 All right, well, I'd like to thank the organizers for bringing together this workshop and a very nice school in the previous week. I wanted to probably take a break from Wilson Loops, and I'm going to talk a little bit about ADS Black Hole, ADS Black String Entropy Story. So this is based on work done with one of the organizers, Leo, and also with my student, Jung Ho, and some work in progress. So I just want to give it a little bit of introduction on this idea of what we can do as an application of localizations to calculate supersymmetric partition functions and then to sort of use ADS CFT to make it a sort of precision test of holography. So just as a quick reminder for all the students in the school, when we talk about supersymmetric localization, it's sort of a tool for us to compute quantities in supersymmetric field theories, supersymmetric gauge theories, and typical things we might consider are like sphere partition functions, Wilson Loops as you've heard about, and then if you compactify on sort of sphere times S1, you can also compute supersymmetric indices over here. So we had some nice lectures on localization. I won't repeat the process of localization, but just to remind you, since we're going to see some building blocks of basically what happens after you localize, you have a partition function that can be written as sort of a classical piece, which is the integral over the localization locus times a one-loop determinant from the quadratic fluctuations. And so it's very cute. You get relatively nice expressions or finite integrals, matrix models out of a complicated field theory. And so I want to use this feature to connect the topologically twisted index on S2 times S1 or S2 times T2 with a black hole or black string microstate counting. This I should mention, of course, is work pioneered by Bernini and Zaffroni and many people in the audience over here. So I feel like I'm talking to the experts actually. So this is the sort of index story. And to bring in ADS CFT and precision holography, we can do similar computations on the ADS side and start comparing ADS answers with field theory answers. And so standard things you can do that's been done for quite a bit is, let's say you look at global ADS, and you can relate that to sort of partition functions on spheres. You can work with black holes in an ADS, and now you have a time-like circle over here, so sphere times S1, or you consider a black string in ADS, and you can have something like sphere times T2, T2 being related to basically the longitudinal directions of the string. So we're going to end up focusing on black string in ADS 5. So let me just sort of sketch the setup. So if you have a black string in the center space, in the ADS CFT side of things, you talk about sort of UV or IR boundary or horizon. And on the boundary, you have asymptotically into the center space ADS 5. And as you approach the horizon, you have ADS 3 times S2. Now if I think about an ADS CFT sort of language, I take ADS 5, and with this topology, I have a four dimensional boundary with topology S2 times T2. So from the UV point of view, I could talk about some sort of supersymmetric field theory on S2 times T2. On the other hand, if I focus on the near horizon ADS over here, ADS 3 brings down a T2 on its boundary. So it makes for a field theory side. You can sort of see things both as a four dimensional field theory on S2 times T2, or a two dimensional field theory on T2. And of course the relation is essentially through compactification of S2. The highlight of the string, of course, is the torus, which should remind you of modular invariance and SL2Z transformations. And we will see that sort of showing up in this game. Presumably it goes all the way up to infinity. What a nice thing about ADS is you can always have sort of nice identifications on a boundary. So it is a boundary presumably with topology S2 times T2. The question is, of course, the fall-off, how fast does a fall-off to approach ADS? I'm afraid I don't know the answer to that, but thanks. So I'm going to end up talking about the index on the S2 times T2 for that sort of reason of exploring the black string. But more generally, this idea of these ADS black objects and the topologically twisted index was pioneered by Beninian Saffroni and many friends in the audience as well. And the general setup is to imagine you take an object with some horizon topology of S2 and then you can turn on some magnetic flux. So magnetic black hole, magnetic black string, for that reason there's sort of an S2 floating around in an anti-decider space. And then when you sort of look on the field theory side of that, by having magnetic flux on S2, essentially you've turned on a monopole background and up with sort of black ground or symmetry flux on S2, has to be done correctly to preserve supersymmetry if you're looking for supersymmetric solutions here. And the end result is that you end up with a partial topological twist on S2 where the monopole flux basically cancels the spin connection and you end up with a system that can be computed using localization, this