 Good great It's great to be here at this fun conference. I unfortunately don't have any Dutch proverbs to sprinkle in I Was thinking about one or two to I couldn't really think of any I guess We do have some jokes about Romanians, but I will probably Get sued if I tell them to use so that's not So this is going to be based on some work in progress with Michal Heller and Natasha Pinzani Volkeva and this is a brief story about fluid gravity duality So the outline is roughly as follows. I will briefly say something about Fluids are hydrodynamics Sort of the standard way then I will Say something about the gravity dual of that which will also be very well everything will be short Then the third thing I want to say something about effective actions for fluids Which is kind of if you want an old approach and then the gravity version of that and then I want to say something about Some new approaches to effective actions for fluids and the gravity dual of that So this was inspired in part because during the last two years or so. There was quite a bit of progress in in developing a new machinery to write down effective actions for fluids for hydrodynamics mostly by Hong Liu and company and also by Mukun Taranjimani and company But this is all done purely sort of at the field theory level But the structures they get are kind of interesting and it's an interesting question to what extent the developments that they found Can be seen at the level of gravity, so I'll try to explain all of this a little bit So first of all fluids, so normally we describe fluids in terms of conserved currents so the standard equation in the simplest case that we write for a fluid is the conservation of the energy momentum tensor and if you Parametrize the energy momentum tensor in a suitable way in terms of the inverse temperature and some unit velocity field and We write this as a perfect fluid Plus higher order where higher order is derivatives of these fundamental objects And you plug that in here and you get a closed system of equations for this velocity field and the temperature And that's the standard approach to hydrodynamics This I guess a Landl diff sheets and you can find this in many places And in particular here, there's all kinds of coefficients appearing the transport coefficients of the theory there's viscosity appearing etc etc One interesting feature in general so this is a bit of if you want to find a logical approach because you write down the stress energy tensor You write down the most general stress energy tensor that you can write down. It's compatible with the symmetries of the problem And then the other thing you insist is the existence of of an entropy current Whose diver it's not a conserved current, but its divergence is non-negative and this guarantees that there will be only entropy production And you can by imposing the existence of such an entropy current You can find all kinds of bounds on the various coefficients that appear in this gradient expansion of the energy momentum tensor Not everything that you write down here is compatible with this extra assumption. This is a very physical thing to impose Any questions about that? No, as you at obviously at the leading order, we know exactly what s is But then you need to systematically correct it also in the gradient expansion and then you have to just find one that obeys this condition basically, right? Is there any function of beta and u? Yeah, at this that those are the fundamental variables here, right Obviously here at leading order There's there's a pressure and a density appearing and those are just but the assumption is that you know the equation of states So these are known functions of beta. This has a counterpart this whole structure as a counterpart in gravity, which is quite Simple and can be made very precise in ADS CFT because an ADS CFT what you do is you take a black hole a rather a Black brain we typically do this in a case where the boundary is flat so that we have translational invariance So the black hole gets replaced by a black brain. Here's the boundary of ADS and then the idea is to this is basically The equilibrium configuration so this configuration corresponds with the first term here This just describes a perfect fluid and equilibrium And then the idea is to put small perturbations on this background Small perturbations and you do it in a particular clever way such that the small perturbations are basically Parameterized by the same variables that appear here And you write down a suitable ansatz for the metric Which is now also a function of these variables and then if you compute the Einstein equations One precisely recovers those equations here in other words There's a one-to-one correspondence between the if you wanted the solutions of the Einstein equations organized in a very particular way So I'm not going to describe in detail how this works But there's a one-to-one correspondence between Einstein's equations applied to a class of solutions that describe a perturbation of a horizon versus standard fluid dynamics and In this language this this entropy here the the zero Component of this as mu here is simply going to be the area of the black hole horizon and the fact that it is non-negative follows from standard gr here And it's important to do all these things with in going boundary conditions Because that's how you guarantee that things fall fall in and the entropy increases here So this is how fluid gravity works and in this particular case since we use a very particular system namely in this case a CFT, so it's one particular gravitational theory So if you do this you do not get arbitrarily three parameters all the parameters are fixed in Particular all the transport coefficients like viscosity and some that that are three parameters