 Okay, so thanks for the invitation to give a talk in this wonderful conference. I'm actually, you should say I'm very impressed with the progress that has been made as evidenced by the talks on the localization ideas. I was not aware of many of these new developments, so it's very exciting. So the talk I'm going to be giving is based on the work I've been doing with the group here. Jefferson, Kim, and Omori, and it is in some sense in the old tradition of, you know, issues that arise in the context of black holes of various kinds, but also connecting to new ideas having to do with super conformal field theories. So these two areas which have been developing, one from the old days and 23 years now and with the other one more recently, the super conformal field theories somehow come together in some form, and this is the context that this talk is related to. So we know that Calabia manifolds, especially Calabia 3-folds, and their singularities are source of a lot of progress and understanding in the study of super conformal field theories, and there are two specific cases that have become important. One is elliptic Calabia 3-folds, and this is the case where we are studying six-dimensional super conformal field theories with 1,0 supersymmetry. So in the context that we use it is in the context of F theory on elliptic 3-folds, and the singularities could be singularity of the base or the elliptic vibration in the F theory sense. And so this one gives rise to 60 conformal theories. The second one for general Calabia 3-folds, and if you look at singularities involving shrinking four cycles inside the Calabia 3-folds, this gives you five of these super conformal field theories with any equals to one supersymmetry. So namely we consider M theory in this case on these three-folds. So these two cases lead to super conformal field theories, and morally we believe, morally, the space of super conformal field theories in six and five dimensions is morally the space of singularities of Calabia 3-folds, elliptic or non-elliptic in this form. So studying them we think is more or less equivalent to this statement, even though we don't have a proof, it's morally seems to be correct. At least it's based on all the models we have been able to construct or understand. Okay, so these are local properties of non-compact Calabias. Okay, non-compact Calabias, you look at where the singularities are, and from there we can find super conformal field theories. So in particular, methods have been devised to compute quantities of interest in these super conformal theories, and it turns out one of them specifically is related to computation of topological strings, so we can compute topological string amplitudes in these backgrounds, and it turns out studying them is equivalent, is needed ingredient for understanding things like super conformal index or their partition functional spheres. So understanding topological string in these backgrounds is a first step in understanding some aspects of these super conformal theories, in other words, some quantitative measure about operators or their partition functions can be understood from these topological strings, and this is in the context of non-compact Calabia. And the question is, can these methods be extended to compact case? The compact Calabia and non-compact ones seem to be rather different, and it turns out that for the compact case, it's very difficult to compute the topological strings. So in other words, even though we know quite a bit about non-compact Calabias for topological string computation, and we can essentially do it to all genus, for compact Calabia manifolds, the computation of topological string is still at a very elementary stage, despite the fact that, you know, about 25 years ago we already computed up to genus two for a quintic threefold, for example, and that has been pushed up further by claims to genus 51 or so, but there's no general computation, so this is still, I would say, not understood. We don't have a good method of understanding compact topological string amplitudes. Sorry? Well, basically it has to do with being able to do certain information about the properties of quintic. So you can take first few coefficients and fix it by one million or another, at the conifers singularity, at infinity, and this and that, so the collection of various tricks helps you to push it up, but you cannot push it up indefinitely. So, anyhow, the main point here is that the techniques are a bit strange and there's no general answer known, and this is despite the fact that people have tried. So compact versus non-compact Calabia in the physics language is gravity versus no gravity. Compact Calabia means you're interested in gravity, non-compact means you're interested in field theory. So this is the distinction and the field theory is easier. Localization techniques that we have been mostly hearing about relates to field theory statements and even the ones that involve gravity in some sense go back to field theory. That is in the sense that we go back to the boundary or something where there's a field theory active. Honest gravitational ones is very difficult and we don't have localization techniques, unfortunately, in those contexts that we need to compute, and in particular, we don't have yet methods to compute topological string amplitudes for arbitrary compact Calabia. Why is that? Well, part of the trick involved in the non-compact case are symmetries. For example, for Toric Calabia three-folds, there are circle actions and people use these to localize computations and get the exact answer. Non-compact ones do not have, sorry, compact ones like the compact quintic and so