 Okay, it's working that well. It's an honor to be here. I thank the organizers to be here for this very special occasion. And now that ADS CFT is a strapping youth of 20, I thought I'd start by congratulating the proud parents. And this is a logo that nicely captures the yin and yang of ADS CFT design. Actually, it's copyrighted to my former student, General Gump. But it turns out that ICTP actually has produced t-shirts with this logo on it. And I thought that it would make a very good present for the MGKPW. But then like ADS CFT, those t-shirts are also very popular. And we are just left with, there was only one t-shirt. And it's of the size that I think will fit Edward. So as for Juan and Igor and Steve. As you saw, I wore one. Yes. And Sasha, next time they visit the ICTP, there is a t-shirt written for us. Okay, so what I would like to talk about is quantum holography. So much of the very nice talks we've heard and also a lot of the work that has gone on in holographies at large gen, which basically means use classical gravity to study strongly coupled CFT. But of course, one of the prime motivations for doing string theory was to learn about quantum gravity. And they correspond to finite end effect. And one can even ask the question to be completely, does holography really hold at quantum level? And so unless one is religiously fanatic, one should keep an open mind, maybe it is valid only at large end. And these are the two developments motivated by these questions about quantum holography that I have been pursuing for the last five, six years. One is localization and supergravity, because if you want to study quantum holography, you need to develop new methods, non-potability methods for computing quantum gravity effects in the bulk. And the question is, can we make sense of the path integral of supergravity, at least for a suitably twisted version? And another thing which kind of a surprise, an unexpected surprise that came out of this kind of investigation, was a very precise connection to mock modular forms, which is again required by considerations of quantum holography in ADS3 CFT2. So this is what I will try to describe. And rather than jumping to the latest thing I'm doing, I thought, I mean, even for this very select audience, ADS CFT has grown so broad that not everybody would have followed this development. So I thought that I will give a broad summary and sort of the recent developments on where the subject is going. And of course, then talk a bit about our recent work. So this is based on some of my collaborators, which are listed here. And the sum of the work that I've described is also based on two new students and a postdoc at ICTP and a large number of other people. I will try to reference them, but if there is an omission, they are all included here. So localization, so supersymmetric path integral can be localized, as we know, to a finite dimensional integral over some supersymmetric saddle points, using this argument of middle net, basically a Grasman integral is zero. And it has proved to be a very powerful tool for studying strongly coupled quantum field theories. And it has really developed over three decades, starting with Witton's work. Then there were works of Necrosso, Pestone in four dimensions and three dimensions and two dimensions. And every few years, we have learned something really important and interesting about quantum field theory, which we could not have learned without these methods. Now, the goal is that we really try to emulate this in quantum gravity. And of course, in quantum gravity, this is fraught with danger. So one has to tread carefully. But basically, we sort of followed our nose and we have tried to develop these methods. And despite the difficulties, there has been steady progress. And my hope is that localization could prove to be as fruitful in quantum gravity as it has been in QFT as time develops and as more people and some of the younger people get interested in this. At a prototype example, I would like to describe n equal to 8 black holes. And then I will describe more general things. So if you look at the 1-8 BPS states and n equal to 8 theory, type II, T6, and a dionic charge vector Qp, then the u-duality invariant is Q square p square minus Q dot p square. And the counting function for these 1-8 BPS states is given, the index is given in terms of the Fourier coefficients of this object, which is a theta function square divided by eta to the 6, a very explicit object. And the index is related to minus 1 to the d plus 1 of c of delta. And the counting function is a Jacobi form, which means for those of you who, the post-ADS-CFT generation, it's a modular, I mean it's basically very symmetric. It's doubly periodic in z, and it behaves nicely under modular transformation. And already with the, you can see the power of quantum holography, because the near horizon geometry of a black hole, that black hole is ADS-2 cross S2, and it naturally leads to ADS-2-CFT 1-duality. And following that, same developed, the path integral defined, the path integral w of delta in the near horizon geometry as a generalization of the walled entropy. And it's a generalization to the extent that it includes non-local effects coming from massless loops, which are essential for various nice properties of that object. And in this case, quantum holography implies two things. It implies, w of delta must be equal to d of delta. Now, that's a non-trivial prediction for the path integral, because this tells you that the path integral must be an integer. And of course, we can do the, write this equation slightly differently. And it implies therefore d of delta is equal to w of delta. But it implies a non-trivial prediction for the index, because the index, d of delta was an index, and there is no natural reason why it should be positive. And it's a non-trivial prediction for the Fourier coefficients of that model or form, or for an index, because index is equal to degeneracy. This all follows from considerations of quantum holography. Under this second prediction, you can immediately check. If you just look at the, put the modular form on a computer and look at the first few coefficients, they are all