events Right: you can have fun proving some simple things, but these simple things are not so easy to prove, even the very simple things. So, for example, the total number of configurations as a function of the boundary length that I call n, you can prove that it grows exponentially with n.

298
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Courant Events Right: So there's a connective constant in the model.

299
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Courant Events Right: And that's non-trivial because of the multiplicity.

300
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Courant Events Right: If you didn't have the multiplicity.

301
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Courant Events Right: the techniques used for well-known random polygon models, like self-avoiding polygons, you could use exactly the same idea to prove this exponential growth spin. Because of the multiplicity, it's actually much more complicated, because you have to prove that the multiplicity is also bounded exponentially by the boundary length, and that's…

302
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Courant Events Right: Missy? It's, okay, but you guys, it's possible.

303
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Courant Events Right: Eventually to do it.

304
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Courant Events Right: There is a parallel behavior.

305
00:43:01.180 --> 00:43:13.209
Courant Events Right: That I predicted in my 2024 paper, and that has now been confirmed by some explicit combinatorial bijective… with using bijective methods by Buddens Unveiled very recently.

306
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Courant Events Right: And there's also, for the pure gravity model, so these are results for the pure gravity model, where you don't couple any matter, CFT. You also have, for the pure gravity, a log correction.

307
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Courant Events Right: To that scale.

308
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Courant Events Right: Maybe that will ring a bell for some of you. These logarithms are usually associated with the C equals 1 UVL, and here we have a sign that the C equals 0 pure JT gravity has some similarities with C equals 1 UV, and it is actually true.

309
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Courant Events Right: And in particular, in the paper by Timbud, it's made very explicit, but also you will understand if we have time from what I said. And that nice picture here is taken from Timbud papers, so it's a typical, you know, he made the simulation, it's a typical configuration.

310
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Courant Events Right: You see that the boundary is very rough, and indeed it would be a fractal.

311
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Courant Events Right: typical boundaries in finance IG will be fractions.

312
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Courant Events Right: Very good.

313
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Courant Events Right: Let me go fast on these slides. I just want to emphasize… A crucial conceptual point.

314
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Courant Events Right: I explained that in oligographic setup with asymptotic boundaries, the group of diffiomorphisms that you gauge are the ones that are small, so they don't act on the bundle.

315
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Courant Events Right: When you go to finite sites.

316
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Courant Events Right: I claim that you always have to gauge both the small and the large details.

317
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Courant Events Right: Quantum gravity and final size geometry

318
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Courant Events Right: Should not depend on some arbitrary boundary coordinate system on the bundle.

319
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Courant Events Right: And when you discretize, when you have a discretized formulation, that's made explicit.

320
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Courant Events Right: Because the data, the combinatorial data in your discretizations do not encode any particular boundary coordinates, okay? You only have metric data, both on the bulk and the boundary. So it's automatic if you have a discretized formulation. But that's…

321
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Courant Events Right: Very important, conceptually, that if you go to finite volume, you have to gauge the large diffuse.

322
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Courant Events Right: All right.

323
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Courant Events Right: So how can we solve that model? And this I published already two years ago, and, you know, people told me that this is very, very difficult, it's nice, but it's very difficult. How are you going to deal with this super complicated problem?

324
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Courant Events Right: But then, okay, already in 2024, I had ideas how to do it, but now I'm going to explain to you that actually.

325
00:45:38.500 --> 00:45:47.069
Courant Events Right: You can sort of… Solve, at least conceptually, these constraints in a very beautiful way.

326
00:45:47.240 --> 00:45:48.000
Courant Events Right: Okay.

327
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Courant Events Right: And the key point will be to go just to conformal gauge. So the key point is just to be old enough

328
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Courant Events Right: to be born in the period where conformal gauge was very important, knowing the, you know, all the Louisville story and string theory story, and realize that, contrary maybe to the standard law, these techniques are completely relevant for JT gravity.

329
00:46:11.110 --> 00:46:13.510
Courant Events Right: And you can formulate GD gravity in this way.

330
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Courant Events Right: Hold on.

331
00:46:15.390 --> 00:46:22.120
Courant Events Right: Git really, I think, in spirit, is very similar to the SLE story.

332
00:46:22.550 --> 00:46:28.990
Courant Events Right: You know, the SLE story is such that, you know, in physics, when I was a student, SLE didn't exist.