here They're fixed in this particular case. They just they are the transport coefficients of this particular fluid that lives here, which is a conformal fluid in particular to leading order one finds The stress tensor of a conformal perfect fluid. It's a scale and variant fluid One also famously finds from this computation That the ratio of the shear viscosity over entropy density is one over four pi But in principle you could do this order by order and compute All these are the transport coefficients as well based on this gravitational solution here Here yeah, yeah, so the entropy here is the size of the horizon area as you expect from If you go to non-stationary coordinate you can see that this is a quadri-vector If you go to general calling external test city to stationarity of the black hole where the area is not Yeah, determine from this Time the stationary time. Can you make sense of the quadri-vector or the covariant object for the Sorry the correlator you mean the vector of the quadri-vector Makes sense to talk about the area in a in a covariant way when you don't know No, I don't think you can completely covariantize it because the whole point is that you do small It's ordered by order perturbations around a particular given background You could in principle try to covariantize the whole set them, but that's kind of complicated because This is not to manifestly as it stands here You also assume that you have small perturbations around a given fixed background And the same is true here So I'm not sure that there is a completely covariant way to organize this somehow But what I will say later might be close to that Oh, yeah, if you do a rigid boost you just If you have this thing in equilibrium and just being stationary then obviously you mu is just one zero zero But then you can do a rigid boost and turn on new mu. That's just straight forward That's actually how you make these solutions you start with the standard one You do a rigid boost and then you make the boost parameter depends on space and time and so on the point you already have a vector Oh, yeah, yeah, that's true. Sure. Yeah Not too leading order this thing is just proper is just Sure, that's the leading order p That's right. Yeah, sorry. Yeah, I should maybe give these Different labels this you mu is is a four-dimensional if you want and then this would be a five-dimensional metric The Einstein equations agree with this field equation Yes With a negative cosmological constant because we're doing a just CFT You can in principle try to play this game in all the types of backgrounds as well But this is the cleanest situation because we have this nice field theory dual living on the boundary Now there are several reasons why instead of writing down these equations. You might want to try to write down an effective action Yeah, an effective action in the number of dimensions where the fluid lives, so say three plus one. Yes Because for example, we know that in general in hydrodynamics there can be both statistical and quantum fluctuations That might be important. You might be interested to know correlations between different points in the fluids and that's the sort of data which is basically inaccessible at this sort of Phenomological level with and which might be accessible if you have a proper effective action Now the first idea is to write down an effective action for fluids are based on the following variables So the first question is if I want to write down an effective action. What variables should I use? so the original idea Which may go Back to the 50s or so. I'm not quite sure is to use as your effective variables a bunch of displacements fields Phi I of T and X Which basically tells you what happens to a particular little volume of fluid as it moves through time You basically track these little fluid elements So at T equals zero phi i is just going to be equal to X i so phi i of X i tracks The fluid element that a T equals zero was at X i and then you follow what happens as you move through time And this could be a nice way to sort of parameterize the fluid But then it's important if you want to write down an effective action for these scalar fields is to know What the symmetries of the problem are The idea is that this thing has the following symmetries It's invariant under volume preserving different morphisms of the phi i's That's the statement that The only thing that really matters of this little piece of fluid is its local volume But actually how you exactly parameterize the volume is not that important and then the idea is to try to write down an effective action For these fields phi i that has this particular symmetry. Well, that's Not that difficult but also net not that straightforward. So for example one can make the following A global symmetry I'm not going to gauge it No, that's not what this means So yeah, so I'm not saying that this volume cannot change as you move through time But I'm just saying that the way I parameterize the local volume on your case about the volume and not about the rest It's a symmetry of the fields, but there's no local parameter Xi is a function of the depends a bit on what you call local It's an infinite number of symmetries, but there's no arbitrary explicit function of x and t here So it's like a field redefinition symmetry. It's like if you write down a sigma model you have different morphism invariance in target space I think we usually call that a global symmetry, but it's clearly an infinite dimensional symmetry It's a bit like that. So if you write down the following object Which involves derivatives of the phi i's and You take the determinant of mu nu and just to take the squared for simplicity if you built a following object For example, if you now this is the sort of things that might be familiar