dissatisfied with positivity. And you can also prove it. Now to go to the first prediction, it requires a lot more work. And fortunately, using the modular properties, there is this beautiful formula due to Hardy, Ramanujan and Radomacher, which expresses this integer, this is analytic number theory. It expresses integers in terms of analytic objects. And that's precisely the kind of thing we need, because we have a path integral, which is an analytic object. And it expresses it as a sum over Bessel functions with some non-trivial phases. And the Bessel function is just, you know, the usual Bessel function, the modified Bessel function with this integral representation. And in fact, if you identify z with the area upon 4, this Bessel function, the first term, c equal to 1, actually captures all the infinite perturbative corrections in one upon area and log of area, are captured by the first term in this expansion. And then the second term captures non-perturbative effect. And our really ambitious goal is that quantum monography, if Hoan is really right, can be really computed. And the Krusterman sums are actually even more intricate. They have rather intricate phase structure, which follows from number theory. And these are highly subleading phases, but they're essential for integrality, and therefore you have to learn to deal with them. And quantum monography again requires that the bulk must reproduce these non-perturbative phases. And as we will see, quite remarkably, it does. So the path integral on ADS-2, if you include the m-th area circle, then it's basically like ADS-2 cross S1. And there is a family of geometries which asymptote to ADS-2 cross S1, which are labeled here by two integers. And they can be thought of as freely acting ZCR defaults of this disk or the BTZ black hole. And they're related to the SN2Z family of ADS-3 that were considered by various authors as a kind of a possible source for non-perturbative contributions to the entropy. And what localization does is that it actually justifies, it was never clear how you could add subleading contributions in a non-perturbative expansion. But localization is what will justify that procedure, as we will see. So localization simply means that you have some integral which is invariant under supersymmetry, and the whole integral collapses to a solution of that supersymmetry, sort of the BPS configurations of that supersymmetry. And those we were able to solve using n equal to 2 supergravity coupled to nv vector multiplets. And they have a particularly simple form. They are completely universal solutions independent of the physical action, and they follow purely from official supersymmetry transformations. And these parameters Ci are to be integrated over. So your entire path integral collapses to an integral over these integers. So these are real numbers. And if you now look at the renormalized action for the pre-potential F, it takes this very nice form, it can be expressed very simply in terms of this. And very reminiscent of the OSV conjecture from the 2004. And I will comment later upon that. And essentially then the path integral reduces to a Bessel integral with the reasonable ansatz for the measure which we sort of made an ansatz at the time when we were doing it. But I will come back to the major issue in a moment towards the end. And if you do that, you get exactly the Bessel function that we were after. And not only that, it's easy to see that the sub-leading Bessels will also follow by looking at evaluating these saddle points on this topologically non-trivial, I mean, this other geometry is labeled by C. So all the entire structure follows very nicely. What remains are the Clusterman sums, and that sort of stopped us for a while. And it turns out, my wife, Joanne Gomm, she realized that the Chern-Samensterns in the bulk are sensitive to the global properties of MCD. So these localizations, instant terms are completely local, meaning they follow from solving a differential equation, and they are insensitive to the topology. But the Chern-Samensterns in the bulk, ADS-2 cross S1 or ADS-3 bulk, are sensitive to the global properties that contribute additional saddle points which are specified by the holonomies of flat connections. And you just follow, again, the procedure. They're closely related to not invariance and lens space, which were studied earlier. And you just follow your literature in Chern-Samensterns theory, and you evaluate them. There are various gauge groups in the problem. There is a U1, SU2 left, SU2 right, and they all beautifully assemble into the Chern-Samensterns action, terms, phases, assemble into this very nice formula. So now let me, before now going into the new things that I would like to talk about, some of the other things, let me just take stock of what has been achieved in terms of quantum holographic localization. Because there were a number of issues which were very puzzling in the community, and there were all kinds of confusions. But I think one by one, we have resolved many of them. First of all, there was a confusion about whether we are computing an index and how can we compare that with the degeneracy. That is now understood by ADS2, quantum holography index is equal to degeneracy. We have shown path integral is equal to Bessel integrals. Or before, sirens give you Hardy-Raman-Rajan-Royalmacher. Chern-Samensterns gives you Clustermam. And then we had looked at only the supersymmetric F-terms. But some recent work by DeWitt and Murthy shows that the D terms are equal to zero. So I think everything, all the assumptions that we had made along the way can be justified now much better. One of the open problem that remains, which I think is an important one, is the computation of the measure from first principle. Because for example, this is, when you do localization, you have to compute one loop determinants around this certain point. And in gravity, the problem is quite complicated because you had to first of all set up a background field quantization and supergravity, which was not done. And the BRST formalism is rather complicated because it's a soft algebra. So technically it was kind of a challenging problem. And this