on, do not have these circle actions. So therefore, the story is much more complicated. Okay, but then there's a connection with black holes. So then there are two classes of theories I'm talking about, six-dimensional super conformal theories and five-dimensional super conformal field theories. And these two relate to one lower-dimensional black holes. Namely, the idea is that you take these theories in six and five, you put it on a circle and then there's a string around, strings wrapped around the circle with the momentum gives rise to a black hole. That's a basic general idea. In the 6D case and the 5D case, you get 5D and 4D black holes in this case, or if you wish, sorry, there are two different versions of it, but let me just stick to the 5D version in this case. You can use this to understand five-dimensional black holes from this construction, which is going to be the main thing I'm going to emphasize in the following way. So you take type 2B, let's start with a simple example. Take type 2B on T4 times S1, or K3 times S1. So this is type 2B compacted by from 10 to 5 dimensions. Or you can think about them as M theory on T6 or K3 cross T2. These are dual to each other. So these constructions are dual to one another. So this is, again, you can think about them, then M theory from 11 to 5 or type 2B from 10 to 5. So you end up with a five-dimensional story. And this is the context or setup for the original black holes that we studied with Strominger back in the 90s. And so these are relatively well studied by now. And the basic setup there, I remind you, was that we take type 2B, we take the three brains, at least one version of it, is that you take a D3 brain wrapped around the Riemann surface inside K3 or T4 times the circle. So you take a three brain wrapped around the circle times the Riemann surface, which sits in K3 times S1 or T4 times S1. And this is T-dual, sorry, not T-dual, S-dual, to the M theory picture where you have a Riemann surface inside T6 or K3 cross T2 and you're wrapping M2 brain around it. So some particular classes of the T6. The particular class will depend on the momentum and the classes you pick here. And they are dual to one another, as I just mentioned before. So this is the general statement that if you take type 2B on Y times S1, it's M theory on Y times T2. That's just the map between M theory and type 2B. So let us for concreteness focus on K3 and consider a genus G curve inside K3. And if you wrap a D3 brain on it, we get a signal model. The marginalized space of the leftover theory is one dimensional. So you get the one dimensional effective theory, which is the signal model on symmetric products of G copies of K3 times R4. And this R4 is the center of mass of the string. And the duality works like this. So if you take, let's say you take K3 cross S1 for this guy. You take type 2B on K3 cross S1. From M theory, you take K3 cross T2. You pick a Riemann surface inside K3. You take a KK momentum around the circle N. This gets related to the same Riemann surface in K3. And the momentum N gets related to N copies of T2 attached to this Riemann surface. So the collection of N Riemann surfaces attached, sorry, N tori attached to this Riemann surface is dual to momentum N type 2B with the Riemann surface like this. So in other words, in the M theory set up, it's just the two dimensional surface, namely this two dimensional surface in this particular example is degenerated to a Riemann surface attached to handles or tori attached to it. Now the holographic statement of this is that ADS three times S3 times K3 is holographic dual to 2DCFT on the symmetric product of K3. So there's this holography statement that applies in this particular case. And the corresponding string enjoys a four comma four supersymmetry. And there's an SO4 rotation symmetry, the rotation symmetry of this S3 and that gets related to SU2 left SU2 right symmetries of the four comma four supersymmetric theory in the 2D. Again, compute the central charge of the left movers and you find the 6G plus six and if you then take the compactify the theory on a circle and take the KK momentum, that's just computing the degeneracy is the same as computing the elliptic genus of that. And the momentum N is the level of that elliptic genus you're computing and the central charge is known. And so from that you get the usual kind of a formula for the black hole entropy, and which agrees with the prediction of Beggenstam Hawking for large N and large G. So this is the old stuff. And we can also extend this to five dbps black holes which are spinning and in that context the SU2 left quantum number captures the spinning part of the BPS black hole. So you can actually keep track in the elliptic genus not only of the level, but also of this SU2 spin. And that is to spin captures this BPS spinning black hole as well. And those are known to work. Okay, now I want to shift gear. Now, the first step is trying to constructs further more things that we can do. And this is in the context of F theory and the spinning black holes. And the idea is this K3 cross T2 that I've been talking about in the context of M theory is not as general as it could be for a black hole. More generally, you can replace that with a Columbia 3-fold. You can consider M theory, not on this, but on the more general thing, Columbia 3-fold. And the black hole will be just M2 brains wrapping around two cycles of the Columbia 3-fold. And for the general Columbia 3-fold, we just have to count these M2 brains and we do not know how to do it. Why is that? Well, the way we did it here was to push it up in one dimension, get a string and compute the elliptic genus of that