 Sure of the day and the third of Julian's honor about developments on quantum with theory and chaos, please Okay, so the third lecture I want to talk about the ergodic phase of quantum gravity And let me recap a couple of points. So first of all We've already argued that this ergodic phase which I which I defined And to be the phase where a random matrix type description starts being useful and appropriate it's actually a very sensitive probe to the structure of the microstates of the underlying theory and that of course in That case gravity is no exception and in particular. It has non-perturbative sensitivity to the microstates and Then I've argued that there's this sort of symmetry breaking story which for reasons I didn't delve into a lot but Alex and I like to refer to this as causal symmetry breaking because it is intimately tied to the causal structure of the theory and That had to do with what I had been already asked about why for example We have this interesting analytic continuation in the advanced and retarded sector at the same time And and so I would like to say that this causal symmetry breaking is another way abstractly of defining the ergodic phase and in concrete terms for example via this flavor non-linear sigma model But there is The question still remains in some sense, what is special about gravity in the story and Maybe we can discuss this tomorrow or if there is time even today So, let me therefore talk about the example right now of two-dimensional JT gravity But keep title boy gravity So this is a low-dimensional model. Obviously, it's in two dimensions Where you have You know, it's let's call it a low-dimensional Low-D model This the ergodic behavior can be established Explicitly so to define it It's quite useful to write down the action I just realized that I should probably call it I because I will later also in Introduce some object that I will call S JT which will be the spectral curve. So let's call the action I So this is a two-dimensional theory of gravity and in two dimensions the Einstein Hilbert term itself is topological In fact integrates to the Euler number Chi of M and it's customary in the setting to call the coupling constant to the Euler number S0 So that's really just a choice of name, but the name comes from the fact that There is a there is a certain connection to this SYK class of models That have behavior at the lower end of their spectrum That includes JT gravity as a mode and there this coupling is really identified with the the generacy of the ground state of SYK So this is the residual ground state entropy there, but it's here is just a symbol We could just call it any coupling lambda for example if we prefer that but then you actually Have the following model you have an integral over these two-dimensional metrics to two-dimensional manifolds And instead of writing down the naive Einstein Hilbert term which gave me the Euler density We introduced this field phi a dilaton which multiplies r plus 2 And then you have a bunch of boundary terms So for example, you have something like the Gibbons Hawking York boundary term And you will have some Holographic renormalization terms So those boundary terms they are important when you talk about boundary conditions and For the first half I will choose boundary terms such that I can impose Dirichlet boundary conditions So Right so and here I could say that this is Einstein Hilbert term and it gives rise to the Euler density and there's this old Mathematical joke that in three dimensions gravity has no degrees of freedom So in two dimensions it has minus one degrees of freedom So you have to add one more degree of freedom to make this theory a trivial trivial again Okay, so that's why you want to have this field phi here so and What I want to do is I And I want to compute the spectral form factor in this theory and and I should say that Part of what I'm talking about today will rely will be in also in the spirit of a review of Seminal work that is due to the Stanford group. So in this case sad Banker and Stanford Conveniently all working in Stanford and of Maria Mirzokhani and actually also in our and oran time who are more or less mathematical people who actually established Essentially that story but from a mathematical perspective previously. Okay, I Will not actually use by the way the point of view of these authors, but I might make some reference to it in words So we want to compute the spectral form factor of this theory And and as I had already said actually in this setting. It's it's quite natural to compute Things like partition functions where you fix the inverse temperature Fixing the inverse temperature is precisely fixing the length of the boundary circle and That will be what these boundary conditions that will choose will achieve so z of beta is That thermal trace e to the minus beta H of the system and then I can reconstruct the Spectral density row of e basically via inverse Laplace transform of The of beta, okay So and the beta Right, so So let's talk about the pathological for Jt gravity and as I said even though I think the mathematical foundations really were laid by by the mathematicians. I am following the treatment of In the seminal paper by SSS so Well, what one would write here is something like the integral over Phi and metric Okay, of e to the minus i Jt and Where Well, this is I guess an equation I want to refer to again Where this is a two-dimensional gravitational theory and in fact one one thing that is nice to note Is that actually as such Being a two-dimensional theory of gravity Also the string world sheet happens to be two-dimensional and you can actually treat The this kind of object you can treat it in analogy Can be treated to the way that we usually deal with the string world sheet It's also not really a perspective that I'm going to emphasize a lot, but I'm going to make comments maybe here and there where it is appropriate. So, okay So the other thing is of course as so z of beta, right? So Basically, we compute z of beta and other quantities with the same boundary conditions Which basically means that for now we study the canonical ensemble of this So That corresponds as I was saying to these seriously