from membrane effective actions Because famously actions from membranes enjoy some notion of volume preserving different phism invariance as well Then this clearly has this symmetry Because if you make the change of variables because of this determinant that you pick up the determinant and then the one drops out And I can principle write down an action that depends only on this s This is an example of an action that would be allowed It's also Presumably is the sort of action that is lowest order in derivatives that you can write down Which is still compatible with the symmetries of the problem And then you could try to add higher order pieces Just some metric so the idea here is to couple it to a metric And then to get the energy momentum tensor by varying with respect to the metric And then the energy momentum tensor that you get you can compare to this particular energy momentum tensor here It doesn't well There's not this is the world so this is a d dx. This is a spacetime I'm probably not saying this right so I should probably put a gi j here. Yeah, yeah No, right the determinant is fine. Yeah, so this is that's right So the s is a density. Yeah, so you just integrate it and you're good Yeah, so that's why you need probably to put in It's probably something if you want to do it. It's probably something like this to get the right dimensions, right? So if you take this action and you compute the energy momentum tensor you precisely Reproduce this perfect fluid piece here and the interpretation of this function f is that f is the free energy and That is s is the is the entropy So if if you take an arbitrary fluid with an arbitrary equation of state P as a function of row in particularly you can convert it into a free energy as a function of entropy density That you plug in here, and then you have an action for the perfect fluid Now that's nice, but this is just the perfect fluid now You would like to see that when you add higher order corrections You are also able to reproduce some of these higher order corrections here If that would work then this would be a nice way to do effective actions for fluids But if you add higher order then the first thing that you see is that You don't get dissipation This is just an ordinary effective action. It's a real action. It doesn't have dissipation So at most it can describe dissipation less fluid And the second thing that if you work through the details that happens is that you get fewer terms Then you can get here even if you demand there's no dissipation So if you demand there is no dissipation you put an equality here So there's an exact conserved entropy that restricts some of the terms that can appear here But there's still many free parameters here, and even those cannot be all obtained in this way So this this particular approach is useful, but it's very incomplete You can get dissipation in this particular theory, but then you have to couple this theory by hand to some explicit dissipative sector As it stands it doesn't have dissipation Now you can ask is how do you get this from gravity this particular action? What's the meaning of the fields phi i that appear? so we looked into that two years ago or so and The way to get this from gravity is as follows the idea what is the sort of computations that you typically Typically doing gravity and this this is very much inspired by an earlier paper by a nickel damn song It turns out that the right way to get these types of actions from gravity is to set up a double-directly problem Which means that we specify some metric here on this slice, so this is the radial direction of ADS This is the horizon. This is the boundary of ADS and the idea is to take two fixed slices at fixed radius To put a metric on one and a metric on the other And to take this region in between and to try to compute an effective action for this setup by finding the bulk solution whose boundary values are g and little g and Evaluating like we always do in ADS DFT you evaluate the on-shell value of the effective action You could think of it for the time being it could be anywhere But ultimately I do want to take it to the horizon, but that's a bit of a singular limit Because in that limited metric diverges and so on In these coordinates, so you have to be careful to do that in the right way So first I'm going to be safely away from the horizon could call it a stress horizon if you want So there will be some effective action But that effective action will be very non-local Because this metric and this metric they they live on two independent spatial slices There's a priori no relation between the coordinates on one slice and the coordinates on the other slice These are independent sets of coordinates and moreover since we are computing an effective action in gravity It's supposed to have a Diffumophism invariance for that metric which just means I choose different coordinates and it is supposed to have a separate Diffumophism invariance for the second metric and There's no local effective action that you can write down that there's these two different morphism invariance Where these things talk to each other and the fact that you get a non-local effective action That's usually a sign that we sort of integrated out light degrees of freedom And if you would have kept the right light degrees of freedom in the problem, you would have gotten a local effective action So what are the light degrees of freedom that you need and that appear in this problem? That you need in order to make this into a local effective action the light degrees of freedom that you need are scalar fields That are a map From one side to the other side There's different ways in which you can argue that these are the light degrees of freedom you can a rough way to argue it is that up here you have