 Thank you very much. So let me just say that it's a great honor for me to speak at this conference. And like everything, pretty much everything in geometry and physics in the last 20 years, this talk will be greatly influenced by work of Maxime and Jan. So this is about connection between Donaldson Thomas invariance and character varieties. So to be more precise, especially the motivic Donaldson Thomas theory developed as I said by Maxime and Jan, and a conjecture formulated by Haussel, Rolette Lié, and Rodriguez Viegas on the cohomology of character varieties for smooth projective curves with marked points. So for the first, I don't know, 15, 20 minutes, I'm going to just set up the stage and review this conjecture and try to write down the formula they proposed, which is a truly beautiful and remarkable formula. So let's set the stage. So C is a smooth projective curve over complex numbers. The genus is going to be G. We're going to have two marked points on C, P and infinity, P, of course, these things marked points. This is not a serious restriction. This is mostly for presentation purposes. I don't want to get tangled in too much notation. And gamma, P, and gamma infinity are the corresponding generators of the fundamental group of the punctured curve. And then we need to specify some more data. So we're going to have to fix a diagonal matrix. It's all complex numbers. I'm not going to keep writing complex numbers. Diagonal matrix and invertible diagonal matrix. Mu is going to be a partition of R. This is the partition that determined by multiplicities of eigenvalues of lambda. So we determine a partition of R. And now, finally, an integer E co-prime with R. So E is just an integer co-prime with R. So what is the character variety then? The character variety is going to be, roughly speaking, the motorized space of conjugacy classes of representations of this fundamental group into GLR. So we set C e lambda to be, as I said, the motorized space of homes from the fundamental group of the punctured curve into GLR, sending gamma P to the conjugacy class of lambda, and sending gamma infinity to a central element, 2 pi square root of negative 1 e over R. And all of this is modular conjugation. So this is a major object of study. The topology of this character variety is very, very interesting. Now, we have a theorem here. So in order to make the notation short, I'm going to refer to the work of Hauser-Lettelier and Rodriguez Viegas as HLRV, saying that for sufficiently generic lambda, C e lambda is smooth, is a smooth quasi-projective variety, of course, over complex numbers. And furthermore, a consequence or empty, possibly empty. But if not, empty is smooth and quasi-projective. And if you compute this dimension, you find out it's some formula where it depends only on the partition mu. So it has some formula I'm not going to write down, but it's only dependent on the partition mu. All right, so now what are we going to study in this context? It's some kind of weighted Poincare polynomial of this variety. So what I want to study, I mean, what I would just say, HLRV studied, was they looked at, since this variety is non-compact, it has a non-scrivial weight filtration and comology due to Dulin. And they studied the, so we have the comology of this variety. C has a weight filtration, W. And you can define, I mean, I'm sorry, the compactly supported comology, I wonder, it still has a weight filtration. So we can define some kind of weighted Poincare polynomial by the following formula, p compact of this. It's a function of two variables, formal variables u and t. It's the sum over dimensions of the graded pieces of the weight filtration on HK compact weighted by this monomial, which looks a little bit funny because it looks like I need to take a square root of u. But then furthermore, but in fact, this turns out to be a polynomial. So there are further results, again, of HLRV, saying that this is in fact a polynomial of, so this PC, it's a polynomial of u and t. So only even i occurs in the formula. And furthermore, it depends, again, depends only on mu, so on the partition mu. So a lot of data that I specified before doesn't actually enter in the, I mean, there's no dependence on the specific values of the eigenvalues, for example. So we can call it, so we can denote it, we can just write p of mu u t without any ambiguity. OK, so now finally, let me get to the point of writing this conjecture. They formulated it. So the question was to determine all these polynomials, for all possible values of the rank and all partitions mu, and they wrote a beautiful formula, which I want to write down here and leave it on the blackboard for the rest of the talk. So this is the HLRV conjecture. So it's written in a very, which is in a way that's really very, very suggestive to a physicist. It's written like saying that the partition function of some system is the exponential of a free energy. Well, I'm using these names. They certainly did not use, you did not use this terminology, but for a physicist it's very natural. So first of all, let me write this here. So what appears in the exponential, it's a plateistic sum. So it's a sum over all partitions. It's a sum over k greater or equal to 1, 1 over k. And then we have what we have here. And let me write the sum in here. W to the minus k d mu, z and w are going to be some formal variables, purely formal variables. So the k divided by 1 minus z to the k, w to the 2k minus 1, m mu of x to the k. OK, z, w and x are