boundary conditions So what we say is that and let us so so you remember also from the previous lecture now These ideas these pictures of time slices basically discs with a circle boundary so let us think about Disc with a circle boundary and let us parameterize the distance from the boundary is epsilon So this is like very large radius in the Notation that the that the Chris was using okay, but I'm epsilon small meaning we're closer and closer to the boundary and What we want to say is that? Phi at this boundary, which we regulate with epsilon is fixed Which means that we call it something like phi Boundary and we have to multiply by one over epsilon squared No one over epsilon Well, and we also fix the metric which is like let's call the boundary Time-connet g u u and this we fix to also a constant here. It's one of our epsilon squared and What this means is that the it fixes the boundary length to be? Fixes the boundary length To be Beta over epsilon squared which is the appropriate thing to fix it to in the canonical ensemble But it only fixes the length We haven't said anything about the shape of the boundary curve So in fact it allows for an arbitrary shape of this boundary curve Which in this business people call it allows for a wiggly boundary So in actual fact we cut out some disc shaped region, but we don't really specify the shape so this thing is Called a Wiggly boundary and if you want of course what we have is we still have topologically a circle It's just we haven't we haven't fixed its shape And so you can actually define such curves in terms of circle diffeomorphisms and the theory of circle diffeomorphisms Via its connection to the Versailles-Rochot joint orbits is actually sort of the mathematically correct way to think about really defining What's going on with this boundary wiggle contribution to the passenger goal? But we can be more pedestrian We can actually just say When you Integrate what remains here on the disc is The degrees of freedom of the wiggly boundary So we just need to figure out what is the action cost of the wiggly degrees of freedom and then integrate over them And that's by the way the analogy Although I'm now realizing it's maybe not the best possible analogy, but over these boundary gravitons and adiastry So however, okay, this this is this is what we have here And and maybe the other thing I wanted to say is that Wiggly boundary so you see this this this ghy boundary term roughly is something like boundary metric Times phi at the boundary times the extrinsic coverage of the boundary minus one Where the minus one is a holographic renormalization type term This is what we would call the Gibbons Hawking York boundary term and we're not fixing so that's another way of saying wiggly boundary We're not fixing case of the extrinsic coverage in particular need not vanish So it need not be a geodesic curve So basically I'm just restating that you can allow these wiggly boundaries and we have to integrate over them so and what you do is that and We need to integrate over the over the wiggles and It's actually a simple case of Putting in those boundary conditions putting in some parameterization for an arbitrary Wiggly curve and plug it into this formula and what pops out is this this famous schwarzen action So need to integrate over the wiggles with The schwarzen action so in other words it costs you a little bit to wiggle the boundary But in a nice way and and so this integral can be done and Well, I will cite some results. I will I will I will give you some results in a little bit Okay now very good, so However We also have something like this defi integral here and the defi integral just gives you a constraint namely that are So it's like a delta function of r plus 2 so in the end you can focus on hyperbolic manifolds And you just need to somehow integrate over the moduli space of the hyperbolic manifold and over the remaining boundary wiggles and so so let's say the phi integral imposes Delta of r plus 2 and so it remains To integrate over hyperbolic metrics and so the result will be something like the partition function. Oh, and I don't I don't know why I Neglected to say this so for example here what we have is one boundary One circle means one boundary, but in general we may consider The case where the boundaries actually a disjoint union of circles. So as circle one Union two union SN So you have some manifold which has you know all these Circular boundaries and of them each boundary has some wiggles and then we need to glue them together and Integrate over the remaining moduli space. So then you get an expression of the form z of beta one up to beta n where each of these boundaries has some fixed length and This is actually going to be a a sum over the genus of coupling constant lambda to the Euler number Which we have seen and of something like Z G comma M Where G is the genus G contribution with n boundaries and to this partition function So Kai the Euler number is the usual Two minus two G minus n where G is the Genus n is the number of boundaries and the coupling constant lambda is equal to e to the minus s zero and It may be appropriate to make that comment now So so from this point of view this coupling constant lambda would be if we were to To be in the mood to treat this like a string theory well sheet This would be what we call the string coupling and it's just parameterized here by e to the minus s zero, okay? Very good, so see GN is then the integral over all hyperbolic metrics on surfaces of genus G with n boundaries so People like to draw sort of diagrams where you have various Numbers of boundaries and you could have you know an arbitrary number in here and then you could put handles also however many you want and so We have basically here. We have our you know s one As to et cetera SN Okay and for each boundary In principle you would have this sort of wiggly cut-off boundary shape here also to take into account Okay, so now There are two distinguished cases. I've already shown you one So before I write down the answer for the for this the general answer This general answer can be phrased in terms of only three ingredients One of the ingredients is this Okay, and in particular also