 So our next speaker is Tamar Yakov. He will tell us about Black Hole Entropy in 5D. Thank you very much. I'd like to thank the organizers for organizing this wonderful workshop and for inviting me to speak on Friday the 13th. And I'll be presenting some work in progress with Morteza Hussaini and Alberto Zaffaroni, which has to do with a five-dimensional version of this Black Hole Entropy. So identification of the microstates that are related to Black Hole's longstanding problems since the work of Beckenstein and Hawking. And there's been some success. I've written some notable example. But in principle, ADS CFT should allow us to calculate the quantities or to count the microstates in a more or less straightforward way. And attempts to do this came about a long time ago. Many attempts have been made and not all have been successful. But in 2015, Benini Christov and Zaffaroni managed to make a match between the partition function of the twisted index on S2 times S1 and the entropy of a 4D Black Hole. And this equation that I've written is the equation for the entropy. So the field theory model in three dimensions is ABJM, which is a Lagrangian maximally symmetric super conformal field theory in 3D. The entropy, which they calculate, is represented by the finite part of this partition function. It isn't an anomaly in the field theory. The partition function is computable at large n on the field theory side using what should be regarded as the effective twisted super potential if you reduce one dimension back to two dimensions and the beta-anzatz equations and some large n techniques. And then the comparison was made to an ADS-4 supersymmetric Black Hole, which was introduced by the authors here, and the comparison worked out. And my understanding is that it's not entirely obvious why the comparison works, but it does. OK, here are some of the technical aspects that went in this calculation and that I think made it successful relative to the previous calculations that involved the untwisted super conformal index and three and four dimensions that I had on the first slide. So first of all, the index that they computed is topological. All of the states that are supported on the two-sphere are ground states of the system. Technically speaking, the one-loop contributions are simpler, and the flavor symmetries in the model that they took are manifest. So ABJM has the flavor symmetries that they needed manifest. There is a complex scalar modulus. It's related to a scalar in the vector multiplet and the wholeonomy around the S1. And the contour for how to integrate that modulus was understood from the relationship to two dimensions. And it's given by the Jeffrey Kirwan prescription, which is also we also heard about in the previous talk. And possibly most importantly, there are supersymmetric fluxes. So flux supported on the S2 for both the dynamical and the background gauge fields. For the background gauge fields, this is needed in order to reproduce entropy of what is a charged black hole. But it's also very important that the flux for the dynamical gauge fields is summed over. And this summation allowed to reproduce the correct entropy. There's also an equivalent deformation, a sort of omega deformation that you can do on the S2. But also importantly and also not as in the super conformal index case, it can be turned off. So it can be turned off. OK. So what I'm going to introduce is a 5D analog of the same type of calculations. And I'm going to just stick as closely as I can to the things that made the three-dimensional calculation successful. So I'm going to consider an appropriate theory in five dimensions I need at least n equals 1 supersymmetry in order to do this. So I'm going to consider something that has at least n equals 1 supersymmetry on a manifold of the type m4 times S1. And I'm going to choose m4 to be Torr-Keylor. And the reasons are that it matches very well with the three-dimensional calculations that shares many of the features. So there is a topologically twisted partition function, which is amenable to localization. We already heard about this type of localization in Maxine's talk in the morning. There are gravity solutions with which to compare. The theory lives in an odd dimension, so the finite part of the partition function is expected to be universal. There's an equivariant deformation and the cross of deformation, the omega deformation on the compact space, which one can turn off. And there are fluxes to be summed over and a contour prescription for the evaluation of the matrix model. And as we heard in the morning, which fluxes exactly should be summed over, what exactly is the contour is nowhere near as clear as it is in two dimensions. But maybe as importantly, most of the necessary localization calculations have already been done. So by these authors and by a bunch of authors who did the S5 partition function and related things. Okay, there are also a number of differences from the three-dimensional case. So first of all, we can consider more complicated topologies. In three dimensions, you basically work on S1 times a Riemann surface, at least if you want to have product manifold. Here I can take any torrent-calor manifold. There is a Lagrangian in five dimensions with N equals two or maximal supersymmetry, but it's not conformal and the strong coupling limit is believed to represent the six-dimensional two comma zero SCFT. So if you think that this only works in the maximal supersymmetry case, you're stuck looking at that case. There are instanton contributions, which presumably go away at least at leading order at large N. And there are some technical challenges that I've already mentioned, which is that the integration contour in some of the fluxes is not well understood. I'm going to make some comments about what has to go right with