333
00:46:28.990 --> 00:46:45.349
Courant Events Right: But we had very nice statistical physics courses explaining these beautiful, you know, physical things that are the easing model interfaces, or the population interfaces, etc. Very complicated, very mysterious objects.

334
00:46:45.500 --> 00:46:57.860
Courant Events Right: Then, eventually, the magic of SLE has been to realize that all these objects are particular instances of a process that is magically and beautifully related just to a Bronian process.

335
00:46:58.330 --> 00:47:11.550
Courant Events Right: And that's what makes it so simple, in some sense. So these super non-Markovian things are actually related to a Bornean process. And then you… with a good parameterization, if you like, you start to really understand what's going on.

336
00:47:11.870 --> 00:47:14.319
Courant Events Right: For this self-overlapping

337
00:47:14.610 --> 00:47:29.370
Courant Events Right: configurations in JT, I think what I will present is the same sort of thing. It will not be a linear process, it will be related to a low-correlated Gaussian free field, eventually, but that's the same idea. With the correct parameterization.

338
00:47:30.020 --> 00:47:35.580
Courant Events Right: Things are very simple, even though the actual geometries that you want to describe look so complicated.

339
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Courant Events Right: So let me describe the result for the flat JT on the disk.

340
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Courant Events Right: So you go to conformal gauge.

341
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Courant Events Right: you can write your metric, as I said, as I just said here, and I just parameterize it with some Q, because it's convenient to be complete.

342
00:47:53.610 --> 00:48:07.559
Courant Events Right: Okay, for those who are familiar with UVL, you recover some sort of UVL convention here. So D squared is exponential to phi over Qtz squared, phi is the UVL field, or conformal factor, that is parametrizing my matrix.

343
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Courant Events Right: Just in parentheses, I'm allowed to go to consider matrix of that form because I gauge the large TFOs.

344
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Courant Events Right: Okay, going to conformal gauge is possible if and only if you gauge the large DFOs. If you don't gauge the large DFOs, so the one that act naturally on the boundary, you cannot go to conformal gauge. Okay, so that's a very important thing to keep in mind. But since I did this, I'm allowed to do that. Now, for flat GT,

345
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Courant Events Right: The condition of flatness is just telling you that phi is a harmonic function.

346
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Courant Events Right: So it's super elementary.

347
00:48:41.650 --> 00:48:49.779
Courant Events Right: if you expand the boundary value of phi in Fourier mode, then the bank value is just obtained by acting with the Poisson kernel.

348
00:48:50.040 --> 00:49:03.920
Courant Events Right: which in polar coordinate is this very elementary and nice formula, okay? So everything is actually encoded in the boundary conformal factor in JT.

349
00:49:04.270 --> 00:49:12.929
Courant Events Right: This is in sharp contrast with UVIL. In UVIL, boundary and bulk fluctuations are independent.

350
00:49:13.700 --> 00:49:16.440
Courant Events Right: In GT, the bulk fluctuations

351
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Courant Events Right: are completely encoded by the boundary fluctuations, because of that constraint. That will also be true for the negative curvature model.

352
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Courant Events Right: But here, I don't have time, so I use this flat mount.

353
00:49:28.940 --> 00:49:31.810
Courant Events Right: Very good. Now, the measure?

354
00:49:32.380 --> 00:49:38.910
Courant Events Right: on the boundary, on this boundary UV field, will be region

355
00:49:39.060 --> 00:49:52.629
Courant Events Right: very explicitly in the way I just said. So, in terms of the Fourier modes, you just have some LeBank piece, and then some action, which is… the quadratic piece in the action is just governed by the directional Newman operator.

356
00:49:53.100 --> 00:50:04.300
Courant Events Right: So, in a Fourier mode, it's super simple. You know, just the sum of n, absolute value squared of phi n squared, then plus Q phi naught. Phi naught is the zero mode.

357
00:50:04.770 --> 00:50:07.259
Courant Events Right: This Q final term is crucial.

358
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Courant Events Right: The beauty of this formula is that it is a PSL2R invariant measure.

359
00:50:12.610 --> 00:50:27.409
Courant Events Right: And it has to, by consistency, because when you go to conformal gauge on the disk, of course, every… all the formalism has been variant under the automorphisms of the disk. So this has to be a PSL2R invariant measure.