this idea of integrating over the wiggly boundary the other in the ingredient will be something that's related to The modular space of genus G Riemann surfaces with n geodesic boundaries And this is this is really where Mirza Hane's work Has been extremely important And and another another geometry like this one, which is the only other distinguished one So let me write this here. So we have our two distinguished configurations, let's say so the first one we already said was the disk and the disk has Oiler number one because it has one boundary it has no genus and I start at two so with oiler number one and this guy I've already drawn And you integrate over this with respect to the Schwarzian action and what you actually find So now it's something like one over beta to the three halves times e to the sum number divided by beta. I Think Yeah And if you want just notation wise we might call this guy Z zero comma one Simply because we have g is equal to zero and is equal to one But the other and this has positive oiler number There is one more contribution that has positive oiler number which has the topology of an annulus so the second one is the topology of the annulus But in this business the annulus is called a trumpet and that's a that's a geometry which has two Boundaries, so therefore it's an annulus Genus zero so it has oiler number zero Okay, and I actually for this game. I need to consider one boundary to be Wiggly and the other boundary will be geodesic meaning that it has Zero extrinsic curvature, and I fix its length to be L just by definition of the notation And okay, so why do people why did people call it the trumpet because? So I think this notation came from the SSS paper because they like to draw it like this where you have you know, maybe this geodesic boundary here and then the The other end of the trumpet they drew much bigger and you can imagine this this might be a trumpet but you know it's topologically the same as this annulus with one wiggly boundary and one geodesic boundary this guy would be Z zero comma two and Also, you can integrate against the Schwarzian and you find that this guy has One over square root beta Times e to the sum number divided by L square root beta now So then let me write the general answer for the general contribution And then we will go back and we will pick out the terms that are actually of interest for us for now Okay, so I'm assembling sort of a number of ingredients and Well, we're we're slowly getting to where we want to be so in general then In general you have then z g n and Yes, actually it's it is good that I wrote it here because this is really the result of all these people together Z g n Can be written as so this is now the actual the actual answer for a geometry like this Where I have if you want n of these wiggly asymptotic boundaries. I have some Genes that comes from you know what I drew here and the way that I Did I assemble this is actually by using? trumpets for each of these asymptotic boundaries and then gluing them to a Riemann surface of genus g which has n geodesic boundaries because those those objects Are in some sense more? well-controlled so I can write this then as a integral of D li li One for each geodesic boundary Okay, so each of these geodesic boundaries are integrate over its length and and I put one of these z o z trumpet So I'm also this is this this is also sometimes called Z trumpet So I take a Z trumpet Which has a functional form that I showed but it also helps us to basically convert And this li into the beta I boundary condition that I fixed Okay, and I integrate those guys into an object which is called v g n of L1 up to L n where the only object that I haven't yet defined is this v g n and this is called a vial Peterson volume Of the moduli space of Riemann surfaces of genus g with n geodesic boundaries and these guys satisfy the so-called Mizahani recursion So the fact that we can actually compute these things which are very interesting geometric objects and Also interesting object as it turns out in two-dimensional and quantum gravity And Is thanks to a very ingenious recursion relation of Mizahani, which also as I said was developed and Put into matrix model technology context by you know I just ask one. Yeah, so your number of weekly boundaries is the same as the number of geodesic boundaries. Yes Is that necessary? And no, I don't think so so and I think you can choose different boundary conditions on the external boundaries you could also in principle consider something like a Neumann boundary condition for an external boundary or you could even I don't know people have done that but give some kind of Geodesic boundary condition for the external boundary and then those things would not necessarily match up But the general configuration that you are taking is a Riemann Sarthasab genus G with certain number of weekly boundaries and certain number of geodesic boundaries, right? the configurations are the configuration I want is the one which has Asymptotic boundaries, which are wiggly boundaries and the way that I constructed all of them are with the boundaries So the external boundaries. I'm choosing them all to be wiggly boundaries because that's the canonical ensemble boundary condition that I want Okay, but if I think of this as a more general formalism that produces answers for geometries I am free to specify other asymptotic boundary conditions. I see. Okay, just to add to this answer So in order to construct this whole partition function one can have some building blocks so for each of the wiggly boundaries as he drew that one can have a trumpet and To construct the general Riemann surface one can have a generic pair of pants decomposition and one can cut Along some circles and for each pair of pants. There would be three geodesic boundaries over which it has to be integrated over Well, that's a question of how you how far you think about the while Peterson volume, which I basically just Put here so indeed if you wanted to develop this Mirza honey recursion Then you would have to think about the different pair of pants the compositions of Riemann surfaces with only geodesic boundaries and genus G all right, so But actually Okay, it's