these in order for us to reproduce the entropy, but I'm not going to solve this problem. The other challenge for us is the correct analog of the Betanzas equations is not clear, and I will produce some sort of conjecture for that and then check it in one particular case. Okay, so this is the relevant four-dimensional topological quantum field theory that is supporting the states that are counting the black hole microstates. In the three-dimensional case, this was the A model in two dimensions, and it's, if you have a gauge theory, it's something that calculates intersection numbers on the modular space of flat connections, and here it's Donaldson-Witton theory, which was already introduced in the morning. It's a twist that you can do that gives you a comological, topological quantum filter. The energy momentum tensor is Q exact, and if you take the gauge group to be SU2, then this produces Donaldson-Witton, the Donaldson-Witton TQFT, and it gives you, computes the intersection theory on the modular space of instantons on M4, the so-called Donaldson invariance, if you put in enough operators to compute all of the classes. There is an approach to computing Donaldson-Witton invariance that I won't touch at all, but perhaps is the right way to do this, which is the low energy approach. The low energy approach is to take the cyber-Witton solution of the SU2 theory, or maybe some theory with a higher rank, and to compute everything using the solution by taking the manifold to be really, really large, the theory is topological, and looking at the low energy effective theory, and for instance, an important paper on this is the one by More and Witton, and there are the contributions, this isn't localization, well, not really localization the way that we think about it, and there the contributions come from an integral over a complex modules, the U-plane, and over some discrete solutions called cyber-Witton monopoles. So I'm mentioning this just to say that perhaps this is the right way of doing this, and not the way that I'm going to present. Okay, so this is the type of manifold I'm going to take, none of the data here is actually important, I just want to show this to you to say that people know how to build Toric-Keylor manifolds in a very, very explicit manner, you glue a bunch of things that look like C2 together, it's known where the Toric action degenerates, there's even a canonical construction for the complex structure and the metric and some canonical coordinates that you can put on this manifold, we're doing everything in a topological field theory, so none of this is going to matter, but I'm just saying this because if you wanted to check things like the Killing-Spinner equations and things like that, you could just take this sort of metric. Okay, so our M4 Toric-Keylor manifold admits in its isometry and when the metric on the four manifold admits in isometry, there's a refinement of Donaldson-Witton theory introduced by Nakrasov, which uses the omega deformation, so we already heard about this in the morning, but you introduce a supercharge that squares to the isometry instead of squaring to just a gauge transformation and on our four, this is the setting for Nakrasov's partition function, which can be used to compute the effective pre-potential of the low-energy theory. But on a Toric-Keylor manifold, use the Toric isometry to localize to the vertices of the polytope and we're going to do the five-dimensional version of this so we're going to be localizing to circles that are supported at the vertices of the polytope but I'm just going to use the five-D version of the Nakrasov partition function at every point. So one important thing is that the Nakrasov partition function contains a lot more information, even in the limit epsilon one, epsilon two goes to zero, it contains a lot more information than just the effective pre-potential in flat space. So this is an expansion in epsilon one, epsilon two and these extra terms H half, F one and G one show up in calculations on curved manifolds and this expansion has been worked out in five dimensions for the five-dimensional Nakrasov partition function by these people. So I think this paper was already mentioned in the morning, the copy that I have is from 2006. This is Nakrasov's conjecture for what the partition function on a Toric-Keylor manifold would look like. This is not a localization calculation and some of the things that I've generously put here like the sum over K and this contour integral were not specified in the paper. So the top part is the partition function in the way that was presented in the morning, which is a sum over basically the R4 partition function localized at the vertices. And on the bottom, this is the limit where epsilon one and epsilon two go to zero and as you can see, there are a bunch of terms from the sub-leading terms in terms of epsilon one, epsilon two of the Nakrasov partition function that show up in this effective action. So the way that you should think of the last bit of this equation, this effective action, is that you need to expand up to, you need to extract the four form from this. So for instance, in the first term, you have to bring out two line bundles. The LAs are line bundles, C1 is their first churn class, and you need to bring out two of them. That would be a double derivative of F0, the usual low energy pre-potential. And those represent, you need then to sum over these line bundles, so those represent dynamical fluxes on the Torre-Keylor manifold. In the second term, you would have just one of these and in the third and fourth terms, those don't depend on the fluxes, but they do depend on the geometry of the manifold. They depend on the Euler number and on the signature of the manifold. I've admitted all of the observables and as I said, and I'll emphasize