these different Translational symmetries of the problem, but by choosing this bulk solution You basically break the symmetries to the diagonal subgroup Definitely need to be a bunch of goldstone bosons and you can interpret these scalars if you want as the goldstone bosons of that symmetry breaking And you would like to construct these in a covariant fashion So for example, you can construct them But a different covariant constructs will be related by field three definitions But an example is for example to just say that you take a Geodesic here which leads in an orthogonal direction to this boundary and you just follow the spatial geodesic until it hits the other side That's a coordinate independent way to define a map from this slice to that slice So if you do this problem in gravity, then there's going to be a local effective action That also involves these objects in this way and you can just compute it explicitly in agency of T what is this too lowest order and Because we have these two different of visions in two different more viscine invariances that appear the basic building block that will appear If you want to write everything in terms of the mu nu variables So sort of in terms of UV variables And what you can do with this map is you can pull back this metric to that side in this way And that gives us a second metric here that you could call h mu nu And if you do this is different morphism invariant in the MN capital letters But it's not and it transforms like a symmetric rank to tensor under little mu little nu So basically what you get is an effective action now Which looks like it's an effective action for two metrics and it's quite easy to write down such effective actions You don't you have even have to use derivatives You for example can just take alternating products of g's and h and take the trace and This is a perfect example of something that you can write down Which is clearly different morphism invariant now and they're both different morphisms on the left brain and different morphisms on the right brain you can even Decide to call this guy m of mu lambda and Then the effective action to lowest order will be a function of the trace of m the trace of m squared Etc. That's that's everything that you can write down. This is lowest order in derivatives Which is compatible with the symmetries of the problem and it's something you can compute explicitly by Literally doing this computation in agency of t you put a metric here metric here you take these fields you plug it all in now this you can compute and Now if you take this effective action and You send the IR To the horizon That you might worry that that's a bit of a singular limit because the metric behaves a bit in a peculiar way If you take this limit, but this action has a finite limit and then you can take the other boundary to the boundary of ADS That's also something you can do easily Then the effective action blows up, but as usual in agency of t we subtract it There's a local counter term you can write down and you're still left with something finite and then you really compute an effective action for this entire piece of ADS between the boundary and the horizon and you would believe based on this This cartoon of fluid gravity here that you'd somehow capture all of this and Indeed it turns out that this effective action that you get from here by taking this limit Is exactly identical to this effective action here In the case where this is a conformal fluid Because obviously we're doing agency of t. So we're describing a conformal fluid So this is how you derive this effective action from agency of t Yeah, how does this action know about boundary conditions? In the UV You mean this one here? No, the one you were constructing Oh, that's if you do this limit You take this boundary condition little g and you scale it as a function of some radius exactly as a radial slice of ADS would scale No, no, it doesn't constrain it just a particular limit that that you take Which is imposing the ADS boundary conditions if you want but it doesn't constrain the action by itself at all The action by itself was was the lowest order You could probably compute it for other signs of the cosmological constant to 12 It was a generic answer felt it for any slice in in Einstein gravity with a negative cosmological constant No reference to ADS or anything You do make a few implicit assumptions because this is the lowest order effective action So you do Have in mind that it's a small perturbation around something because in general if you set up this problem This might not be a good problem. This is your desicc might cross there might be caustic I don't know what but as long as you do it order by order around to give an okay background that you'd be fine You could write down a separate piece that only involves the infrared metric But that will not give rise that just that first all that's not as effective action If you compute that it doesn't appear. It's by computation. It does not appear It could it does appear for example here if you want need to subtract a counter term if you go to the boundary So as a counter term it could appear but typically these sort of standalone pieces They they haven't they make no reference to this interior here So they would be the same regardless of the metric that you put here And also so you could take the limit where this metric is identical to that one And in that limit you would like the action to vanish because there's nothing left And that those sorts of arguments very much constrain those sorts of one-sided local pieces So these two metrics are static or they can depend on time They can depend on time, but this is just the lowest order part of the effective action So to compute that you can already derive this lowest order piece by