string. For the general Columbia 3-fold, for example, for Quintic, there is no string around. There's nothing we can do. We need a string in one higher dimension and there is none. There's no apparent one and there's no natural way to do it. So for general Columbia 3-fold, we do not know how to compute the black hole entropy. This is embarrassing. After 27 years from those computation, we still don't have any idea. For example, Quintic, the usual Columbia 3-fold Quintic, the BPS states give you black hole, the counting of these M2 brains wrapped around the cycles of the Quintic, give you black holes. We do not know how to count it. That's a bit embarrassing, I would say. It's a challenge for this community. We need to figure out how to compute this. But if we want to get a string in six dimension, there is a trick. There is a trick if the Columbia is not generic but special. If the Columbia is elliptic, then you can do something. So if the Columbia is roughly speaking the product of the elliptic curve times a four dimensional base or more precisely elliptic vibration over the four dimensional base, in this context, we know that if you take M theory on these manifolds and take the limit of this tiny, take the limit where this T2 becomes tiny, this is dual to F theory, which basically sees the base only. And the extra circle, if you add the extra circle, you get back the 5D theory again. So in other words, you can push the theory up one dimension to six dimensions on this base. In this context, then we can get off the ground because in the context of F theory, we can consider D3 brains wrapped around cycles, the Riemann surface inside the base, very much like the K3 story. So in the K3 case, we had K3 times something and we take a Riemann surface in it and then we got a string wrapping D3 brain around it and we take a momentum around the circle and that gave us RBPSA. The same thing is true here. So in other words, the elliptic vibration is good enough to do the same thing we did when it was strictly a product. We don't need a strict product. The idea works even if it's just elliptic vibration. And again, this gets mapped in the M theory language to M2 brains wrapped around the Riemann surface, which is glued to N copies of the elliptic fiber at N different points. So now, however, the amount of supersymmetry is reduced. If you wrap it D3 brain around the Riemann surface, instead of getting a 4,4 supersymmetric theory, you get a 0,4 supersymmetric theory. You get half as much. It's not surprising. Now, it is not difficult to compute the central charge of this string. So we can do the same game we played with K3. In that case, we had a symmetric part of K3. Here, we just need to count what's the central charge of it. It's easy to compute it, and you can compute the central charge of the left and the right movers. And using that, you can compute the prediction of the black hole entropy exactly the same way you did for K3 cross T2. And this was also done further in a paper with Haigat and Murthy and with Vandoren and myself a few years ago. And also, there are some related works which suggest that we can actually talk about this as a holography, a more general holography, in the sense of ADS3 times S3 times a base being dual to some conformal field theory with 0,4 supersymmetry. Now, this base that people have mostly studied is T4 and K3. But now, it could be any base of the Kalavia 3-fold. So it's much more general. And the price we pay is the lower reduced supersymmetry for the conformal theory, which is dual to it. So there's a general statement of this form. And of course, it doesn't mean we know what the CFD is. There's got to be a CFD. But how practical, how simple do we know it? Is it as simple as the symmetric product of sigma and the symmetric product of K3's? The answer is no. It's not that easy. And in general, we do not know exactly what the CFD is. If we know exactly what the CFD is, it's equivalent to doing black hole computations exactly. Nevertheless, what we learn is that if you take the elliptic genus for these corresponding conformal field theories, namely, these conformal field theories associated with these strings, that is, if you take their elliptic genus, then that's the same thing as computing the topological string amplitude on that Kalavia 3-fold. They're computing. So from the elliptic genus perspective, it's like the type 2B side, the K3 cross S1 and all that. So if you compute that, we know that's the same thing by duality as M2 grains wrapped around the cycle, which gives you the topological string amplitude, which is in turn captures the black hole partition function. So there is this relation of three things here. There is a string, 0 comma 4 string. There is topological string on Kalavia 3-fold. And then there's a black hole. And they're all related. Well, knowing one gives you any other one. So they're equivalent statements. So it doesn't mean any of them is easy. They're equally hard. But at least there's some reformulation here. Z5. So again, thank you, Atish, for making that URL on board. So this is five-dimensional black hole we're talking about. Z of 5D black holes, more precisely, 5D spinning black holes. So the 5D spinning black holes has two things. One has a charge, and one has a spin. The charge gets dualized to scalar classes for the Kalavia topological string partition function. The spin gets related to the coupling constant of topological string. So you have the full parameters to capture the degeneracies of the black hole in terms of topological string. And this is related to the elliptic genus calculation. OK, so we are reduced to the question, how can we come up with