again, the contour for A and the exact sum are not known in all cases. This K is an integer flux, so you, in principle, if you don't know the stability conditions, you should sum over all of the line bundles. In principle, what this sum gives you is a theta function, right? It's a sum of an integer squared and the first power of an integer times some horrible function of the scalar modulus A, so the result or the effective action for A is a theta function. Those theta functions appear also in the low energy approach, but I don't know what the relationship is between these and those. Okay, so I'm going to be using this to compute a partition function in five dimensions and here is the setting for five dimensions. I'll take a Wilbine, which is derived from this, say, this canonical metric that I showed before. I'll add one more extra dimension whose size will be two pi beta and I'll define some vector field which just corresponds to the directions of the torus action. And the five-dimensional metric that I have is not quite the product metric of this time direction with M4. It's the product metric with these off-diagonal components in E5 mu that represent the omega deformation. So this is just the omega deformation in five dimensions. And if you want to preserve supersymmetry, all you have to do, so I said requires additional work, but the only work that you have to do is to twist. But I'd like to present one slide on what the modern approach is to preserving supersymmetry on these types of spaces. The modern approach is to couple the theory to rigid supergravity. In principle, you have to choose which supergravity you want to couple through their different flavors, but one nice alternative is to take what's called superconformal tensor calculus that seems to capture all the flavors. And I'd like to apologize that I don't have an appropriate citation. So you could, in this situation, start with the five-dimensional while-multiplet, and you would need to find, if you wanted to preserve supersymmetry on this space, you would need to find fixed points of the gravity, no indilatino equations. But we're in a very simple situation where I can simply set these additional fields in the while-multiplet T mu nu and be mu to zero. The variation comes down to this generalized killing-spinner equation. I also don't need eta, and all I need to do is to choose the r-symmetry connection for the 5D, at least N equals one theory, to be proportional to the spin connection, and that's going to preserve a spinner. The spinners are symplectic myrona, and I've written them as matrices. So this preserves one spinner. At least SU2, so for N equals one. The supersymmetry algebra is almost the same as the usual one for the omega background in four dimensions, and specifically it's the same as in the 5D contact manifold case. So I've written it here in twisted variables, which are the most convenient to use in this situation. So I redefined some of the spinners, and I redefined some of the auxiliary fields to get these simple supersymmetry transformations that look homological. If you had hypermultiplets, you would also need to twist the scalars, and there's a slight subtlety there that Maxim already mentioned, that has to do with the spin-C bundles instead of spin bundles, and it's not really going to matter for me. Let's just say that you can put this on any M4, which is to our calic, whether or not it has a spin bundle. Okay, this is the necessary slide on the basics of localization. I've identified an appropriate fermionic charge. I can now choose a V, which is positive semi-definite, and deform the theory by QV. Q should square to zero on V, but it can square to some isometry or whatever as long as V is invariant under that isometry. I deform the action, and the resulting path integral is independent of T. I can add some Q closed operators, but what I really want to do is take T to infinity, where the measure localizes and is very small for anything where QV does not vanish, so it localizes on the zero set of QV. Semi-classical approximation becomes exact, but you have to sum over all the saddle points, the moduli space where QV is equal to zero. In the example that I've taken, this moduli space consists of a few pieces. Yes? This is, so far, this is a general N equals one 5D gauge theory, general 5D N equals one theory. I haven't shown any particular example. Okay, so the non-trivial saddle points for a gauge theory come only from the vector multiplets, the fixed points of this equation for this fermion in the vector multiplet, and they come in three classes. There's a, I'll call it a bulk modulus. This is the VEV for sigma, and also flat connections for F. If you were looking at this as a six-dimensional one comma zero theory, then they would both be flat connections. So this is the constant bulk modulus, the one whose, there are instanton contributions that are isolated at these fixed points or fixed circles with the equivalent action on M4 degenerates, and there are fluxes. So the existence of these fluxes was worked out in the papers that I've cited here, and they are the heroes of my story, and I can't give them up. So without these fluxes, nothing would work. In principle, there's one flux for every equivariant divisor, but this is subject to topological and stability conditions dictated by the manifold. Okay, so I haven't shown you an actual localization calculation. That's because all the ingredients were already there, and this is the expected result. This, I will mention to what extent this result is a conjecture, but we didn't add to it. So there is a five-dimensional necrosis partition function. It depends on one more parameter relative to the one that was shown in the morning. This is beta. It's the size of the extra circle. Inside this partition function, I'm going to include