taking a time independent metrics That's that's enough to fix this form of the action But then by symmetry principles it must be valid for general metrics at this order In the derivative expansion now this again by construction because you fix the horizon metric this describes a system where there's no dissipation It's even kind of difficult to extend is to higher orders Especially if you want to take this limit of this IR brain going to the horizon That's kind of a complicated thing to do Which can include dissipation by imagining that so ultimately we want to couple this to this Black brain here. So perhaps the full thing that we should include Maybe the full action that we should have written down is this effective action Plus some sort of black hole or infrared effective action Whatever you want to call it, which is just a function of G and Which describes all the horizon physics of the black hole dynamics So this might be if you wanted to include dissipation the thing that you want to do where somehow this this knows about the black hole dynamics and Then what you would like to do in this particular case is to integrate out G or to go to a saddle point This might give you a dissipative effective action for a little G Because now I've properly coupled the whole thing to an actual dynamical black hole If you include a dynamical black hole then we should be back roughly in this original fluid gravity setting And you would imagine that then we are in a system where there is in fact dissipation Yeah You could also imagine trying to do this at a stretched horizon and but at the end it Since you match two pieces together, it shouldn't really matter where you do it You can do And in fact, that's a good point because if you do this computation what you find is that this This action doesn't really make sense That doesn't exist as a simple action which has to do with the fact that in the physical system you want to impose in going boundary conditions So this does this action doesn't really exist But the field equations that you get from this do make sense And if you work out what this is these field equations precisely impose in going boundary conditions But you cannot this no longer works really at the level of the effective action So now you can only do this at the level of field equations And if you do this at the level of field equations, then you do reproduce this dissipative hydrodynamics As you should because we already knew that if you couple gravity to in going boundary conditions We reproduce that's the basic starting point fluid gravity And that it reproduces the standard dissipative stuff. It's also fun to do this in Euclidean signature as a side remark if You do this in Euclidean signature, then the horizon gets replaced by the tip of a cigar And then you might think that this infrared effective action is nothing because it's just going to be the contribution from the tip of the cigar However, if you write down effective actions in gravity with a boundary There's also this Gibbons-Hawking boundary term that you need to be careful about and although if you take the limit Where you go to the tip of the cigar the Bullock action vanishes the boundary action does not And if you work out what the effective action in the Euclidean cases What you get is not this but there is an extra little piece and Then integrating out is G what it does for you. It's a precisely Lejean that transforms F and it replaces the free energy by the pressure Wait which boundary and the tip of the cigar is no boundary Now that you have to so you put this slice here Well, this is some distance epsilon you do the computation and then you send epsilon to zero You cannot first go all the way. This is the way we do it We have this non degenerate metric G then you couple it to a very little tip you add it to You integrate out is G and then you take epsilon to zero Both have a boundary term which cancel obviously Now this this directly in in order to have a well-defined directly problem in gravity you need this extra boundary term Otherwise the directly problem is not a good problem in gravity. So you need to have this extra term This extra term was included in the effective action that I wrote down But if you couple it to this little tip of the cigar you also need to include it here and that makes this thing non-zero even in the epsilon to zero limit and That basically Performs a legion the transform of this F and replaces it by the pressure And that's nice because we know that the Euclidean partition function of the system should not be the free energy It should be the pressure. This is how that comes about and Finally a nice application of these types of actions is also That these in going boundary conditions. They're extremely simple boundary conditions on this effective action So in some sense what one does is We have complicated dissipative transport in the UV and Now we started this entanglement into complicated non-dissipative transport from the UV to the IR and very simple dissipative transport in the IR This is very much in the spirit of semi holography where you have a very simple infrared sector which you then couple to Standard propagation between the IR and the UV and then you get something complicated in the UV and the same is true here so dissipative UV is And this is complicated, but non-dissipative and this is simple and This is also complicated. So this is very much in the spirit of something called semi holography You started to write down an action for the dissipative part, but then you tell us If you if you want to include dissipation Yeah, then I can only write down equations because it's not possible to write down an effective action which imposes in going boundary conditions If you're just on one side, which brings me to the next