topologically twisted index. And so Beninian Hirschdorf and Saffroni studied this topologically twisted index for S2 times S1 in regards in relation to magnetic, they charged black holes in ADS4. And this has been extended to various cases, dynamic black holes, hyperbolic horizons, magnetic black strings, and so I'm going to end up going to the magnetic black string case. But first, I do want to give you a little bit of a discussion of the black hole case because it's kind of a beautiful story and it sort of sets the scene for the black string. I want to give you two examples. The first is the magnetic black holes and M theory on ADS4 times S7 mod CK. This is the ABJM theory. And then I also want to show you something about magnetic black holes in a massive 2A on ADS4 times S6. And after that, we'll move on to magnetic black strings on ADS5 times S5. The black hole story for all the students that attended the school, if you're talking about the sort of three-dimensional terms of matter theory on S2 times S1, Albertle gave us some very nice lectures on this. The building blocks of the index after you do this localization process, which I'm not going through, is that you have some set of vector multiplets, some set of carom multiplets, and the different components are in the vector multiplets. You have a prediction function that's built about an angle over the cartontaurus and contributions from the different carom multiplets. And so depending on what theory you're talking about, ABJM or massive 2A or something like that, you're basically putting these components together and calculating the topologically twisted index. So for the ABJM case, or for the M-theory on ADS4 times S7, you end up with this ABJM model. This is the current assignments matter with two gauge groups at level k and level minus k with four bifundamental fields, A1, A2, B1, B2. And you just go to the previous slide and look at all the components at index. You just put everything together. You end up with the first UN, the second UN of minus k, and the A1, A2 fields and the B1, B2 fields like that. That's quite remarkable that when you actually do this, you can actually calculate and you end up trying to evaluate this topologically twisted index, this partition function using the Jeffrey Kerwin residue. I'll just give you a pretty picture. I hope it's a pretty picture. So if you study it numerically, you find out that you can find a solution to what may be thought of as the beta assets like equations for eigenvalues. So there's two sets of eigenvalues. There's the first UN, and they sort of go along. This should be like that. There's a second UN kind of going on like that. I just picked certain parameters for the chemical potentials just for illustration over here. And it's not completely obvious from this picture over here, because I just fixed n equals 50. But the real part of the eigenvalues are here. In the way that was set up over here are centered at pi. And they only go up to some values related to the minimum and maximums based on the deltas over here. They don't wrap around. But in the imaginary direction, they scale with n to the 1 half. To work on this, you put it into the beta potential, you do a calculation. You find out that the topologically twisted index has a leading order term. That's n to the 3 halfs. That's sort of what you expect for maybe GM. This is black hole microstate counting, but you still expect n to the 3 halfs. I didn't actually write out this coefficient F0, but that is known. That's what Alberto showed us last week. And then you find sub-leading corrections. Scaling is n to the 1 half, and then minus 1 half log n and so on. And I have to say, we tried quite a bit to figure out what this. How can we calculate sub-leading corrections? This is precision holography. We want things to match, not just at the leading order, but add sub-leading terms as well. So we spent some effort trying to figure this one out. Didn't succeed, yes, absolutely. How do you know that? I do know it had to match. Well, the question is, the index computes things with a sign, right? Minus 1 to the F, and so you can get cancellation on index. Why do you expect this to maybe match black hole micro-state counting? Or something like that. I don't have a precise answer for that. I hope the experts might be able to say something about that. But it is a little bit interesting because there are examples when you just do like super conformal index, ankles for AMOs, large n limit or something like that. You got order one scaling because of the large cancellations. So somehow this topologically twisted index appears to actually capture the features that you want. And then there's some argument perhaps that the ground states over here are all sort of like bosonic ground states or something. And so there's no minus fermions to cancel. But I don't have a precise sort of understanding of that. You could wonder, even if the leading order works just by construction, if some minus 1 to the F cancellation might give you trouble at these higher order terms, and so that's, to me, I don't have a full