360
00:50:27.820 --> 00:50:32.609
Courant Events Right: For reasons that are completely different from the PSL2R invariance of the Schwarzel.

361
00:50:32.860 --> 00:50:35.410
Courant Events Right: Here, we're for the flat model.

362
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Courant Events Right: And this PSL2R will be present also in positive or negative curvature, but it's not… it's not related to isometric group of a public space, it's related to the… to the fact that it's the automorphism group of the disk in conformal gauge, you have that. So this simple measure you can check is PSL2R invariant.

363
00:50:54.100 --> 00:51:02.920
Courant Events Right: you have that term, you can add also a bulk cosmological constant term, and this is the magic term. So this S lambda.

364
00:51:03.120 --> 00:51:07.150
Courant Events Right: It's the integral of exponential 2 phi over Q d2Z.

365
00:51:08.080 --> 00:51:13.800
Courant Events Right: So let me emphasize that term, because that's really the new… the fundamentally new ingredient in JT.

366
00:51:14.060 --> 00:51:19.649
Courant Events Right: Curiously, it's never been considered before. It could have been considered also in the context of UVL.

367
00:51:20.040 --> 00:51:21.960
Courant Events Right: What are the features of that term?

368
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Courant Events Right: 2 exponential, 2 phi over Q, is the area not renormalized.

369
00:51:29.230 --> 00:51:33.779
Courant Events Right: In UVL, you will have a renormalized operator, like some exponential gamma phi.

370
00:51:35.050 --> 00:51:39.059
Courant Events Right: Here, it's the classical guy, unrenormalized.

371
00:51:39.610 --> 00:51:43.030
Courant Events Right: Why am I allowed to do that? It's because in JT,

372
00:51:43.490 --> 00:51:46.330
Courant Events Right: The bulk is a smooth, here is flat.

373
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Courant Events Right: This capital phi is just a harmonic function, completely smooth.

374
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Courant Events Right: So this operator is protected, if you like.

375
00:51:54.970 --> 00:51:56.570
Courant Events Right: is unrenormalized.

376
00:51:57.250 --> 00:52:06.240
Courant Events Right: It's a marginal unprotected operator that you can add in your model. You could have added that in UVL as well, by considering

377
00:52:06.420 --> 00:52:10.220
Courant Events Right: on the disc, the harmonic extension of the boundary unit.

378
00:52:10.810 --> 00:52:13.110
Courant Events Right: And build that term and add it.

379
00:52:14.330 --> 00:52:23.660
Courant Events Right: Basically enough, I don't think that's… this has never been considered. People didn't realize that this term was a perfectly good marginal operator that you can add in UV,

380
00:52:23.860 --> 00:52:35.050
Courant Events Right: And it's conformally invariant, it has all the properties. So it doesn't exist on the closed space surfaces? Nope, it doesn't exist on the closed surface, absolutely. It's on your… you need the mark, absolutely.

381
00:52:36.480 --> 00:52:38.730
Courant Events Right: But no, in JT, this guy…

382
00:52:38.920 --> 00:52:48.709
Courant Events Right: is the geometric area, okay? So it's the physical area, the geometric area, and that's why you think about it, if you like, and you realize that you can write it this way.

383
00:52:49.470 --> 00:52:51.390
Courant Events Right: Alright, what time is it?

384
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Courant Events Right: 23.

385
00:52:54.570 --> 00:52:58.209
Courant Events Right: 23. 23, okay, so I have 7, 8 minutes.

386
00:52:58.530 --> 00:53:00.529
Courant Events Right: Two questions. Huh?

387
00:53:00.810 --> 00:53:18.920
Courant Events Right: Including questions. Including questions. So I'll be fast. You know, I'll be fast. It's okay, I think I can do it 5 minutes. So, all right. So, you know, I think I've shown enough to give you the flavor that, you know, all these old UV techniques are now going to be imported in JT.

388
00:53:19.240 --> 00:53:24.040
Courant Events Right: With some new flavor, with some similarities, but also some differences.

389
00:53:24.300 --> 00:53:27.970
Courant Events Right: The bulk is smooth, but the boundary is rough.

390
00:53:28.160 --> 00:53:46.580
Courant Events Right: And it's fractal, so you would… on the boundary, it will be the normal… actually, the boundary guy is exactly as in UVL. So you'll have renormalized boundary length beta, et cetera, et cetera. You can compute… you can use KPZ arguments to compute, for example, the post-door dimension of the boundary.