this 0 comma 4 to DCFD? How can we practically have a concrete realization of it? Now here it is that I'm going to fall back to the discussion I had for super conformal field theories. For super conformal field theories, in sixth dimension with 1 comma 0, which are the ones which are local singularities of Kalavia three-folds, we do know quite a bit about the conformal field theories that live on the corresponding strings. If you take the D3 brain wrapped around these two cycles which are shrinking, you get a string. And very often, or at least in many cases, we know how to describe these strings in terms of some quiver 0 comma 4 quiver theory. So for some classes, we know exactly how to describe these conformal theories. However, the problem is here we are not just dealing with local singularities. They are glued together using a compact geometry. So that's what's the new challenge. How do you use this local ingredient and piece it together to get the full string that could appear for wrapping not just one cycle near one singularity, but mixing different cycles and different singularities together and describe it as a 2D CFD with the 0 comma 4 super symmetry? So you can take examples of the form T2 times C times C moded by Zn times Zm. So if you take some Zn times Zm symmetry where the Zn and Zm act on the symmetries in some determinant of 1 equals to 4, determinant being equal to 1, where, of course, n and m cannot be arbitrary because you have to have symmetries of T2. So if you do this, so first of all, I have to explain. Why are we putting T2 here? Well, because we are talking about F theory. We're doing 60. So this C times C is the base of the F theory. And T2 is the electric fiber. And so if we want to describe a class of simple super conformal field theories in sixth dimension, you can take the products of this type. And you get, depending on their intersection, various kinds of conformal field theories that have been studied a while back by various people. But simplest ones, for example, if you take Z2 times Z2 singularity, you get, so these are some examples of what we call the conformal matter in a collaboration with Michaela and Company, where we had some kind of a simple description of how these singularities can be related to a blown-up geometries of the F theory. For example, if you take Z2 times Z2 version, you get, first of all, low psi, where there are singularities in the base, their curves. Where there is a curve, you get a gate symmetry, or more precisely, a global symmetry, which becomes gauged if you actually compactified the corresponding D7 brain. So these will be intersecting D7 brains carrying SOA times SOA. At their intersection, you get a conformal theory. In the Z3 times Z3 case, for example, you get E6 seven brains, two E6 seven brains intersecting. And at their intersection, you get a conformal theory, which can be blown up in the base to give you three P1s. And these numbers denote the normal bundle with a minus sign in the geometry of the normal bundle of the corresponding P1 inside the base of the F theory. So these are some examples. Z2, Z3, Z4, et cetera, depending on which one you get, you can get a 4, G2, E8, E7, SO7, et cetera. This is the geometry of local singularities under global symmetry. So in other words, for example, this thing, Z2 times Z3 is a conformal theory. This one gives you another conformal version, and so on, with the corresponding global symmetry. So for example, this one has E7 times E7 global symmetry, which corresponds to the gauge freedom living on the seven brain. Of course, this is a non-compact seven brain, it's a global symmetry. Okay, so this is the nature of the local singularities. Now, however, this is the local one, but now we are interested in compactifying it to a fully compact model. So for example, a Calabiah like this. Can we describe this one? For each pair of fixed points, we get the same kind of story before. But now the global symmetries get gauged. In other words, these lines that I had here, instead of being non-compact, previously they were part of the C, instead of they being either this C or this C, they're two C's, and each one of them, each one of them is in one of the two C's. But now we are modding that out, first of all it's on a torus and we are modding it out. So each one of them becomes actually a sphere, T2 mod Z2 or T2 mod Z3, et cetera, and it's finite area. That means the gauge coupling on the corresponding seven brain is not zero, but actually finite, therefore we are gauging the corresponding symmetries. So it's like super conformal field theories with global symmetries that we are gauging. So in particular, let's just take a simple example, take T6 mod Z2 times Z2. So if you think about this T6 as T2 times C times C, each one of them remember I told you has a conformal theory which is O minus one over P1, this is what's also known as the E string conformal theory, we heard about it yesterday. There are 16 fixed points on T2 times T2 mod Z2 times Z2. Each one of those fixed points gives you a conformal theory. So it gets 16 conformal field theories like this. Each one of them is at the intersection of two of these singularities, which has an SO8 symmetry. It turns out that every four of them are in the same SO8 global symmetry, but this SO8 symmetry that is common to each one of them is now gauged because the corresponding space is finite in volume. So it looks like you're taking all of these and you're gauging horizontal SO8 in this particular form for each group of this and the same with the corresponding vertical one. So you have horizontal and vertical SO8s that you gauge. Each gauging turns out to correspond