point So this is all the the obstruction to write down effective actions here has to do with the fact basically That we're looking at the one-sided black hole But it turns out you can do much better if you go to the two-sided case which in field theory means that you use a Calde-Schwinger formalism and Once you double everything which is also what you do in standard finite temperature field theory Which in and from a gravity point of view means going from the one-sided black hole to the two-sided eternal black hole Then it turns out you can do much better. So in field theory The idea is as follows There's some initial time which is roughly minus infinity and there's a thermal density matrix here and Then we evolve it with the insertion of some sources here And this is basically a Calde-Schwinger contour. So one way to write all of this is as the trace of u2 dagger a2 Ro not u1 a1 Where you want a1 will be e to the i the integral a1 times some operator o1 And similarly for a2 So this is a standard way in a standard approach to finite temperature field theory Where you have to basically double space time You are on the two pieces of this Calde-Schwinger contour and by taking suitable linear combinations of operator insertions here and here You can compute retarded advanced etc etc correlators in the theory and again here, so this Just is a sketch this describes the coupling of a source to operators and again you could imagine computing a partition function But now it's a partition function of two sources And that sort of partition function in principle contains all this information Again, if you do this computation in a finite temperature system, this this thing will in general turn out to be non-local You need extra degrees of freedom to write down an effective a low energy effective field theory in this two-sided case as well And what you need is basically two times the set of goldstone bosons That I had here. This was a single set, but here it turns out you need twice as many It's not so easy to prove that rigorously starting from an actual path integral writing down this Calde-Schwinger path integral And trying to argue that really these are somehow effective light degrees of freedom. That's not easy at all But there are other ways you can argue that these are the relevant in the finite temperature case These are the relevant light degrees of freedom that you need to keep in order to describe the suitable low energy effective field theory Now in principle what you would like to then write down is this effective action But there is you need some guiding principles to write down this effective action So there's a whole series of things and because of time I don't have time to say too much about it, but one thing that you can see from this particular way to do the computation Is that if you bring this u2 dagger around in the trace Then you see that if a1 is equal to a2 that the partition function is basically equal to 1 That this is equal to 1 and another thing that you can instead of bringing this around you can also take this one and move it through to the right and then you see that if a1 t is equal to a2t minus i beta because this finite temperature density matrix it basically means that you shift imaginary time by the inverse temperature That in this case z is also equal to 1 so these are two features that this effective action will have and then so one idea was to try to Guarantee these in an effective action in some way using some symmetry principle now if you want that the Certain things in your theory decouple and one way to guarantee is to argue that the the operators In this particular case it's o1 plus o2 Because if you if you just add these two things then this is the structure that you get So you see that you want that correlation functions that only involve the sum of o1 plus o2 that they should all vanish If you write it in this way Now you can postulate that the reason that correlation functions that involve only o1 plus o2 vanishes because there is some BRST Operator and this is the BRST of something No, no, no, I just took to the sum of the difference of the two and rewrote it in a trivial way With up to a factor of two or something So it just means that the fact that if these are equal it's sorry I'm saying it in the wrong if this is equal It should vanish so these correlation functions should vanish So suppose that we would have a theory that has some BRST symmetries and suppose that this operator would be BRST exact Then correlation functions of this operator would vanish identically Not all correlation functions need to vanish Because for example if this operator is not BRST closed then we cannot say anything about correlation functions of this operator And similarly this other combination Could be the Q bar of something which is Q bar is a different BRST operator and if you put this all together Then we get some then you get to a proposal to write down an effective action for this whole system in a suitable with lots of BRST symmetry It looks like a Bit like what is known as a so-called aquavariant topological field theory Strangely enough it just comes it just basically postulated by observing that one has these Symmetries of the problem that certain operators have correlation functions that vanish and then you can try to sort of engineer that using a BRST symmetry and using These BRST symmetries and setting up a large formalism where you also then need fermionic partners of all these operators You can write down interesting effective actions, but it's not yet enough to fully constrain the system and Then basically once you then you need to introduce further ingredients and these have still not been derived At like a really fundamental level because in the work of Ranjiamani at all They need