391
00:53:46.580 --> 00:53:57.369
Courant Events Right: So this is the formula. Nu is the inverse of the Ozdorf dimension, and C is the central charge of the CFT that you couple to JT, if you like. So C equal to 0 is the pure JT.

392
00:53:57.630 --> 00:54:01.929
Courant Events Right: You see that SQL0 plays the role of SQL 1 in UV.

393
00:54:02.690 --> 00:54:13.249
Courant Events Right: something I had advertised weekly before. So pure JT is the case where the other dimension is 2, and the parameter gamma equal to 2, so it's the critical value.

394
00:54:13.810 --> 00:54:15.679
Courant Events Right: For this low correlated field.

395
00:54:17.130 --> 00:54:25.370
Courant Events Right: Right? Very good. And you can use this continuum formulation to draw some typical configurations.

396
00:54:25.630 --> 00:54:28.020
Courant Events Right: So that's the top…

397
00:54:28.610 --> 00:54:35.770
Courant Events Right: picture, and you compare with what Tim got from his combinatorial, and I think that's pretty good.

398
00:54:36.220 --> 00:54:39.409
Courant Events Right: So this is for fugitive.

399
00:54:39.710 --> 00:54:46.679
Courant Events Right: Here, you know, you have two other pictures. C equal minus… so, when C goes to minus infinity.

400
00:54:47.030 --> 00:54:49.680
Courant Events Right: Exactly as in UV, that's a semi-classical limit.

401
00:54:49.960 --> 00:55:05.699
Courant Events Right: And indeed, you know, here, there's a picture for typical configuration at C equals minus 125. You look, you see that the boundary is smoother, much smoother than before. C equals 12 is in this, very dangerous interval between 0 and 24.

402
00:55:05.700 --> 00:55:19.239
Courant Events Right: Where everything explodes still, okay? So, it's like the interval between 1 and 25 in UV, you have some analog situation here, and it's some sort of exploding universe. I can analyze this picture, but I don't have time.

403
00:55:19.340 --> 00:55:22.079
Courant Events Right: Okay, now these are slides.

404
00:55:22.250 --> 00:55:26.090
Courant Events Right: That we're giving you more information on how you derive those results.

405
00:55:26.450 --> 00:55:29.190
Courant Events Right: I just want to mention that

406
00:55:29.380 --> 00:55:42.919
Courant Events Right: A crucial ingredient for the background independence of the model comes from the result by Collagen Guilleroux and Norang Guillpe in 2007 on the properties of determinants of direction to Newman operators, so I think it was quite nice.

407
00:55:44.060 --> 00:55:51.479
Courant Events Right: But that's not what I want, I don't want to finish on that. So, I want just to finish, okay, here I presented some sort of path integral.

408
00:55:51.620 --> 00:55:59.769
Courant Events Right: presentation, but of course, there is a CFT associated with that. So, let me call it the JTCFT. And this JTCFT

409
00:56:00.190 --> 00:56:07.229
Courant Events Right: plays the role for JT as the UVCFT plays the role for UV quantum gravity, okay? And this object exists.

410
00:56:07.470 --> 00:56:21.159
Courant Events Right: The crucial… so here I have the action. The crucial insight is that this bizarre field phi, which is widely fluctuating on the boundary, but which is smooth in the bulk.

411
00:56:21.480 --> 00:56:28.100
Courant Events Right: can be written in terms of ordinary CFT objects, just by writing phi equal psi plus chi.

412
00:56:28.550 --> 00:56:37.050
Courant Events Right: where Psi is a normal Newman-Coulomb-Gas scalar field, And Kai.

413
00:56:37.240 --> 00:56:41.680
Courant Events Right: Is a time-like scalar field with directionally boundary conditions.

414
00:56:43.540 --> 00:56:49.409
Courant Events Right: So, cyber sky is some sort of, like, a light-light direction in string theory. It's like an X+.

415
00:56:49.810 --> 00:57:00.489
Courant Events Right: But with bizarre boundary conditions, the time-like piece chi has directional boundary conditions. And this particular combination is such that when you compute, let's say, the two-point function.

416
00:57:00.930 --> 00:57:03.480
Courant Events Right: The piece from Newman in the book.

417
00:57:03.870 --> 00:57:09.619
Courant Events Right: That is widely fluctuating will be canceled by the minus sign that comes from the time like pi.