nice that we can write down this general formula and it will be useful But I want to know and pull it back again and talk about some simple some simple examples that that will Make contact to this special form factor story and I should say Yeah, no, I will say that okay that will come in natural very good so So then the JT special form factor So and what we find is then that in in pictures Z of beta All right, that's equal to and all of the geometries that have one Wiggly boundary by the way So it will be this plus e to the minus s zero the thing with one handle plus etc Okay, but this Z zero one the leading contribution is Precisely what we calculated That's the disc with the wiggly boundary and So that means that if we if we take just this so So from this From this alone We would get the we would get the equivalent of the answer that I argued happened in the three-dimensional case So we would get something like Z of beta one Z of beta two would be a product where I have this Times this where that guy might have fixed boundary length beta one and this might have fixed boundary length beta two And of course I would have corrections where I multiply for example this guy with this guy or this guy with some other guy But they are they are suppressed in e to the minus s zero and they actually don't change the qualitative story That I want to say now to have a qualitative change. We have to do Something else, but first let's let's let's say what this story is So then if I now look at what is the answer for the disc that I had Then I find that f beta of t Basically will be this Z zero one of beta plus it Z zero one of beta minus it and looking and I like to consider this just to see what it looks like at late time I take t much bigger than beta one beta two If I look at what that what that gives me it gives me One over t from each of them So one over t cubed and this guy gives me the usual Beta over beta square plus t squared so that gives me the promised that gives me the promised decay Which is one over t cubed Okay, and so that gives me the Semiclassical part of the spectral foam factor if you want and in particular we notice that this we noticed already that this is in trouble with uniterity at late times But actually What what really happens is that if you take this recipe what you really should say is that it's this guy so What really happens is that Z of beta one the of beta two Computed in JT according to the path to go that we just established This has our contribution that we just talked about plus each of the minus s zero times a geometry which has now two external wiggly boundaries and no genus so G Z zero comma two and And that thing can be calculated again with the ingredients that I showed here And and what you find is we find that Z zero comma two of beta one Beta two is actually one of a two pi Times the square root of beta one beta two Divided by beta one plus beta two And if I again expand this for late times This guy now gives me some one over two pi times t linear and t over beta one plus beta two So this is exactly the linear ramp Contribution or in chaos terms. This is what we have called the spectral rigidity or you know It's spectral rigidity is some sort of Version of level repulsion because you see all the levels repel each other So if I push on one then all the levels we be pushed So they tend to actually arrange themselves in a fairly regular pattern and the fact that these quantum chaotic systems arrange the spectrum In a fairly regular pattern. It can't be exactly regular by the way, but fairly regular pattern That's what people like to call spectral rigidity and is behind this this ramp behavior actually. So this is what We have found here So in other words what we have managed to do is we have managed to cover two Epochs if you want of the spectral form factor We have calculated this decaying part the e sorry the one over t cubed and now we have found that it gives way to in Fact this linear rising ramp as it should as we have expected Okay, but this linear rising ramp for now keeps going So this is this is like the z 0 comma 2 contribution and this is if you want the semi classical semi Classical contribution Okay, and So that's very nice. Of course. That's that's I think a beautiful result So in other words There is there are configurations which we call Euclidean wormholes. So they are connected contributions that and Have the same number of boundaries as the product of two disconnected space times here and that sort of saved the day that sort of Quantum fluctuate in at late times to at least go in the right direction But the linear growth as I already argued is not all and in fact what we you what we were quite excited about Asking about is this plateau? So that's what I want to talk about next But I see that there are any questions. Yeah, so what you are calculating here presumably is not just the product It's some kind of averaging going on right Because if it was just a product you couldn't have gotten the connected Well, I didn't do any averaging. You would have seen me do it I didn't do some sleight of it, but no no, but the question is extremely good in the sense that what I have done is I have Have looked at the JT pathway will only I Haven't I haven't done if you want a boundary computation and the JT pathway will at least in the interpretation that I that Maybe is argued for by these people Tells you that there is this contribution here now what you could say is that now if you go back to What does this mean from from the boundary perspective there it seems to imply Some sort of average at this point Yeah, because you could have individually calculated zeta beta 1 and zeta beta 2 and you will not see this diagram at all Right. Yeah, but what I could have done is I could have calculated In an individual theory Z of beta 1 Z of beta 2 and then subjected to this quantum chaotic type Analysis where I told you that I need to do some sort of effectively some averaging And One certainly viable interpretation is that what JT gravity at least this level has access to is only that part of the computation But underlying it is some microscopic theory But those are very subtle questions, which I think we