is, again, work to appear. This is a private communication. But I think these people have made what looks to me to be very nice progress. And they have set up the BRST formalism for the background field quantization and supergravity, and then you can compute the determinants. So I think now localization and supergravity is sort of coming to a level maybe one can, not yet, but that's my hope that it will eventually in this coming year or so it will be there. And there is very interesting question about duality invariance because you can distribute the charges differently with the same duality invariance. And they appear this calculation differently. And it's not at all obvious a priori that the answer will be duality invariant. And what Gomesch realized that actually the duality invariance implies non-trivial number theoretic identities like Selberg identity for the Klustermann sum. And it's a very interesting problem to obtain them from the bulk again by considerations of this John Simon's thesis. He has made some progress in that direction, but I think again it's a problem that requires further work. Then how do you get the sub-leading Bessel functions? That problem also, for example in n equal to 4, that problem also has been now recently addressed by Gomesch and Murti and Reiss. And they come from the instant terms of the topological stream. Okay, so let me now quickly summarize. How much time do I have? I was counting on you being as gentle as you were the first time. Okay, so let me... Okay, if I can take maybe a few minutes extra. So the path integral, what is I think as surprising from the path integral which is a complex analytic object, yields an integer, and it accounts for all kinds of... I mean it accounts for the non-potability states actually whose mass is higher than the string scale. So in this sense it's really an IR window into the UV. And if you... There are generalizations that one can consider and it really calls for some kind of topological supergravity along the lines of what Sparks and Wafa also mentioned in their talks. Okay, maybe I will come to this transparency towards the end. This was a kind of more philosophical comment which I'll come to this time. Now let's go to the n equal to 4. Meromorphic Jacobi. So in n equal to 4 the counting function turns out to be a meromorphic Jacobi form. So if you remember our counting function was theta square upon eta to the 6. Here this theta square is in the denominator rather than in the numerator. And that means it has poles. So Fourier coefficients now depend on the contour. Which one should we choose? It will give you a different answer. The contours move around under modular transformation and therefore modular symmetry is lost. And this really bothered me because... Again because of considerations of photography because ADS-3 requires that there should be a modular symmetry because this modular symmetry can be identified with the modular symmetry of the boundary torus. So the loss of modularity would have been a disaster for a holography and I didn't see any way of restoring it. And it turns out, talking to a number theorist, Zagaya about this, it turned out that his students had solved that problem going back to Ramanujan's work from 100 years ago and that led us to a collaboration and it led to this decomposition theorem that I will now describe. So what we proved was that a meromorphic Jacobi form has a canonical decomposition, psi m finite and psi m polar. This is called a Mach Jacobi form which has no poles and all the poles are residing here and they are called the apple lurks some in the max literature and you can think of them as basically some kind of an elliptic average of the... there is a pole and you just average over... on a torus you have a plane you just average over it. And then the contour depends on the modular in a precise way and the modular space splits into chambers consistent with duality. There's a very beautiful physical interpretation in terms of black holes that there are walls in the modular space on the left hand side there are single centered black holes on the right hand side there are single centered and double centered black holes and the Mach Jacobi form counts the single centered black holes the apple lurks and counts the multi centered black holes pole crossing corresponds to wall crossing and residue corresponds to the jump in the degeneracy. And the non-trivial part of our theorem was that the Mach Jacobi form I mean of course you can always divide a function in this way the non-trivial part of the Mach modularity is that that Jacobi form admits a modular completion which means that by adding a small piece to it you can make it modular but that piece is non-holomorphic and therefore that completion is denoted with a hat it satisfies a holomorphic anomaly equation and of course the reason they were mysterious because Ramanujan did not tell us anything about this holomorphic anomaly he just devised the Mach modular forms and they said from the goddess of Namagiri and he said that they have nice modular properties but now we can actually verify them if we add a correction term then we can actually verify that the modular form is so Mach Jacobi forms therefore are holomorphic but not modular and the modular completion is modular but not holomorphic so there is this tension between these two things between holomorphic and modularity and again it looks mysterious but now I will discuss that there is a very simple physical derivation of it which I will describe in two final slides but basically this analysis of the DMZ paper led to we analysed all these infinite family of Meromorphic Jacobi forms with double poles and a related family with single poles and basically they yield most Mach modular forms known to humankind all of Ramanujan's examples the generating function of Hurwitz-Kroniker which was studied by Zagye the new forms that appeared in the Moral-Monsha and an infinite class of new examples so it's a kind of a nice bonus that comes out of considerations of quantum holography but then the question is can we really take holography seriously and derive this Mach modular form and the shadow from the near horizon geometry so this is