to an O-4 geometry because you have a T2 mod Z2s that you're modding out and the corresponding T2 has four fixed points and the normal geometry for it would have an O-4. And therefore, you would have a geometry which is fours gluing every four of these. So if you look at these four, it's gauged by an SO8 and that's one of these horizontal or vertical ones. So you see there are eight fours here in this picture and these eight fours corresponds to vertical and horizontal directions on my previous graph. So you get the quiver which is like this and everything is glued together in this way. So you're taking these local ingredients and gauging it in this particular way. So this is giving you the full non-gravitational sector of this theory. How do we know we're not missing anything? Could it be that we are missing something? Well, we can check for example that the gravitational anomaly cancels. So we are not missing any ingredients. If you compute the anomalies for the number of hypers minus the vectors plus 29 times a tensor minus 273 it better be zero and you can check. You can put these numbers and you check that the corresponding things that we think there are actually canceled. So we have actually gravitational anomalies and so on cancel as they should. And one can also check the mixed anomalies, gravitational and gauging anomalies, cancel, et cetera. So we seem to have a perfectly complete model. Similarly for T6 mod out by Z3 times Z3, in that case each one of these fixed points is the conformal theory which we call the E6 times E6 conformal matter. And at the intersection point you have the geometry which you blow up it gives you 131 and there's an SU3 gauge symmetry on top of the three. So and now the corresponding P1 that you're gauging gives you a node which is a six and that has an E6 symmetry on it which is you're now gauging. Again you compute the conformal anomaly cancels, sorry, gravitational anomaly cancels as it should. These are highly non-trivial checks that we are not missing any ingredient. That the full non-gravitational sector is just given by these conformal theories whose global symmetries have been gauged in a particular way. So we have a nice compact description of a quiver of a full non-gravitational theory. This doesn't happen very often for us in the context of string theory. We always have there's a gauge theory here, there's something there. We don't have such a simple description of them but now here it looks like some simple gauge theory in some form. And you can do similarly for Z6 times Z6, Z2 times Z2, the same kind of games but here for example in the Z6 times Z6 case you get these conformal theories which are gauged by G2 or F4 or E8 and so on depending on which ones they are. So you get more interesting patterns of which conformal matter with which gauge symmetries and again each one of them you check the conformal, the gravitational anomalies and the mixed anomalies all work out beautifully. So this tells us that we have got the non-gravitational sector of it correctly. At least it's strong that's correct. For the Z2 times Z2 we have a concrete candidate for what the corresponding string is. As I told you, we know something about the corresponding strings obtained by wrapping these three brains around the cycles of this geometry and for the O minus one theories in chains with O minus four theories we had studied this and various other people had studied this, what kind of gauge theories you get. It turns out you get the product of SPN and OK gauge theories and so there's a concrete proposal about what is the corresponding string living on the D3 brain wrapped around our pre Riemann surface inside this geometry which is exactly what we needed to do to get the elliptic genus of the compact kalabia or the partition function of the black holes. So the description will be something like the potential duality would have been something like ADS3 times S3 times the base which in this case is T4 mod out by Z2 times Z2 being dual to some kind of a quiver in terms of O k's and n's where k's and n's depend on the cycle inside T4 mod Z2 times Z2 which you are wrapping. You know the information about the black hole fluxes which is going to be the fluxes given through S3 is encoded in terms of these particular n's and k's. Now one check is that we can compute the anomalies for this quiver theory. The 2D quiver theory we can compute the central charge of the left and the right and we can recover exactly the prediction based on the geometry. So the thing that we expected based on geometry that you should get the correct central charge C left and C right. You can recover them directly from this quiver theory so that's reassuring that we have a quiver which gives you the right central charges so at least the leading growth of the black hole should work. Without actually computing it we know the leading one should work. So we would be saying that the topological string on T6 mod out by Z2 times Z2 which is a partition function of black hole should be the electric genus of this quiver. It would be interesting to check whether this is just beyond it's just a leading behavior or it's actually exactly correct. There are hints that this cannot be exactly right and part of the reason it cannot be exactly right is that the quiver theory we write has more symmetries than the corresponding D3 brains. We have extra SU2 so we have we should have only an SU2 times SU2 but we have an SU2 times SU2 times an extra SU2 which is strange. So from the viewpoint of the holography you should have only SU2 times SU2 but we have one extra SU2. So that means there's some ingredient which is not quite right in this