absolutely everything, including the classical piece, so that my result can be written as a product of a bunch of the classical partition functions at the fixed circles, a sum over semi-stable fluxes, and an integral that I will tentatively identify with the JK residue contour in the space of this dictated by this bulk modules, this complex bulk modules. And I've shortened AL, so this inside the necrosis partition function, this bulk modules is shifted by the values of the fluxes, and that shift is proportional to epsilon one and epsilon two, so in principle, it goes to zero once you turn off epsilon one and epsilon two. And we can write this down explicitly, only in some cases. And regardless, a direct large end computation of this type of partition function is extremely difficult. Yes, it's the Z classical of the theory that I'm considering, but I haven't determined which theory that is. This was computed by Necrosse. Yes. For any chosen field content in action with n equals one supersymmetry in five dimensions, I can write down the, yes, the twisted n equals one field there. No, no, there is an action. The localization only works when you have an action. I didn't bother to write one down because this works any time you have an action, but of course you need it. Yes, of course, and I will show a specific example. So far, I've just discussed how to compute things and then you need to choose your favorite theory. Okay, so now is the point where I actually have to do some sort of calculation in the large end limit. Some calculations were done in the four-dimensional theories and the references that I showed before at rank one and were compared with Donaldson with an invariance. So that is the evidence that I'm aware of that the sum over flux is actually correct and this counter-prescription as well. But I would like to do large end and I'm going to use the same trick or the same very helpful guide that was used in Brian's talk, which is to use this nekrosov-Shatashvili approach and to think about what it took to identify the large end limit in three dimensions. So in three dimensions, there was this auxiliary quantity, the effective twisted super potential of the two-dimensional theory, if you had reduced the theory from three dimensions and for two-dimensional massive gauge theory, you can solve the partition function by solving this equation for the vacua, this beta-onsets equation for the vacua. You have to first identify w tilde, but since w tilde is one loop exact, you can get it directly or you can get it from the calculation of the partition function. And the Krosov-Shatashvili identified this with equations arising in integrable systems and they showed how to get the appropriate w tilde from higher dimensions. And very importantly, the large end limit is tractable when you use this equation because you can take sort of the large end limit inside w tilde first and then identify the solution to this equation which is dominant at large end. Okay, but we are not in two plus one, we are in four plus one and a reasonable analog of this equation in four dimensions is the effective pre-potential. The effective pre-potential is what comes out of the nekrosov-partition function when epsilon goes to zero. And it's reasonable to think that it plays a similar role on a compact twisted four manifold. And indeed, nekrosov and Shatashvili reduced a compact four manifold to a two manifold in two different and important cases and found that you get the twisted chiral superpotential from the effective pre-potential. So one of these cases was the nekrosov-Shatashvili limit on R4. This is the limit where you take one of the equivariant parameters, the one that's going to rotate the plane in the remaining two-dimensional theory to zero. So there's still two-dimensional Poincare symmetry, but the other one is kept finite. So the theory on the other plane is effectively compact and you can evaluate what is the effective twisted two superpotential in two dimensions. And it's at leading order in this equivariant parameter, it's just related to this effective pre-potential in four dimensions. That means that in principle, if you wanted to solve the two-dimensional theory now on a, put this topological theory on some remand surface, you would have to solve the equation that I've written. Those, that equation determines the vacua of this two-dimensional theory with an infinite number of fields. The other context in which nekrosov and Shatashvili did this is in a twisted compactification from four dimensions. And here I've written down the equation for the n equals two-star theory in four dimensions. This is a compactification, I'm sorry I didn't write it. This is a compactification on the sphere with flux through the sphere. And here the effective twisted superpotential is again related to the four-dimensional pre-potential. But very importantly, it's the twisted superpotential is the derivative of the four-dimensional pre-potential with respect to several parameters. The equation that I've written down is sort of unfolded in the two-dimensional case. That's why there are both magnetic and electric fluxes. But I've written it down this way because it looks duality invariant in four dimensions. It looks invariant under s duality. So the top equation involves w proportional to f and the bottom equation involves w proportional to the derivative of f. And I'm going to claim that you need to take both of them seriously if you want to work on a twisted four-manifold with the equivalent deformation and still find the, define a reasonable set of beta-unsets equations that are going to give you the partition function at large n. And it goes without saying that this is a conjecture. Okay, so now I'm getting to an actual quantum field theory and comparison to black hole entropy. So the black hole