another symmetry is some mysterious u1 sub t symmetry. So they Whose microscopic origin remains obscure, but there is there seems to be some sort of extra u1 symmetry in the game That they invoke and that seems reasonable, but it's again hard to argue exactly where it comes from And you and friends they use roughly some sort of z2 symmetry and that's z2 symmetry of Hongyu and company Is a bit better to understand where that comes from it has to do with the fact that here You also have the KMS conditions that are that are in post on correlation functions And the z2 symmetry that you at someone use is basically descents from the KMS condition of the original thermal field theory Either way you can then write down effective actions You can reproduce Disapative hydro now these effective actions will in general be complex And you can prove the second law based on all these things that I just said so it's interesting to see whether This entire formalism all these BRST symmetries And also these extra symmetries whether we can understand where all of this comes from somehow in gravity What this all means is there a gravitational interpretation of This particular formalism to write out effective actions for finite temperature field theory As far as I understand, so that's supposed to take some interacting theory and then you do this doubling formalism Yes Is it clear that you can write back in some sort of BRST for an interaction theory? So the fact that these correlation functions vanished is an exact statement Yeah, but I mean the original action is fixed. Sure. Yeah. Yeah There's no yeah, so there's no a priori reason That this BRST formalism should work, I think you could imagine this is one way to get in The question is whether any action that has these particular symmetries that these things decouple whether any action that has these features Always admits a formulation in this BRST language. That's the question that you I guess would like to answer Yeah, but I do not know I do not know what the original action is So the idea here is as as as usual is that I have a UV theory that is I have maybe a Lagrangian I want to write down the low-energy factor field theory. It's probably very difficult to do the loops and everything by hand So what you want to do is to use symmetries and what have you not as much as possible to constrain the form of the low-energy Effective action. So here we are agnostic about the UV physics We just want to constrain as much as possible the form of the low-energy effective action and then you can ask whether Any theory that has these properties always admits this formalism or not No, no here it's this this BRST symmetry the structure is proposed as a way to incorporate these particular features of the theory I Think that you might try to prove It's an interesting problem to prove that any theory that has these symmetries can always be represented in such variables But I do not know if you can prove anything like that Certainly at the quantum level that that might be a bit of a challenge That's another reason why it's interesting to see whether we can see a hint of all this structure in say a g of t Because that might give some more physical insight into where all this stuff comes from The first one, yeah, not the second I don't think that by that by itself doesn't buy you too much I think No, but the second equation is Characteristic with that. Yeah. Yeah. Yeah, but the first one is generic. Yeah That's right. Can you find this cue for any You could try I don't know if it buys you anything there might be some some tautological way in which you can always take any low-energy factor field theory and stick and some BRST and so on so that you Don't essentially change the theory. I'm not sure Do you relate to No, no, that's not here. No, that's a separate ingredient that you need to impose There's so there's more here So there's more symmetries here with so ultimately one uses these goldstone bosons and this volume preserving different phism inferences indeed also used in the game No, well, there's different level you can also gauge fix that Are not depends. There's different formalisms depending so one of the reasons why it's quite difficult to compare these different series of papers is That one series uses and gates fixed versions and uses this you want to enjoy the uses partially gates fixings and the z2 and so on and That makes it very difficult to see how these are connected They keep more symmetries. This is more gates fixed Yeah, good So the conclusion The conclusion is the following Suppose you want to do this in gravity It will be a little bit of a Romanian conclusion, but that's okay. I guess so So if you have the eternal two-sided black hole Written in this particular way then we can try to repeat the thing that we did before on the in the one-sided case And try to set up this kind of directly problem But now there's four slices there's an a G left a G left a G right and G right Moreover, we want to actually engineer a bullock version of this particular Calde Schwinger computation and the way you do that is So here this state is inserted at some initial time We can take the time to be zero. It does not matter And then you want to have this thermal thermal state you can do that by gluing a Euclidean geometry to this Laurentian geometry It's it's not very often done in AGCFT to take the boundary to have a mixed signature of Laurentian and Euclidean But in principle you can do it And then the game in town is to find a bulk solution that also has a mixed signature Laurentian and Euclidean And you just need to make sure that when the Laurentian and Euclidean connect that there is some that Fields behave in the right way. This should not be like a singular energy located at the interface or something