understanding of that. It is kind of curious that we tried our effort to get to enter one half, to figure out, study it numerically. You realize that the log n coefficient looks very much like minus one half. So we had a conjecture that in fact this is minus one half. And more than just that, it appears quite universal, independent of the fugacity sort of chemical potentials that you turn on. And if you're a little bit careful, you can try to do a one loop quantum super gravity calculation. And we're able to reproduce that from the super gravity side. If you're a little bit careful in realizing that these black holes are actually embedded in ADS-5, and not just the horizon. So somehow the embedding of the black hole in ADS-5 seems to be important to get this coefficient to match on the gravity side. Well, I'll give you one more example, then we'll move on. This is the massive 2A case that was sort of promoted by a Guarino, Jafferys, and Varela. And here you just have a single SUN gauge group at level k. The level's not balanced, it's just plus k, there's no minus k. And so you get into the 5-3rd scaling out of this model. The index again has a contribution for the single SUN vector multiple. And then from the three matter fields over here. Again, many friends in the audience worked on this calculation for the topologically twisted index. Due to calculation, I'll just give you the picture one more time. This time there's a single set of eigenvalues, disappear approximately on a straight line, but you see higher order corrections showing up when you compare the solid line is the leading order result, the dots are the exact numerical results. So sure enough, they were able to get the leading order term over here, correct analytic function of behavior and everything like that. If you look at some leading corrections, you find it skips a few orders of n to the 1-3rd, and the next term, this is numerical evidence, is n to the 2-3rd, n to the 1-3rd, and log n. So there's a question in the ABJM case. You can find minus one half log n with reasonable numerical support. Question is whether we can figure out anything to do with the log scaling over here and then turn that around to an ADS type calculation. That's sort of work in progress. Well, that's the warm up. Now I want to move on to the black string story, or the S2 times T2 index. And so let's consider one more case, what's the building blocks over here? This is ADS5 CFT4 at the starting point. And so we're talking about four-dimensional A-Mills theory on S2 times T2. And it goes one language, the topologically twisted index is again broken up into vector-multiple contributions and chiral-multiple contributions. Because there's a T2, everywhere you should start thinking, modular functions, and sure enough, theta functions, the decadeta function, things like that, you have everything written quite nicely in terms of functions on T2. Unfortunately, they're harder to work with in some cases as well. So these are the building blocks of the index. We want to assemble it according to n equals 4 A-Mills on S2 times T2. And so let me mention the framework of this calculation, or the framework of the holography, is an ADS5. We're going to have a manically charged black string, an ADS5. Yep, the string over here, it's easier to actually construct. Or what you usually do is you go down to some U1 truncation. So U1 to cubed, sometimes called the STU model, which means there's three U1 fields, there's three charges. You can turn on three magnetic charges. It's a string, so the electric charges are a little bit different in five dimensions. So it's basically just turned on three magnetic charges. And we can call that N1, N2, N3. There's a constraint from supersymmetry, which you can realize is the sum of the charges in this convention is equal to 2. You end up with a near horizon solution. And a near horizon solution was basically studied by Benini and Bobov. So you know more or less all the properties you want to know at the horizon. The full magnetic string solution in full ADS5, well, you have the pressure equations, you can integrate them numerically in certain special cases. You can get some analytic solutions. There's reasonably good evidence that these strings actually do exist as supergravity solutions, but the most complete information we have is basically at the horizon. So on the field theory side, because we've turned on these magnetic charges for the string, we have magnetic fluxes on S2. The torus itself has some ultra-parameter tau. Then in the field theory, you can also turn on chemical potentials. This times there's three deltas, delta 1, delta 2, delta 3 for the three U1 fields, which means the topologically twisted index, in general, would be a function of the chemical potentials. The background magnetic charges implementing the twisting plus the flavor, and then the torus parameter tau. You put together the topologically twisted index, and basically here it is with one vector mulplet and three carbon mulplets, and it equals one language. Again, you calculate. So you use the Jeffrey Kerwin