should discuss Either in the discussion session today or at the very end Because they go right to the heart of some of the important conceptual implications here But for the time being I want to maybe push the technical develop a little a little bit further Okay, well probably I have not understood the notation, but even in JT gravity. I understood from that formula that Z of beta 1 beta n is obtained by taking n boundaries Yeah, so This is in JT gravity. So z of beta 1 Indeed you've wrote that formula, which is a sum over over the genus So I have not understood the it seems to me that this should be z of beta 1 comma beta 2 if you if you write 0 of beta 1 and I mean, I think we're just we're just arguing about notation here. So yeah What I'm saying is that the quantity Z of beta 1 Z of beta 2 evaluated in JT gravity I Interpret that to be the instruction to calculate all JT configurations, which have two boundaries What is it different between 0 of beta 1 0 of beta 2 and 0 of beta 1 comma beta 2 One of them is just a connected contribution and the other one. Oh, okay. I see I see Okay So let me let me make a couple of comments. So one is that Okay, so the interesting question that I would like to answer today still is how do we think about the plateau? but but I want to point out something About the let's say Non-perturbative nature of this question So So the ramp is a need to the minus s zero effect that is a Non-perturbative effect from the point of view of this two-dimensional gravity, but of course in the actually identification here You see the coupling constant here Lambda is just e to the minus s zero so in some sense I could translate this into a perturbative question and again if we compared this to the well-cheat computations that That I shall extend is teaching us about this would be sort of a perturbative expansion in the coupling constant The plateau on the other hand so so it's still from the 2d quantum gravity point of view It is still a non-perturbative effect. Okay, don't get me wrong, but the plateau is of the order e to the some phase Times e to the s zero Okay, and so so so okay, so people sometimes call this non-perturbative For the reasons that I just said then they would call this guy doubly Non-perturbative and to illustrate this just a little bit more if we wanted to to do a preview of what what is done in higher dimensions and one one Morally good way to think about this is that let's think about the microscopic theory of something like s y k S y k with n fermions has a Hilbert space dimension that is 2 to the n so the entropy is n So indeed this guy is an e to the minus n type non-perturbative effect So if you were to think about some analogy in n equals to four these are like e to the minus n type effects and those guys are e to the Phase times e to the n effects so again doubly non-perturbative effects from that point of view Okay, now Right, maybe I don't want to go on a little too much with that There is also something regarding this non-perturbative completion of matrix models, etc. But maybe maybe tomorrow Okay, but notice maybe the last thing I should say is that this is kind of of the form e to the One over GS although there is a phase here, so I'm not quite sure how to think of the phase From from your perspective, but it's like e to the s zero is one of a GS Okay GS with what okay, I called lambda here Very good. So So let me make some comments on higher dimensions Just just preview type comments So the story here is is two-dimensional and And it relied on the fact that there is a good e to the minus s expansion which really comes back to the fact that the Einstein Hilbert time gave us the Euler density gave us the Euler number and we could arrange this thing in a nice topological expansion that thing that structure will not be at our disposal in high dimensions And it will be difficult to have a precise direct analogy, but in this Current perspective this Delta which goes like e to the minus entropy generally Is such a parameter and in fact this flavor non-linear similar model Actually is an expansion if you remember an omega of a Delta, so it is an expansion in each of the Eat to eat to the well here. I'm calling it e to the s Okay, so organizing things in terms of the inverse level spacing is some sort of analogy to this Genus expansion which carries up to higher dimensions And Now the question in higher dimensions that is equally subtle is also What about these wormholes in high ID and Actually there, you know, some candidates have already been written down many years ago by Mother Senna and Maas and More recently many people have been thinking about it, but in particular also Kotler and Jensen have Suggested that there may be some solutions, which are not quite saddles But they have you have to put some some additional constraint and you might call them constrained Gravitational instantons, so that might be the right way to think about it and Finally the question that already has been As by Ashok is sort of this like factorization versus ensemble Which is even more of importance in higher dimensions and why I will Think talk about tomorrow But there are many ideas here again Coming from for example the Stanford group, but also by Andreas and Thomas Merton's and other collaborators, so the idea is You know indeed how do you recover Individual quantum systems from this kind of description and in high dimensions that is actually a more pressing question For a reasons that I will explain tomorrow Okay, so let me now talk about the plateau. So that's indeed the last thing I want to cover today So We've had we've so we've had some success now in understanding some of the ergodic physics of Of two-dimensional gravity and so we have already established level repulsion if you want but what what is an exciting prospect or an You know something that we should really want to do is to actually also Get to this plateau and as I said this plateau is particularly interesting because it's it's it's very much dependent on The discreteness so it's sort of if you get the blood plateau then you have Enough resolution in this theory to talk about individual microstates so let's Let's do it and so This I'm using the