like that so this formalism was developed in some detail for example by scan there is an bolt-on race So this you can do and now we need to extend this whole problem in the following way We now sort of get some Euclidean boundary data. Let me call this G E And G E here and now again we can try to Analyze this particular system. So this is the bulk dual of this particular computation and This will give rise to an effective if you now integrate out this blue region You get an effective action Which is a function of G left G right G E G left G right and G E and Then in principle there's goldstone bosons That you also need so that's a very complicated beast that in principle you can compute and now it's a two-sided story To lowest order again this this can be computed but now we have this what's now really different and This is where all the interesting stuff happens is in the infrared which is this particular region here because now in Principle we no longer have a horizon within going boundary conditions. We just have an interesting partition function to compute In particular this trace here Which means that these things here are identified is simply the statement that a little slice here is identified with a little slice there But this in principle should be an interesting infrared partition function now And if we compute this infrared partition function We would again be in exactly the same situation that we have a simple infrared partition function and It's somehow all the dissipation and all the fun comes from this infrared effective action Which is coming from this near horizon? mixed Lorentzian Euclidean piece here now that computation turns out to be very simple you can take this infrared limit So let me say a few words about this infrared effective action and wrapper It's quite simple because in the infrared limit when you approach the horizon at the geometry becomes approximately a rindler But rindler space and the horizon they scale in a different way in the near horizon limit, which means that the horizon Largely decouples and this is also one of the explanations for some of the Symmetries and in particular this different morphism inference that one sees is that because the horizon largely decouples the only remnant of the horizon is basically the area That's some scalar field, but for the rest you can do an arbitrary different morphism the volume preserving different morphism And this is sort of the origin of that But because the horizon largely we basically left with a one-plus one-dimensional problem and it's literally this picture here There is a radius and there is some time and now in this one-plus one-dimensional problem If you look at that one-plus one-dimensional problem, you have a bunch of gold stones They're just a bunch of massless gold stones and what you find is that the left moving part of that gold stone is Related to a1 minus a2 and the right moving part is related to this a1 t minus a2 t minus i beta and indeed If we have a solution of a scalar field where the left moving field is identically zero the action vanishes because the kinetic term Is d plus phi d minus phi and if this is equal to zero we are in a situation where the right move is vanished identically and the Effective action is also zero so that suggests That we should interpret the left and the right movers if you want to make a contact with this beer esteve formalism We should interpret the left and the right movers as the cue of something Which suggests that in order to make a contact with this beer esteve formalism is that we should replace this one-plus one CFT in the in this infrared region by probably by an n equals 2 comma 2 one plus one CFT Because this is precisely the sort of structure that you get in in a super conformal field theories Now up here. There's no reason here to do that You could say it's string to your super gravity or something like that But there's no obvious reason to do that But this if you want to have a bulk incarnation of these beer esteve symmetries then this seems to be what we need to do So there's an interesting thing to think more about is whether this is a natural thing to do in gravity or not is to replace in The infrared near the horizon to replace the conformal field theory by a super conformal field theory somehow and interpret the supercharges as these beer esteve symmetries And to see if that's really the right thing to do you can in principle set this olive without using these beer esteve symmetries but These effective actions are particularly nice in that language. So this is clearly something that needs to be understood further So it's an interesting connection between all these things and I will stop here. Thank you If I put boundary conditions all the way here and you also identify Basically what you do is you you could almost view it as collapsing the horizons on top of each other But what you do is you you basically identify this here and here It turns out that that by itself is a well-defined problem with a unique solution already So there's no freedom. There's also somehow no room for things to go here. They would have to come out here again So these these if a one is a two for example You can see that that's precisely the case where things are purely in going and they are purely outgoing here in this way And they're kind of identified like that So there's not there's nothing here which sort of can absorb things anymore. What about the Euclidean piece? Yeah, so the whole solution of these you have mode Euclidean mode solutions here and Laurentian mode solutions here And you can explicitly solve them all in terms of the boundary data here. They're explicit solutions. They're Unambiguous so it's a it's a well-defined infrared problem that makes it nice and you can quite easily solve it