residue, and you re-express this. In terms of a sum over solutions to what you might call the beta onsets like equations of various expressions from the vector and the chiral mulplets, and then you have this Jacobian factor over here. And the Jacobian factor is really obnoxious. That makes everything quite a bit of a challenge. But you take what you get, basically, and you work with it. And what you find is kind of interesting. Here's the beta onsets like equations. We're at the eigenvalues, if you want to call them that, are basically these Ui's and Uj's. And so U1, U2, U3 up to Un. So you want to solve these expressions over here for the U's, basically. What we can find, or what Hossaini, Nedlund, and Zephyruni discovered, is if you want to avoid the complication of these Jacobi Thetas, what you can do is you can expand what might be called the high temperature limit. This is the limit of the shrinking time-like circle. So this is not a thermal theory. It's a supersymmetric theory. But I sometimes call it temperature just by analogy. And so you're in the limit of the shrinking circle. Now you have modular transformations, tau goes to minus 1 over tau. So this tau goes to i0, can be replaced by tau goes to i infinity. You end up with simpler expressions over here. You solve it, and you find out, let me just give you the pictures, the equations over here, that the eigenvalues can be evenly distributed along the time-like circle. U bar over here is just a constant, sort of an average value, so to speak, plus this evenly spaced along the time-like circle related to this tau over N factor. Now in the black hole case, in the S2 times S1 case, it was more complicated. It was those sort of lines and curves and things that I showed you. There you solve the equation. You find what you might consider sort of like a saddle point or something like that. You say, I'm done. I'm happy. I calculate with that. That seems to be the dominant saddle. It gives me the right expressions. Well, here it's interesting. You should remind yourself that this is a torus. You should remind yourself there's SL2z going on over here. So if I had a solution this way, what happens if I do an S transformation? Tau goes to minus 1 over tau. Well, it flips like that. If you actually check, this also solves the beta-onsage equations. This does too. So does that. Does that. It's very fun. This is doubly periodic or quasi-doubly periodic. You realize, if you distribute your eigenvalues evenly around the two cycles of the torus to form sort of a sub lattice or a smaller torus, it also does solve the same beta-onsage-like equations. In effect, if you really want a modular variant answer, you need to keep all these solutions. So this is an example where you have not just a single saddle, but you have a sum of saddle points or a sum of solutions that you need to take into account for your modular variance. And so you can label the solutions, if you wish, by three integers, actually only two independent ones, m, n, and r, m and n. Our divisors are capped to n, n is the SUN theory. And roughly speaking, m is how many last points you have horizontally, and n is how many last points you have vertically, r is related to a shift over here. So your original torus has larger parameter tau, and then your sub lattice or your smaller torus over here can be described by larger parameter tau tilde. This is going to show up several times m tau. So it's basically a fraction m over n times tau, but it can be shifted as this picture shows over here as well. Not to say we haven't proven that this is the complete set of saddles or a complete set of important saddle points, but this appears to be a minimal set that's needed to enforce modular variance. Let me just, all these are related by SL2Z, absolutely. This picture or this equation here? I'll give you an example. Yeah, I'll give you an example later on, which hopefully will make it a little bit more clear. I'll give you an example very soon, actually. But anyway, let me just give you an equation before I give you an example. The idea is that the topological twisted index really should be thought of as some over sectors. Each sector is labeled by these three integers, two independent parameters. And the partition function in a single sector is just exactly what you might think it would be. It's just the original object with the UIs substituted for their saddle point expressions. You play with it a little bit, you find out there's a function of the original torus tau and a function on the smaller torus tau tilde over here, where this is all built out of some sort of a Jacobi form, if you wish. As I say, we keep track of two modular parameters, the original torus tau, and sort of like an orbifoda torus and the smaller torus tau tilde. And you're summing over sectors and just basically summing over different tau tilde's over here. Here's the picture. Basically, well, I'll get to the picture. Another way to think about this is you're calculating an index on S2 times T2. If you compactify an S2, you end up with an index on T2. Trace minus one to the F on T2 is essentially