one thing that we realised in this process is that Mach modular is closely related to non-compactness and if you're computing an elliptic genus of a 0.2 SCFT with a compact target then you get a Jacobi form but if it is compact then there is a famous argument of Witton that the Bose Fermi cancels in pairs the right-moving Bose Fermi cancels in pairs and you get the left-moving right-moving ground sticks and left-moving oscillations but if the M is non-compact the spectrum is continuous and then the Bose Fermi cancellation may not be exact because the density of states of the continuum may not be precisely matched and in this case the elliptic genus so defined would be non-holomorphic and that basically is interpretation of Mach modularity from the point of non-compactness and now going to holography we know what should be the near horizon non-compact SCFT and if you just look at the attractor equations the near horizon geometry of the brain system has a non-compact abnut space with a factor whose radius squared is equal to M and it's very easy to see that the lattice sum of this asymptotic circle gives precisely this term which is the shadow and the density of states can be computed using scattering theory and we have not completely dotted the eyes yet but following these ideas one can really provide a physical derivation of the holomorphic anomaly and therefore a physical derivation of Mach modularity and once again it is inspired by holography because without holography we would not have gone down this track and this is some work that I will describe I mean that will come out soon so I think I do I have time for two philosophical comparisons okay this goes back to the discussion that happened that is quantum gravity dual or emergent and I think there are two points of views that were expressed I think this I would say is the old school the pre-ADS CFT point of view that ADS quantum gravity is exactly dual to CFT M theory has its own rules of computations with an as yet unknown non-portrait formulation and it's just a special corner of some platonic elephant of M theory that we don't know and the second point of view which is I think much more popular nowadays is that ADS quantum gravity is just emergent from CFT and any reasonable CFT gives a non-perturbative definition I mean we don't yet know what exactly is reasonable but that's the kind of belief that it will give you a non-perturbative definition of quantum gravity and of course my color coding here is not entirely impartial but I would say that our competition seemed to argue in favor of one and the theory is really not just UV complete but UV rigid and since it's a little bit of a contrarian point of view let me say why I think so for example in this ADS-2 examples all integers d of delta define a valid CFT-1 but only a very sparse set among them has a dual ADS-2 so the rest of them are in some kind of a swarm plan only those which come from Fourier coefficients of a modular form actually correspond to a proper ADS-2 dual moreover we have independent rules for computing the w of delta we did not rely on some rewriting of the CFT and then trying to interpret it as gravity we just said there is a definition of semi-classical quantum gravity so anywhere whenever we claim to have a non-perturbative dual of a CFT we should check whether the bulk has semi-classical rules of quantum gravity and whether whatever answers we get agree and what we saw was that w delta equal to d delta really required some precise details of n theory and such as John Simon's terms so from that point of view I would argue that this is the point of view that I would take that this is the elephant of the n theory this is just the left ear ADS-2 is just the left ear of this elephant which is a mammoth it's a mammoth and my personal bias is to believe that if the left ear is there then there is a bigger beast attached to it and the left ear of the beast is not the whole beast but okay I mean this is a I will stop there I think this is a good point working n equals 8 written in n equals 2 formalism which also contains hyper-modestates now doesn't that require to understand off-shell hyper-multiplets so this is the issue that I talked about major is that first of all we know from the attractor mechanism that the hyper-multiplets are not excited so we don't expect them to modify the instantons but they can contribute to one-loop determinants and to study that problem I think linearized off-shell you just require one-loop analysis and this is an issue that these people have tried to address and my belief I mean it's not yet completely done but my belief is that we would be able to address that issue so if I understand correctly these configurations you integrate over are not solutions they are not solutions of equations like in any localization situation I thought that in order to compute something like the graviton determinants you need to have a solution of equations aha very good so that's related to yeah so that was one there is a conceptual question how do we even set up which I don't have a full understanding of how do you even set up localization and gravity because the q supersymmetry is supposed to be gate symmetry so what exactly does it mean and the one way to set it up is that you split the graviton use the background field method as we do so you have a background field you have a background field plus quantum fluctuations then you choose a gauge for the background field and then you choose a gauge for the so you develop the whole VRST for both these gate symmetries independently and then what you find is that at the end of it you're left with asymptotic part of the gate symmetry is left over which is what you use for doing the localization so basically now you have an ADS to background field which is completely well defined and the quantum field is fluctuating in that background but which asymptotes to ADS too and you choose that asymptotic ADS to define the localization in the background metric so that's I think that's the way David and these people have tried to solve this problem I have not fully gone through the work but that seems to be the right direction to develop ok let's thank Tateesh again