418
00:57:10.100 --> 00:57:22.739
Courant Events Right: And that's why phi is not zero, but is smooth in the bell. So that's how you make link between this bizarre object and normal, let's say, scalar field in CFT, and you have an action

419
00:57:22.740 --> 00:57:31.060
Courant Events Right: with the term proportional to capital lambda is the guy I emphasized already a couple of minutes ago, this new area term.

420
00:57:31.060 --> 00:57:43.440
Courant Events Right: Which is a smooth, but you are also, for the curved model, so the term proportional to eta Q, eta is plus 1 for the positive curvature model, minus 1 for negative, and 0 for flat.

421
00:57:43.490 --> 00:57:47.879
Courant Events Right: So… For the curved model, you also have this additional

422
00:57:48.080 --> 00:57:59.279
Courant Events Right: Marginal chi, so that's another marginal operator, proportional to chi, and this exponential term that you have to add to go to the curved model.

423
00:58:00.630 --> 00:58:01.550
Courant Events Right: Alright.

424
00:58:01.990 --> 00:58:03.980
Courant Events Right: Do I have one minute to conclude?

425
00:58:06.310 --> 00:58:12.729
Courant Events Right: Yes? Yeah? I'm a little? Okay, so let's do the last minute, some philosophical remark.

426
00:58:13.060 --> 00:58:18.530
Courant Events Right: You could say, okay, well, okay, so we have the infinite geometry picture that's a Schwarted.

427
00:58:19.030 --> 00:58:28.270
Courant Events Right: No, I try to convey the complications and interests to go to the finite geometry pictures of JT, so that's completely new.

428
00:58:28.410 --> 00:58:30.040
Courant Events Right: It is natural to ask.

429
00:58:30.290 --> 00:58:38.139
Courant Events Right: Well, is the infinite geometry picture can be obtained by some limit, or the finite size picture?

430
00:58:38.350 --> 00:58:44.230
Courant Events Right: And I think yes, but that's quite sub… How could that work?

431
00:58:44.470 --> 00:58:59.150
Courant Events Right: So, the idea, first of all, is that, okay, if you want to go to infinite geometries, you have to inflate, if you, like, favor the large geometries. A simple way to do that is to turn on this bulk cosmological constant.

432
00:58:59.150 --> 00:59:08.610
Courant Events Right: Lambda and pec lambda goes to minus infinity. Obviously, that will favor very large areas, because remember, lambda is the term in the action that is proportional to area.

433
00:59:08.610 --> 00:59:14.830
Courant Events Right: So if lambda is very large and negative, that favors very large geometries. So taking that limit

434
00:59:14.940 --> 00:59:19.249
Courant Events Right: Probably is going to inflate my geometries.

435
00:59:19.690 --> 00:59:31.889
Courant Events Right: The betta S, I don't have time to explain how I get this formula, but this beta S of the Schwarzan theory will be related to the microscopic parameters of the finite size theory in the way I indicated here.

436
00:59:32.040 --> 00:59:35.530
Courant Events Right: But what I just want to emphasize is that if it works.

437
00:59:35.850 --> 00:59:38.550
Courant Events Right: It has to be in a hydrodynamic way.

438
00:59:39.000 --> 00:59:42.569
Courant Events Right: It is clear that even if the geometry is super large.

439
00:59:42.890 --> 00:59:52.169
Courant Events Right: On distance scales that are of the order of the curvature length scale, you'll see the fractal structure. So if you zoom in, you will always see the fractal structure.

440
00:59:52.700 --> 00:59:56.849
Courant Events Right: However, the claim is that because of properties of hyperbolic space.

441
00:59:57.000 --> 01:00:01.710
Courant Events Right: When you have very large geometries, there will be a separation of scales.

442
01:00:03.060 --> 01:00:08.439
Courant Events Right: The geometry… the binary will become much, much, much, much longer than the curvature length scale.

443
01:00:08.590 --> 01:00:12.319
Courant Events Right: And you can imagine that on those very large length scales.

444
01:00:12.520 --> 01:00:18.500
Courant Events Right: the boundary is smoothed out. There's an average notion of smooth boundary, and

445
01:00:18.900 --> 01:00:29.040
Courant Events Right: maybe it wiggles smoothly and gently, and maybe the fluctuations of these large-scale things is given by the Schwarzene action.