picture and so there's something that we need to understand better but seems like we are very close to some kind of a compact description and it'd be interesting to check the details of that. Anyhow you get expressions that you know we heard about yesterday in talks about what kind of expressions we get when we try to compute these. Elliptic genus for these quiver theories but I don't bother going over it. Needless to say it's not easy enough, well it's not easy enough to check it because the topological string on these has not been computed and it's not completely trivial to compute these either but anyhow so it'd be good to do some checks but anyhow let me move on. So for the next thing I want to take the following challenge. But here the story was for the elliptic Calabias of what the idea was. To try to basically stitch together conformal field theories using gauging of global symmetry so that's a summary of the 6D story I told you. How about 5D? Well if you take a generic Calabia the shrinking four cycles gives you 5D super conformal field theories they also have global symmetries and it might be that when you stitch them together similarly you can get the description of a compact Calabia based on these conformal theories in 5D. So in other words is there a similar story in that context? So let's talk about it turns out that the simplest one to talk about in this case is not the Quintic three fold but the mirror to the Quintic three fold. So you take the Quintic three fold and you take its mirror which means you take the Quintic itself and mod out by Z5 cubed. So as some of you may not know this so you take Cp4 a degree five equation in it and that's called the Quintic. And if you mod out all the coordinates by fifth root of unity subject to the determinant being one and up to projectivization gives you Z5 cubed and that's what's called the mirror Quintic. So if you take that geometry what is interesting about this has a lot of singularities. So typically the mirror Quintic people don't study it as much other than in the context of mirror symmetry is that it has many singularities. Z5 cubed these has a lot of fixed points and so on and these fixed points are quote unquote headache. You don't want singularities. Well actually not so for us. For us the more the singularities the better. Singularities are the bread and butter of super conformal field theories. So therefore we like singular manifolds. We like to singularize manifold as much as possible to pieces which are just singular because we understand singularities. They're conformal theories. So we want singular things as much as possible and that's why mirror Quintic is better than Quintic for us. Quintic is smooth, nice, there's no singularity whatsoever. So we can't have a handle. Singularities give us handles on Calabias. So we want to singularize as much as possible and this is as much as we can do. We mod out by Z5 cubed. It turns out for every three coordinates vanishing any triple of coordinates vanishing we get the conformal theory and so there are basically five choose three which is 10 conformal theories labeled by i, j and k and for every pair vanishing you get the curve of singularities with an A4 singularity. So for every pair you get an A4 singularity but in M theory that means SU5 gate symmetry. So you have SU5 gate symmetry for every double of these guys, pairs of these guys and for every triple you get the conformal theory. So what happens is like this. So for example look at this triangle here. This is one, two, five. One, two, five means Z1, Z2 and Z5 equals zero. That's a singularity which is a conformal theory. Z1 and Z2 equals zero gives you a curve off of it which has SU5 global symmetry which is global symmetry of this conformal theory but actually becomes gauged later on. So there's a one, two line, there's a five, two line and there's a five, one line. So in other words this conformal theory has SU5 cubed global symmetry. And similarly for all these triangles. So each one of these triangles gives you a conformal theory which is attached which has global symmetries which now again get gauged because it's compact. Just like the previous example. So you take each of these triangles which are well known conformal theories in 5D with global symmetries but now you gauge their global symmetries in a very particular way and this is I think the most accurate picture of, you have seen these fancy pictures of Calabria and so on. This is a real picture of the Meruquintic. I mean this is literally what it is and I like this because it's simple and nice. You can see where the ingredient is. You have conformal theories and they're gauged in this particular way. Okay so given this what is it good for? So we have put it into pieces. First of all are we sure this is the right picture? Well let's do some first count. So each one of these ingredients like this one to five is dual to web diagrams, PQ fiber and web diagrams which looks like this. With parallel lines going like this which give you SU5 global symmetries living on this and these are the three global symmetries I was telling you about. And every triple of them, so for example this SU5 you can think maybe of this one too is connecting three different conformal theories together. So one conformal theory from here let's say with the other one here, the other one here. Each one has SU5 global symmetry and you're gauging them. So some SU5 gauging of these global symmetries. And you're doing this for every pair in this particular way. We want to compute topological strengths for these amplitudes. First of all do we have the right degrees of freedom? Well there are 10 conformal theories because