calculating the elliptic genus. So you should end up with an elliptic genus or a Jacobi form of weight zero. And let me give you, here's the picture for N equals six. There are 12 sectors over here. And the original solution, and eigenvalues disappeared along the timeback circle over here, there's another obvious one, which is the S transform of that. But in fact, there's a bunch of other solutions as well, labeled by M and an R, which is given by these numbers over here. What I highlighted is tau tilde is given by the complex structure of tau tilde, given by the red and blue arrows over here. So what happens to monitor covariance or monitoring variance? Well, if you look at T transformation, tau goes to tau plus one, these are the orbits of those 12 sectors under tau goes to tau plus one. Probably the easiest one is the starting point over here and they're all distributed vertically like that. The top eigenvalue, well, over here is essentially the same as down there, but this top one over here will get transformed up here to tau plus one, so to speak. And so everything gets translated this way, this way, this way, that way. And then the S transformation, tau goes to minus one over tau, maps the various sectors into each other. Again, the easiest one to see is the vertical one mapping into the horizontal guys. The guys that are a little bit slanted over here are a little bit more difficult to sort of work out. As you see, these sectors depend on having these two integers M and N, the mole play to capital N. And so if you have SUN, where N is a prime number, then there's only two ways to do it, one times N or N times one. But if you have composite numbers, which can be factorized in various different ways, then you have a more complicated sort of set of sub-sectors. What is tau tilde? If you have a sector labeled by M and R tau tilde, something like M, tau plus R over N like that, I'm sorry. This is not SL2Z in what sense? No, no, no, this is not a transformation. No, this is just defining the tau tilde parameter. No, no, sorry, that's not the transformation. What happens? Well, you see what happens here. Now I do, tau is my tourist parameter and there's only one tau. You take tau, the tau plus one, tau tilde in some sense goes to tau tilde plus over N or something like that. Another way to say it is if I'm adding one to tau over here, it's equivalent to adding M to R over here. So it takes a sector M and R into M and then R plus M. Well, then you can shift the big tau by one so you can mod that by N down here. All right, well, with this in mind, let me just sort of, let me move on to the high temperature limit. Anytime you have these sort of modular functions, the natural thing about the high temperature limit is something like a Cardi limit, is to do a tau goes to minus one over tau transformation. And so what happens is this is a Jacobi form of some set of indices. And so the transformation of the partition function is sort of primes. All the primes are the S transformed version of the original guys. And then as a Jacobi form with some index M, it transforms with this exponential factor over here. When you do that, you try to work out the expression. What Hossini-Nedlini-Saffroni showed is if you assume the index is dominated by this vertically distributed sector over here, which I call one N zero over here. The S transform of that brings it down to the N one zero sector over here with different parameters. Any study, the high temperature expansion of this guy, this guy's very easy to look at. He end up with basically a C function that when extremized can be related to the central charge of the string. But this story cannot be complete because you have a bunch of saddles or a bunch of solutions. You should really sum over all of your solutions. If you do that, you end up with an expression that looks kind of like this. It's probably too, or it's not that ugly, but it's got a bunch of parts to it, of course. There's chemical potentials are delta. You can choose a convention where delta one plus delta two plus delta three add up to two pi. It's nicer to scale things by two pi. So little d's are just the delta scaled by two pi. They add up to one. And you end up with the contribution from the vectors. The tau dependent contribution is d one minus d. The tau tilde contribution, x one minus x. The tau tilde contribution has this extra n prime involved in its expression over here. And then the second line over here is the obnoxious one that comes from the Jacobian. And there's this function alpha of the chemical potentials which we have not been able to write down a nice expression for. So we just left it as alpha, basically log determinant of this matrix is up to some prefactors just given the function alpha of d. So if you just looked at the dominant sector, the one n zero sector, you transform that as the n one zero sector which gives you a capital N over here. This is n squared. This little thing over here is the number one. So x a equals to d a in the dominant sector. And then this term combines it with that term with a n squared minus one n