446
01:00:29.940 --> 01:00:42.420
Courant Events Right: Okay, so that's the claim, and, you know, you can present the conjecture very precisely, because there is actually a very natural candidate for the emerging reparamidization mode, so, you know, I think that's a super

447
01:00:42.420 --> 01:00:52.499
Courant Events Right: nice problem for both mathematicians and physicists, you can consider f of theta defined in that way, okay? So that's a very rough

448
01:00:52.590 --> 01:00:55.219
Courant Events Right: Object, that's a distribution object.

449
01:00:55.340 --> 01:01:02.229
Courant Events Right: No, the claim is that, in the limit I indicated before, the law of this guy will go to the Schwarzene.

450
01:01:04.570 --> 01:01:08.329
Courant Events Right: So, some… from very rough to very smooth.

451
01:01:08.600 --> 01:01:14.100
Courant Events Right: And that is how the reparametrization could emerge in that limit.

452
01:01:15.350 --> 01:01:25.469
Courant Events Right: If you think about it, it means, you know, that the reparimetrization, which is the gravity mode, emerges, it means that there is a fundamental coordinate that emerges.

453
01:01:25.500 --> 01:01:36.389
Courant Events Right: And the gravity is described by this re-parametrization of the phenomenal coordinate. Here, the phenomenal coordinate is just the angular coordinate that you have in conformal lineage.

454
01:01:37.230 --> 01:01:40.110
Courant Events Right: So that's emerging time, literally.

455
01:01:40.730 --> 01:01:49.979
Courant Events Right: You know, if what I've said is correct, you go to a… in a hydrodynamic way, you go from a model where you didn't have any

456
01:01:50.170 --> 01:01:52.770
Courant Events Right: Privilege coordinate on the boundary.

457
01:01:53.400 --> 01:02:07.209
Courant Events Right: to something that was described originally as a model where you didn't gauge the boundary reparameterization. So this boundary coordinate has to emerge in that picture. That's a model of emerging time.

458
01:02:07.900 --> 01:02:08.870
Courant Events Right: Detroit.

459
01:02:09.110 --> 01:02:11.279
Courant Events Right: And it tells you that, you know, this…

460
01:02:11.340 --> 01:02:29.919
Courant Events Right: Bondary gravity on very large distance cache is a smooth, it's hydrodynamic. It's given by hydrodynamics. And the TT bar, for those who know what it is, my claim is that all these TT bar approaches to finite geometry nallography, they are actually given not the microscopic theory, but this hydrodynamic guy.

461
01:02:29.990 --> 01:02:32.210
Courant Events Right: Okay.

462
01:02:32.400 --> 01:02:33.680
Courant Events Right: deal with punishment.

463
01:02:34.360 --> 01:02:44.889
Courant Events Right: Yes, it's… yeah, that's Euclidean. But, you know, I think this story has also a real-time picture. But yes, I understand things very well in the Euclidean, absolutely.

464
01:02:46.090 --> 01:02:48.150
Courant Events Right: Okay, thank you very much for your attention.

465
01:02:53.250 --> 01:02:55.289
Courant Events Right: Questions or comments for Frank?

466
01:02:56.210 --> 01:03:04.180
Courant Events Right: Yes? So the TDCFT that you mentioned, do you understand how it can be obtained from a limit of time like Livial to space like Luvial? No.

467
01:03:04.870 --> 01:03:19.599
Courant Events Right: So I, I just… I don't think those… those… yeah, I don't think it's, it's very similar, right? So, oh, sorry. So the claim in the physicist literature, whereby combining time like Luvial and space, space-like visual, the usual Livial.

468
01:03:19.600 --> 01:03:31.260
Courant Events Right: and do recapination rescaling, using the common parameter and equals to zero becomes something like this. No, because… so what you're referring to is a formal limit

469
01:03:31.400 --> 01:03:34.800
Courant Events Right: Where you reproduce, the Schwarzene.

470
01:03:36.080 --> 01:03:56.060
Courant Events Right: This has nothing to do with this. No, but there's a recent work, actually, I also test this directly in 2D, so in a fold, where you combine the time-like fluvial and space-like fluvial, and you will see this kind of… But you think gamma goes to zero? That's right. So that's a semi-classical, and that would be related to Schwarzene-like physics. Yes. Here, it's supposed to be something much more general.