perspective indeed as I said earlier that comes from work with Outland with under Hayden and Eric Valinda and So What what we want to learn how to define so we want need to Learn to work these spectral determinants, right that that the one minus H that e2 minus H Well, yeah, okay. Yeah, any number we really Okay, that e And minus H and maybe it would have been good to give this is some prime One so that I only have to write any of them basically Is that the notation I chose? No Actually, I had okay. Sorry. I call these guys x1 up to xn and Then these guys y1 up to yn and we need to learn how to work with this in Jt so So the the approach that we chose to take is one which has been Referred to so people use the word of Universe field theory in a sense that you want some theory that allows you to create These boundaries and to create an arbitrary number of them and then calculate correlations of them now a Okay, this is this is kind of like what Sakura was saying earlier So if you think of this as a string world sheet theory then this this structure is actually a string field theory But if you say string field theory then most people leave the room so I refrain from saying it Let's call it more evocatively universe field theory, okay, so So what we need to introduce is basically we need a gadget that is We need something that is defined non-perturbatively and it can produce the Jt genus expansion and It produces So this is in some sense when I expand it around a certain vacuum expanded around or around a Certain standard saddle point, but it can also produce these e to the i alpha e to the s0 effects And in this case, I will expand around Some alternative saddle or perhaps what is the most sort of smoking gun? Smoking gun evidence or the smoking gun derivation is that I can actually use it to derive The non-linear flavor at the flavor non-linear sigma model Okay, so I can actually go back all the way from This theory which produces the Jt genus expansion and I can use it to derive Fully explicitly this flavor non-linear sigma model that I described My out of time Much better Yeah, so before going into this description Yeah, so you told us how to compute all these Partition functions which are exact but integrals on the geometries and one can list the geometries and Sum them so what prevents us to compute this plateau in the in the original formulation Yeah, so there is two answers to this so one is that first of all it's an asymptotic series and In some sense the plateau Is a non-perturbative effect in this asymptotic series? So you can't directly compute it But of course you can try and actually people have done this with success To do some kind of resurgence technique to this series So this is for example some work that Ricardo Skiapa and collaborators did moreover, however after We calculated this using this Very nice well, I think very nice non-perturbative structure there were some beautiful papers actually again and one of them Andreasin is involved in and another one is just by the Stanford group Which which basically? Do some I understand it anyway to be some resummation technique again on this expansion Working for a particular eigenvalue that is very close to the edge and actually extract again the plateau From from indeed knowledge of this expansion. So it is possible Yeah, I think that having like this sort of fully non-perturbative structure here in in particular the one that gives you The flavor nonlinear signal model and that also allows you to derive an action of this causal symmetry Then it allows you to see the full symmetry breaking pattern To mine to my mind anyway still is very nice to have because it really is a proof of this full quantum Ergidicity of the of the system in a very explicit very direct way But indeed there are Clever techniques where you don't need to go to this full universe field theory and you can still arrive at the plateau Yes, if you did use the language of string field theory presumably these are the instant on effects, right and in String filter you understand why you should add the instantons I mean here it looks like you are adding additional component of the wall sheet to boundaries, right and It's a little strange that you start with closed wall sheet and suddenly yeah, I asked you that question So yeah, are you going beyond what you originally started with that? It's not really jetty gravity, but jetty gravity with some Completion so in well what what another way of saying it is that you add different boundary conditions Yeah, and it's some of our all of those yes, yes, yes, you add different boundary conditions and again They are what is appropriate to study the theory in a micro canonical ensemble where you fix energies and don't fix the the length of the boundary circle, so I would say again from from the Quantum chaos type Mindset this is very natural to do because those are exactly the spectral probes that we've calculated Before and you only get them if you introduce those boundary conditions But they do define some brains on which universes can end Which you add to the theory I mean yes I think I would really be interested in discussing that with anyone of course, but it is true that if you're Somebody who really lives truly in the two-dimensional universe and that's the question that we need to ask when we go to higher D Is there some absolutely completely intrinsic justification for adding these additional boundary conditions and I I don't actually know the answer to it But I think it is a very interesting question Okay, so right, I have I guess another so 15 minutes or so so Yeah, I probably won't completely finish, but almost and so we there's still enough time tomorrow. I think and to talk about higher dimensions I want to do a bit more than just talk about high dimensions tomorrow because I also want to sort of wrap up things In a way that is very suggestive really that quantum chaos plays a very important role even in sort of the architecture of space time But I think it can be yeah, we can we can do this okay, so I'm Very good. So I Am only going to give a sketch So it starts by recalling that the spectral density of JT we didn't actually compute it, but It follows from the disc