that I'm going to take comes from the black string solution by Benini and Bobev. It's the gravity dual of the 6d2 comma zero a n theory compactified on sigma g1, sigma g2 and t2. I'm going to take both of these g's to zero so as to get a Torr-Keylor manifold in four dimensions. And I've written down some of its properties. It's a truncation of SO5 maximal gauge supergravity in seven dimensions. It contains two U1 gauge fields and two real scalars and the solution itself interpolates between ADS-7 and ADS-3 times this compact four-manifold and this four-manifold has fluxes on it and we're going to identify these fluxes with flavor fluxes in the superconfined field theory. There's also an ADS-6 black hole that you can get by compactifying one of the directions on ADS-3 and you can also give arbitrary momentum along this compactification direction and it was examined in this paper by Khristov and who showed that the black hole entropy is related by the appropriate cardiform to the central charge if you took it of the black string, the dual to the black string. Okay, so this is going to be my example. I realize that this example is in six dimensions and not in five dimensions but there's a five-dimensional field theory, the maximal n equals two field theory which is supposed to be a Lagrangian description of this six-dimensional theory. So that's the Lagrangian that I'm going to use. I've specified the gauge group and I'm going to take some holonomies that in principle break this to n equals one star but I'm not going to take any soft masses so you should think of this as the n equals two theory but possibly within a deformed version of this. The n equals two theory where the deformation comes about just because I'm putting it on a curved manifold. There's a unique action for the n equals two theory in five dimensions, I'm sorry? This is n equals two superannuals in five dimensions. That's correct. Okay, so you're asking why don't I correct everything with the instantons? There is no action for the two zero theory. The theory that, okay, I'm sorry. The theory that I'm considering is the theory in five dimensions with maximal supersymmetry. That theory has one vector multiplet with the usual action and one hypermultiplet in the adjoint representation with the usual action. The, this theory is supposed to, well, this theory is infrared free but, okay. But at strong coupling, it's supposed to describe the two comma zero theory and what people I think mean by that is that this theory has instantons, those instantons are instanton particles from the point of view of the six-dimensional theory and at the strong coupling fixed point, there is some action that describes the six-dimensional theory with six-dimensional Poincaré invariance that includes, in addition to the 5D Lagrangian and infinite number of terms that involve the instanton fields. However, all of those terms are q-exact, so it doesn't matter what they are. Again, that's q-exact. The, I'm not proving anything beyond what was conjectured before, but. What? The conjecture is that at least for BPS quantities, you can capture everything from the 5D Lagrangian, but from the, again, even tricep squared is not going to matter. That's also. So what do I know about this conjecture? You can look at supersymmetry enhancement on S4 times S1. You can look at the S5 partition function. You can look at S5 mod a discrete subgroup and all of those compare well with what is known about the two comma zero theory and that is the evidence that I'm aware of. And then I'm going to present some further evidence. Okay, are there any more questions? Okay, so as I mentioned, I will compare it to this black string solution and I'll also compare it to the ADS6 black hole solution. Okay, so after all of these necrosis partition functions and everything like that, I've taken the large end limit and this is what remains a very, very simple partition function that involves the non-equivariant limit of the classical and one loop determinants for the n equals two theory, which you can also think of as a, as this type of a, a hoary supersymmetric quantum mechanics where the target space are these, the space of ground states on sigma one times sigma two. Just to explain the notation, X is a dynamical variable that represents the complex modulus, the dynamical complex modulus and Y is a fugacity for the, for the U1 flavor symmetry, but Y is in, well, okay. In principle, this fugacity is complex. You can have the real mass and you can have the chronomy. I'm only going to take the chronomy. So delta is the, is the flavor fugacity. M and N are dynamical fluxes that should be summed over that are related to the two S twos and S and T are the background fluxes for this U1 symmetry, which as I said, I'll identify with the fluxes in supergravity. Okay, so let me concentrate on the first equation that Nacrosse and Schatashvili showed, the one that had to do with Nacrosse and Schatashvili limit or the partial-equivariant deformation and which would tell us that we need to, in order to find the putative vacua, we need to take Df, Da to zero. This is one of the constraint equations in order to find the vacua. So I've written down F, this is the effective, it doesn't look like the four-dimensional one because it includes all the KK modes from five dimensions. So this is the effective low energy pre-potential that comes from compactifying a five-dimensional theory. The polynomial that I wrote there is actually rather important and represents just like in the three-dimensional work of Kloset and Kim and Willet, this represents a gauge invariant regularization of that sum. So you have to work it out very, very carefully. And Minahan, Nedelin and Zavzin did a similar computation to the one that I'm trying to do here on S5 and found that at large tuft coupling, the