471
01:03:57.000 --> 01:04:02.819
Courant Events Right: The picture you're mentioning, too, is supposed to be a limit of that, probably the limit I've just described is hydrodynamically.

472
01:04:02.970 --> 01:04:09.279
Courant Events Right: But this is a much more general, fundamental picture. I'm saying this is describing

473
01:04:09.430 --> 01:04:19.769
Courant Events Right: any JT configuration of finites, including the finite size, for any curvature, in an exact way. And how would you say it's related to the

474
01:04:20.040 --> 01:04:24.169
Courant Events Right: I don't know.

475
01:04:24.260 --> 01:04:37.009
Courant Events Right: I have never understood in any rigorous way even the link with the Schwarzan. I think, for me, this limit was always very formal. Indeed, you see, for example, you can match the Schwarzene field theory correlators.

476
01:04:37.010 --> 01:04:47.859
Courant Events Right: with the UV coil in that, so that's very nice, okay? Actually, I think maybe the first derivation of the Schwarzan field theory coordinators were done via this sort of ANZATS.

477
01:04:48.040 --> 01:04:53.529
Courant Events Right: But I don't think there is any deep understanding of why this is true, but anyway.

478
01:04:53.930 --> 01:04:56.199
Courant Events Right: This refers only to the Schwarzene.

479
01:04:56.340 --> 01:05:13.099
Courant Events Right: So this is indeed sort of confusing, because yes, you've heard about sort of UV-like things in, before for JT, but that has nothing to do with this. So this is really the finite size, finite cutoff story, and that's supposed to be an exact description, and that's…

480
01:05:13.100 --> 01:05:23.780
Courant Events Right: derived from first principles, because there is no… onsats here. You don't understand, for example, how extract the data? How do… Not at all.

481
01:05:23.780 --> 01:05:44.320
Courant Events Right: So, you know, this is completely new. So it's very… it's not… I don't think it's fair to say that this… these things from the time, like, the plasma, space, because there, people understand. No, I mean, I agree. They could still be related. This CFT, you know, you have to study it, I mean, but it is a CFT. The point I wanted to make is that these CFT tools…

482
01:05:45.040 --> 01:06:02.310
Courant Events Right: work and can be applied for GT with unfinite sign geometries. There will be a bootstrap, it's completely conformally invariant. You can use all your favorite tools of CFT to try to work out what it is, but it's completely new, nothing is not… I hope simple things can be done easily now, so…

483
01:06:02.680 --> 01:06:03.530
Courant Events Right: Yeah.

484
01:06:06.320 --> 01:06:10.779
Courant Events Right: This is why you are saying that, or…

485
01:06:10.980 --> 01:06:20.969
Courant Events Right: Okay, let me, repeat it in a similar way.

486
01:06:21.410 --> 01:06:29.060
Courant Events Right: So, the Schwarzene field theory description It's no more holographic description.

487
01:06:29.230 --> 01:06:34.559
Courant Events Right: For which, by definition, The coordinate on the battery is fixed.

488
01:06:35.000 --> 01:06:47.129
Courant Events Right: Okay, so you do not gauge boundary reparametrizations, you don't have that, no one. So you have a fundamental time, let's say, of the boundary. That's what I call the time, that's the coordinate of the boundary.

489
01:06:49.110 --> 01:06:57.019
Courant Events Right: That's, you know, in olography, that's… so the battery is the battery. Tan is the battery coordinates that

490
01:06:57.440 --> 01:07:08.199
Courant Events Right: Here, in our two-dimensional bulk model, the boundary is one-dimensional, so you just have Now, the finite geometry picture?

491
01:07:08.310 --> 01:07:11.060
Courant Events Right: It's based on a completely different starting point.

492
01:07:11.440 --> 01:07:17.790
Courant Events Right: I consider finite-sized geometries, And I gauge Large, if you want.

493
01:07:18.700 --> 01:07:31.940
Courant Events Right: So, for the finite-sized geometries, there is no boundary coordinate that is privileged. By definition of the model, everything is built like… so at the microscopic level, you don't see those things at all.

494
01:07:32.450 --> 01:07:37.059
Courant Events Right: Now, the claim, that hasn't been proven. It's a non-trivial claim.

495
01:07:37.260 --> 01:07:43.350
Courant Events Right: is that in the limit where I inflate, My finite geometry picture?