one-point function This is actually the famous e to the s zero times the hyperbolic sine of 2 pi times the square root of the energy and we take this as an input and And we use it to define a spectral curve, which I call s JT So now this s is the spectral curve and that is an auxiliary equation, which I call H of x and y Such that I take y squared minus We have some taxes of pi. Okay, who cares really but anyway one of a 4 pi squared times sine squared of 2 pi squared of x and X is sort of like minus e So that's why this became a sine squared and so Okay, I am going to sort of say a bunch of words which are useful to those who know them and maybe also to those who Want to look it up? They give you a little bit more of the background Where this machinery comes from but I could actually just write down now an action for you This is the theory and then argue that it does what I want But so the one thing that I wanted to say is so actually this defines a Calabi or three-fold By an equation, which is actually you have to have two more complex erections u v minus h of x y is equal to zero and The theory That one actually defines is a string theory whose target space is that Calabi are And this string theory for us Basically is a theory of two-dimensional universes Which are described by JT gravity in two dimensions? Okay, so technically speaking so this has to do with with The the the target space of a Calabi are three-fold of a topological B model string theory But what we use is a slightly simpler version of this we call it KS theory because they the theory here is called Kodaro Spencer theory and This is actually just the dimensional reduction of actual Kodaro Spencer theory which lives on this Calabi out and to the spectral curve of Jt and you see the spectral curve of JT is embedded in this Calabi out via this equation But as I said I can just write down the action so the action is actually oh and I Think names associated with this are like check out the oguri Waffa Bershatsky That kind of that kind of Developments and those kind of authors so it's a Define Ej of E to the minus the action of this Kodaro Spencer theory or KS KS theory plus One will add some sources now So where I KS? Right, this is a two-dimensional theory because I've reduced it to this spectral curve of JT has this What will turn out to be a twisted carol boson? Phi It has some Bay wedge d phi Plus it has a cubic interaction and actually Very intuitively this cubic interaction is basically the pair of pants which allows you to construct all these Riemann surfaces So and phi is a twisted Carol boson on SJT J is some Current that I add on shell they are identified by the equation of motion J is d phi and In this cubic interaction depends on on the coupling constant Lambda Equal e to the minus s zero Yeah So is this the string field theory whose wall sheet theory is JT? Is that the claim? Yes That's exactly that's exactly the claim But we're not saying it because then people leave the room So What is important for us is that there's a translation to perturbative JT gravity in particular We can write everything that we wrote before which is the Dirichlet version of the JT partition function Which depends on beta one to beta n We can write this as some transformation of the N point correlation function of this J field in this KS theory Where the measure is just e to the minus beta I? Zi squared for each of the insertions and DZ I over 2 pi I and I have to choose some contour for this inverse Laplace transform So the the claim is And this this is this can be shown of course that So studying the correlation functions of this J in this Kudaira Spencer theory Precisely gives you all of this the exact answer that I had here before I erased it for the Canonical partition function of Z JT or the connected contribution of the N partition function Correlation, okay, so however Yeah, I can go a little bit But it allows more so so this is basically sort of we touch base with what we have done before But now what can we do? There actually exists operators in this theory Which we have argued Well, which are These determinants so these operators Basically, you should think of the twisted carol boson d phi itself as sort of the spectral resolvent And so you integrate it e to the phi is the insertion of the determinant so but also allows to compute that type Operators and in particular So Particular for us Something like that x minus h is actually just the vertex operator e to the phi and the inverse determinant that Why minus h is e to the minus five Okay, again As you probably appreciate of course for the You know sake of time I Don't have the time to Derive this completely. I will give our references and also those of other authors that are relevant on the on the web page There is obviously, you know some development to make these arguments. I'm just sort of stating it as a fact Okay, the It's it's a combination of you know defining this theory working with its itself non-perturbatively and recovering The appropriate JT answers in the appropriate perturbative limits So for example here, you might want to know trying to recover Precisely computations where you specify Neumann boundary conditions of someone on the boundaries and again It is possible So these are like The brain vertex operators. Yes, this x and y are the same as those No, sorry, that's actually kind of bad notation So I should have called them maybe Z1 and Z2 and This depends on Z1 and this depends on Z2 And you see because the spectral curve here This is like an equation In x and y and I will solve it for x for example And then the spectral curve will be a double double sheeted cover of the complex plane And that will be the target space of this theory and so there then Z1 and Z2 will the refer to coordinates on that spectral curve Okay, and this dead prime is what? Oh, no, no, it's that to the power minus one ball so now with this technology and One can show the following and this is the key equation The key equation is that the correlation function of e to the phi Of x1 e to the minus phi Of y1 and there has to be some normal ordering procedure e to the phi of xn e to the minus phi of Yn in this case theory Can be written