squared from here and a one from here. And well, maybe I'll write that down in the next after this. But in the dominant sector, you just get some n squared minus one overall factor over here. The real complication is to ask, I say it's the dominant sector to n one zero sector. Why do we know this dominant? We had to actually consider that a little bit. And so what you discover is this is log of the partition function to taking log or trace over the log of this determinant. If the determinant, if the Jacobian itself is some order one expression or something like that, I'm looking for a one over beta term into high temperature expansion. If the determinant is just sort of order one, you take a log of that, nothing bad happens and you just get nothing skating as one over beta and then you just get alpha equals zero like that. The other possibility is the determinant is a little bit degenerate. The determinant goes to zero, log of zero is kind of bad. But of course what happens is that the determinant goes to zero like e to the minus one over beta, something like that. Then taking a log of that gives you a coefficient of one over beta, which gives you a non-trivial alpha. In general, this determinant function alpha is piecewise linear. We just try it for special cases for n equals two. The two one zero sector has nothing, has zero over here but the one two r sector, which is one two zero or one two one sector, there's two of them over here. Gives you a piecewise linear function kind of like that. There is a region over here where alpha is equal to zero but then there's other places where it's not. N equals three is a little bit more complicated. For prime n, there's a bit of a pattern to it. For composite n, it's something we haven't been able to work out in a nice way other than just looking at pictures. N is composite, it's a bit of a mess. Nevertheless, for these cases that you study and in certain regions, like if there is a region where alpha more or less doesn't contribute and you might be able to show that the dominant saddle really is dominant. And so, as I said, the dominant contribution and the one that was looked at previously is what we call this one n zero sector with this n squared minus one. Prefactor pulled out. Now showing that this is the dominant saddle point really is equivalent to showing this type of expression over here. This is for any m prime m prime r prime which is one of these other saddles over here. You wanna show some expression like that. So if this alpha is equal to zero, then you're done, right? Because anything's bigger than equal to zero. Well, not necessarily, but I mean, if you actually work that out, it's okay. But the problem is if alpha is a positive number, then you wanna show that this inequality still holds. And so it appears it's not universally true in all possible cases, but at least in the physical cases where there's a nice black string in ADS dual, it appears that this is true. We have not entirely been able to prove that, but it does appear to be okay. So the simple guess that there's a single dominant sector, one n zero appears to be the correct sort of physical result. I'm just gonna wrap up a little bit. If you do assume this is what Hossania, Netland, and Zeferoni did, if you just focus on that single sector, you end up with a central charge function, which when extremized gives you the central charge in terms of the magnetic charges of the string. Good, so some of the final thoughts. It actually, we started this project as sort of ADSCFT people or holographic people saying, can we figure out the large n limit of the index? But before we figured out the large n limit, we got sort of trapped studying the modular properties of the topologically twisted index. So it'd be interesting to go back to the large n limit and ask what happens. Now, of course, in this high temperature limit, you get this one over beta and some central charge that scales as n squared. If you just calculate for arbitrary beta at large n, what's the result? You can see some modular function, presumably. Basically elliptic genus at large n, something like that. In terms of these log corrections that we've been focusing on, you can also ask, is there a log correction? Is there a log n correction over here? I don't have an answer for that. As I said, you would expect large n to still have modular covariance. That's basically it. Let me just leave you with the parting thought that when we've done these sort of saddle point type calculations, whether for saddle point matrix model evaluations or for the topologically twisted index, in many cases, we just find one saddle point, we're happy. We think either that's the dominant saddle point or that everything else is sort of related by symmetry. There's not much else that's going on. In this T2 case, there might still be only one dominant saddle point, but if you wanted to enforce modular variance, you really have to consider that there are multiple solutions to the saddle point equations or multiple solutions to the beta outsides equations. So that's my take home message for you. Thank you.