eigenvalues in this equation were pushed apart and found a solution for the dominant eigenvalue configuration and we found the solution as well. And I wrote it down here. The solutions are very, very similar. And in fact, if I plug this solution back into the pre-potential, I will recover the S5 free energy. However, that is not what we're trying to calculate. The pre-potential is not the effective action for our theory. But if I did this, I would recover the S5 free energy so it at least shows me that we're on the right track. This relationship was also noticed in three dimensions. So if you plug the saddle point configuration for the three-dimensional twisted index into the effective two-dimensional twisted super potential, you recover the S3 partition function, the free energy that's computed on S3 at large M. So the same thing happens here. However, I've only used one equation. If I tried to now use this equation to compute what I actually wanted to compute, which is the twisted index, I would find, this is the way that you write the twisted index in order to put in the solutions of the beta-alans and equations. I would find that there are still an infinite number of solutions. There are an infinite number of solutions because there's one such solution of the profile for the eigenvalues for every flux sector. I should explain. One of the fluxes on one of the two S2s has been summed over in order to reproduce this form, which is very, very similar to the S3 calculation. However, the other flux is a parameter. And for every value of the other flux, there is a 3D computation to be done. And there are an infinite number of such fluxes and as long as you don't determine what they are, you can try to do some minimization procedure. You won't get the right answer. But thankfully, the second, the Krasov-Shatashvili equation that I showed you, that had to do with twisted compactifications, tells you what you should do. There is also a piece of the twisted, twisted chiral superpotential in two dimensions that's proportional to the second derivative of the pre-potential. And if you take that piece and solve the relevant vacuum equations, you get a constraint on the flux. That's this flux N. And I can put in that constraint and recover the partition function that I wanted to see. So this partition function depends on the size of the circle, the tuft coupling, and delta one and delta two are this fake way of writing two deltas in terms of one delta. They satisfy a constraint, so their sum is like two pi. There's actually just one delta. There's this T and S that represent the fluxes on the two spheres. And this matches, including the dependence on delta, the trial right-moving central charges was computed by Beninian baller. So right now, we didn't say what delta actually was, and what we've computed here is something that has to do with the black string and the two match. The relationship to the modular parameter on the boundary of the black string is the one that I've written down here. It has to do with the size of the sixth dimension. Okay, but we wanted to recover the microstates. And to recover the microstates, we need to move to the micro canonical ensemble, so we need to do a double transform. And I'm defining a quantity here, which is the appropriate index. The log of the microstates is equal to this index, but only at the critical point, which has to do with taking a saddle point approximation for this integral with respect to delta and beta tilde. This is the same procedure that was used in three dimensions. And if I do this here, I recover the expected Cardi formula for the number of degrees of freedom of the black hole. This is what currently works very, very well in this situation. And so to summarize, 5D twisted indices are a direct analog of this BHC computation in three dimensions. And we can compute these indices using localization on a Tor-Keyler manifold, up to some conjectures that have to do with summing over fluxes and the exact contour integral prescription. However, we expect the contour integral prescription to be related to the JK residue. And we expect the sum over fluxes in principle should be determinable from some work in the mathematics literature if you can read it, specifically this citation of Kuhl. And I've shown a matching to black hole entropy and to the properties of a black string in some very, very specific example. Now, notably, this example had to do with the six-dimensional theory. It's had some caveats of its own related to the Lagrangian, but on the other hand was on a product manifold from the four-dimensional case. So it was both harder and easier, but the matching works. And there are a few points left to sort out before this can be done on a general and four, some of them having to do with these stability conditions, the identification of the JK contour, and perhaps a better understanding of why these bettons as equations are the ones that are relevant, why you have to take both of them, and what is the physics behind the fact that this determined the correct vacua? Not so much at large N. Maybe one of them is a large N artifact, maybe they both are. Or maybe both of them can be used to determine the partition function at finite N, like they can in two dimensions. And it would be intriguing to compare this to the way that you do the low energy calculation, which looks very, very similar, but counts completely different things in four dimensions. So this is one of the areas that I would like to go forward with this. And of course, to compute, there are other five-dimensional theories with N equals one supersymmetry that really live in five dimensions, for which you can compute these partition functions in comparison with six-dimensional black holes. And we're in the process of looking at some of these. Thank you very much.