exactly by exact manipulations So we can call this maybe by definition e of x This is one of these generating functionals where the excess now again is the graded vector That is made by the x1s up to the xn's in the fermions and the yn's y1s to the yn's in the bosons and this guy can be written exactly as a integral over a graded n slash n matrix dA Times e to the minus e to the s0 gamma of a and plus e to the s0 times the super trace of x times a where gamma of a Well, take some form for JT gravity is the JT potential if you want we can call this the JT flavor potential I don't want to write it here. It just has some inverse trigonometric functions and so on. It's a it's a relatively It's a relatively simple expression But it's an explicit expression and what is what is interesting to note For for those who who know this it behaves like a cubed for small a which is precisely the way that the edge physics of JT or SYK or if you want some sort of generalized airy model behaves so it behaves Exactly the way you want and there is some very explicit form of this but maybe one more comment that can be made is that for example in the color space language So the original way that SSS defined the JT matrix model It's actually very awkward to write the potential they never explicitly wrote a potential But for good reason because it's very awkward in the flavor language You can just write the potential and it's not too bad of an expression And and so now okay, we're just one step away. So now this is one one Step before we go to the flavor non-linear signal model. So maybe just to As the final expression Let me write that this now you can analyze and it breaks the causal symmetry Okay, so this leads to causal symmetry breaking and In fact, it gives a It gives a Sigma model which is of the type a 3 E to the and here I'm going to write down The action e to the minus i Row of e so that the row of e that I wrote there in the beginning times the super trace of x times a And and the target space is simply n slash n is the generalization I mean so what what I've done here also I've taken the liberty to generalize to any insertions of determinants and even let's say And we we just get the fully general the general guy. So let's say n is even Okay, so this is indeed now the Flavor non-linear Sigma model For two-dimensional quantum gravity and for any part of the spectrum So if you want I'm of course, you know with with quite a bit of machinery That I didn't have the time to go into detail, but this is a proof That two-dimensional quantum gravity is fully hard quantum chaotic has a fully ergodic phase And that we can rewrite The full non-perturbative completion of JT in terms of this This this proto Sigma model and then reduce it to the a supersymmetric Sigma model for a particular symmetry class So in view of time I lent here I'll have some more comments on this tomorrow and then we'll talk about higher dimensions But I've already taken up enough of your time. Thank you So can this a be thought of as a open strings living on the instantons this matrix a What is your integrating at the end? and So I I'm not sure about the the instant on the answer is that that's maybe true But something we should discuss in detail, but what I can tell you is that there is this This Calabi out and on this Calabi out I can have compact and non-compact brains and Actually, the different boundary conditions can be can be understood as open strings Like open JT universes that end on one or the other of these brains. I mean there are many of them But one or the other types of these brains So the best analogy that I have actually is one to Leoville Where the non-compact brains would be like FZZT brains and the compact brains are like ZZ brains and you can have mixed boundary conditions and in fact, this is something which I I found quite fun is that I Wrote very early on. I wrote this psi a mu type thing In this framework, those are just chan pattern factors Where the open string adds ends on one or the other brain? And so the compact brains the ZZ brains would be the color space brains and the non-compact brains The FZZT brains would be the flavor type brains and the non-linear flavor sigma model if you want is an effective action that we've That we've that we've derived for the non-compact brains I said, but if there are no non-compact brains, there should be a Spatial direction on which the matrices depend so these are some kind of path integrals Yeah, they're wrapped on some cycle in this Calabi out Okay, so then they're like the instantons right I mean I have that the I mean I'm happy to they don't have extension in any Up the non-compact directions. Yeah, I guess. Yeah, okay, so then I'd call them the instantons So I wanted to ask about this So this you said this is a non-perturbative definition of JT is it unique I'm I don't think so But I haven't completely I haven't completely or we haven't completely resolved that issue Yeah, I don't think it is. I unfortunately don't think it is unique But like within your formalism, I think that you cannot see that so I think that The different kind of completions that you could have they actually in at some point in derivation Translate to different choices of contours and what is true is that from from if I just use my sort of chaos goggles There is a particular contour that I need to choose and that does give you a unique completion But again intrinsically from the JT point of view. I don't think there is uniqueness Okay, so if there are no other urgent Is this true is it true that this club your has infinite dimensional homology and It could be true. I think you know, it looks like the cure for the infinite genus sure this side It does have infinite genus because because of this because of the pulls of this Zeroes of this sign so it does indeed have infinite genus. So yeah, yeah, I guess you're right. Yeah And this is somehow okay It is somehow okay Okay, so before the discussion which we start in 10 minutes we can take maybe a 10 minutes break Don't be too over time to come back So just 10 minutes to stretch your leg and then you will have the chance to ask questions to the three lecturers of today Okay, let's thank again Julian