there are 10 of these triangles. Five choose three. Each one of them has six holes as you can see. And each hole gives you a degree of freedom the breathing mode. So there are six times 10 degrees of freedom corresponding to these conformal theories. Each SU5 has four relative cartons. But there are 10 SU5, five choose two. Four times 10. And each one of these SU5 has a coupling constant which we have to take to be identical. And that's an extra one giving us the 101 scalar parameters of the mirror quintic. So the older degrees of freedom of quintic, the scalar parameters are now captured by properties of gauge theories and conformal theories. We understand one has a coupling, one has the SU5 cartons, and the other one is the breathing mode of these conformal theories. They are actually meaning in terms of objects we understand. So it's divestifies the 101. But now we have to understand what the SU5 gauging is. There is a notion of U1 gauging that was studied a while back. And this involves, for these kind of diagrams, we use what's known as the topological vertex formalism to compute topological strings. But for gaugings, we don't have a similar formalism. Except for the U1 case, there was a proposed version of the topological vertex which you take the usual topological vertex and take it mirror. And it turns out to be the analog of the U1 gauging. The standard topological vertex will give you some expressions like this in terms of shore functions and so forth. And the mirror one for the U1 case was studied a while back by mathematicians and also by some physicists. And the question is, what is the SU5 gauging similarly to that? So you want some kind of an SU5 gauging in the story. And following work that Hayashi and Omori did, one can extend the U1 version to an SU5 version by some methodology, giving you some proposed formalism of what SU5 gauging is. We cannot be sure. It's kind of like a guesswork of what this gauging means. So you can try to generalize it kind of by hand what this could have meant. And this would be one natural proposal. So you can check if it works. Namely, let's take a case where all of these cycles are equal given by the scalar parameter gamma. And let's say that the size of this P1 corresponding to the SU5 is given by L. So we take all of these little sizes equal to gamma and all of these sizes equal to L and count how many curves do we get. If you count this for, if you just have Ls with no gammas, you can compute the various genus corrections and you can compute these would-be topological string invariance or more precisely the BPS invariance for any given genus. Or for the L and gamma, these are the same L and gamma we're talking about here. You can compute the degeneracy for the low degrees just by just taking the topological string computations using this SU5 gauging means to come up with this answer. Well, is this correct or not? Well, we asked our mathematician friends, Dave Morrison and Sheldon Katz to check for us. Surprisingly or not, this was not done before because there's too many parameters, 101 and nobody tried to do this. But thanks to our friends, they actually helped us. In fact, they did the computation after laborious work and they found the answers. This is what they got. Okay, well, yes and no. So you see the first row matches perfectly well, but that's not a surprise because that part is the same as the conformal field theory that I said we understand. So this was the easy part. It better have worked than it does work. The second line, you see some similarities. The order of magnitude is correct. The signs are correct, but not exactly right. Okay, so that's slightly embarrassing. What happened? Well, first of all, we didn't have to work because we were just guessing what SU5 gauging means. You can go back and try to think what SU5 gauging is and geometrically you can actually realize what SU5 gauging means in terms of delpetsos and their blown-ups. And so you can remodify what this rule means in this language and then armed with more understanding of what geometry is, do the computations again, refine it and get their answer. So you can understand how you correct your computation to get their answer, at least for the low degrees. We can reproduce it using this hodge part of localization, geometry, this and that. We do not know the full version of how you correct it to get the full answer for all of them, but this is a strong hint that it should be possible. So I think we are epsilon away, I hope, from an answer for what the compact collabia topological string is. We still have not gotten it there, but 25 years after the work with BCOV, I think we're almost in a place where we can say we can compute it. So we have to understand this SU5 gauging a little more clearly. So let me conclude. So we have seen that we can stitch together super conformal field theories in five and six dimensions and try to get a complete description of the non-gravitational sector of collabials. And we can do these for two purposes, for concrete holography, perhaps, and also all genus computation for topological strings and collabia threefolds. Leading growth works, but there are some hints that this is not the full answer. And there's one thing one should perhaps say is that these techniques should somehow be, of course, understood in terms of some kind of localization at the end of the day, which is the topic of this workshop, exactly what is not clear. It sounds like it might be the same sense as SYZ. There's a torus vibration, but not necessarily torus action. And so those kind of ideas might be some kind of a some kind of a suggestion of how it could be. Okay, I'll stop. Thank you.