496
01:07:43.840 --> 01:07:48.080
Courant Events Right: There will be a hydrodynamic, mood.

497
01:07:48.290 --> 01:07:52.089
Courant Events Right: That will govern the long wavelength fluctuations.

498
01:07:53.490 --> 01:07:55.540
Courant Events Right: I will no longer zoom in.

499
01:07:55.650 --> 01:08:12.200
Courant Events Right: to see the fractal structure, I will remain at length scale much larger than the hyperbolic curvature length scale, and on those length scales, I say, okay, the boundary curve looks very smooth, and I will re… I will discover that the dynamics is better.

500
01:08:12.360 --> 01:08:15.139
Courant Events Right: Bion mode, which is a repariat research.

501
01:08:15.470 --> 01:08:20.490
Courant Events Right: And eventually, in the limits, the claim is that it will match with the Schwarzene theorem.

502
01:08:20.819 --> 01:08:29.580
Courant Events Right: Schrozian theory, again, that was defined by having a phenomenal timing. So there's a phenomenal Hamiltonian, if you like, in the Schrodinger theory.

503
01:08:29.960 --> 01:08:34.719
Courant Events Right: Whereas the Hamiltonian final volume at the microscopic level is zero.

504
01:08:35.160 --> 01:08:44.140
Courant Events Right: So if there is a limit that gives the Schwarzene, it has to be in a hydrodynamic kind of effective long wavelength object, and there will be.

505
01:08:44.890 --> 01:08:51.599
Courant Events Right: So, the emerging hydrogen, hydrodynamite, and associated with its own notion of time.

506
01:08:51.750 --> 01:09:05.940
Courant Events Right: So that's the sense… Yeah, I won't… I won't remain a little bit serious undergo, so I will not make such things, okay? But it's a model.

507
01:09:06.460 --> 01:09:23.499
Courant Events Right: for emerging time, and I don't know any other, okay? So, you know, well, I don't want to go beyond that. I think it's a cool thing to see how some Hamiltonian evolution can emerge, starting from a model that doesn't have such a thing.

508
01:09:29.439 --> 01:09:35.619
Courant Events Right: It's a nice question from a non-expert. If I remember correctly, it's a… Here's so good.

509
01:09:48.830 --> 01:09:53.090
Courant Events Right: In terms of the panic comparisons.

510
01:09:54.480 --> 01:10:04.889
Courant Events Right: the rule for anything like that in the area of study, that the synergy gamma goes to 4 or gamma. Yeah, exactly. So, you know, it's… this signal…

511
01:10:05.170 --> 01:10:19.059
Courant Events Right: This symmetry is also present in the VIN, but it's not really seen it yet. So, in the same way as in the VIN, you have such a thing, and you have it here, but it's not from the symmetry.

512
01:10:19.600 --> 01:10:27.700
Courant Events Right: They've been driving, but I don't think it's really valid. As far as I understand, for example, you know, the area operator, I'm talking about building.

513
01:10:27.800 --> 01:10:31.489
Courant Events Right: The early ambulatory, like, these are exponential gamma phi.

514
01:10:32.120 --> 01:10:39.810
Courant Events Right: And, you know, I'm also open nature, where gamma goes 2 over gamma, that's the same dimension. As far as I can understand my beats.

515
01:10:40.180 --> 01:10:45.740
Courant Events Right: France told me that, only one of these guys can make sense of it.

516
01:10:46.180 --> 01:10:48.380
Courant Events Right: And the other guy, not really.

517
01:10:48.590 --> 01:10:53.849
Courant Events Right: So I don't think there is really an exact strategy.

518
01:10:54.080 --> 01:11:02.110
Courant Events Right: Yes, for SLD, yes, but for legal, as far as I understand. Yes, but in your case, they are not editor.

519
01:11:02.680 --> 01:11:20.809
Courant Events Right: Yes, because… Q is also electric gamma in the usual way. So Q is 2 over gamma, it's gamma over. So you will find, you know, just from that… but I… at least at my general of understanding, there is nothing Jim I can say about this.

520
01:11:21.590 --> 01:11:25.760
Courant Events Right: That's something to me in my opinion, but I, you know, I don't…

521
01:11:32.990 --> 01:11:35.019
Courant Events Right: But it does time to drag it in.

522
01:11:38.880 --> 01:11:41.089
Courant Events Right: We've only had 11.