formal variables. x is actually an infinite set of formal variables. The m mu's are the monomial symmetric functions. So this is what appears in the exponential. Is the dependence on genus? Oh, I'm sorry. You're absolutely right. There is dependence of genus in here. Oh, no, it's because I just forgot the p. I apologize. Sorry, I need to have a genus. And the polynomial depends on the genus, certainly. I apologize. This was just an empty formula what I wrote there. OK, so yeah, so it depends on g. OK, so yes, it depends on g. OK, good. So this is one side of the formula. And then what is the other side? The other side, it's the way it's the monomial symmetric functions associated. m mu is the monomial symmetric function for the partition mu. And x is just an infinite set of variables. So you just imagine x is just x1, x2, dot, dot, dot. So it's all formal variables here. All right, so this is one side. The other side, quite remarkably, the other side is just a pure combinatorial object. It's a sum of generable partitions, but other partitions. I mean, well, the same partitions, but they have a different nature. So there is a rational function I'm going to write in a minute associated to any partition. And then it's weighted by modified McDonald polynomials. So I'm going to explain all these quantities here. First of all, this is easy to write. It's some rational function, which is a product over all boxes in lambda of z minus 2a plus 1, which a is the arm length of a box, minus w to the 2l plus 1 to the 2g. L is the leg length of a box divided by z to the 2a plus 2 minus w to the 2l, z to the 2a minus w to the 2l plus 1. So indeed, as you see, it's explicitly dependent on g. So let me just write this in words before I explain what these functions are. So h lambda twiddle is they are called the modified McDonald polynomials. Now, I know very little symmetric function theory, so I couldn't give you a definition of that function over there. Fortunately, I don't have to give you a definition in symmetric function theory. What I am going to do is give you a geometric construction for these functions, which is due to Heyman. So for our purposes, it's much more illuminating to think of them to employ the geometric construction for this modified McDonald's, which is the result of Heyman. So it involves the Hilbert scheme of points in C2. It's sufficiently generic here. How effective is it to complement finally many points or many points? There is a precise condition that I'm not remembering, but there is a very specific condition here. I think it might be a complement of high dimensional subspaces, not just points. Yes, it's explicitly given in the paper. OK, so now let's discuss a little bit the modified McDonald's functions, because this is going to be important for what I have to say next. All right, so now this is, as I said, according to Heyman. So we're going to look at the following diagram. This is the symmetric product S of C2. This is the Cartesian product, and this is an n factorial 2-1 cover. Now here we have the Hilbert scheme, hn of C2. And this is the Hilbert Chao morphism, collapsing whichever cycles there. And now we complete the square, and we get what's called the Isospectral Hilbert scheme. So this is the Isospectral Hilbert scheme, which is, again, it's generically an n factorial 2-1 cover, but something pretty bad happens over the non-trivial fibers of this morphism here. And in general, this is, as the scheme is not even reduced, but what Heyman proved, so there is a structure theorem, what he proved is that this H-twiddle and C2 reduced, if we take with reduced scheme structure, it's irreducible and Colin McCauley. And that was very important, because what he had to, he did next was to take a push forward of the structure here down here. And if you endow this with the reduced scheme structure, what you get down there, it's a nice vector bundle, which is known under the name of Prochese bundle. And that place, that's really the main geometric ingredient in the construction of the McDonald polynomials. All right, so now let's let P, P is the push forward of OH and twiddle reduced. So now this turns out to be a rank n factorial vector bundle on Hn. And not only that, but it carries a fiber-wise action of the permutation group of n letters. So it's an action of Sn by naturally, by construction there. All right, so now the other thing we need is there is not a natural action of a torus on the Hilbert schema point, which comes from the just the skating action on C2, the diagonal skating action on C2. And it's very well known that if we classify fixed points here, what we get is a bunch of monomial ideal labels labeled again by partitions of n. All right, so now to get this McDonald polynomial, what we need to do is look at the fiber of P over a fixed point there, and look at it. And there is a simultaneous action of Sn and an action of T, and they commute again naturally by construction. And what we get, what we learn then, is that if we restrict P to a fixed point, this decomposes into a sum of characters, obviously. The compose over a sum over partitions. Let me write this like this, and I'm going to explain the notation. These are the irreducible representations of the symmetric group. And these are just representations of T. So it has a decomposition like this, which is OK. It's all natural. All right, so now finally, what is the McDonald polynomial in this context? Let me write this is a theorem. Maybe I can write it here again due to Heyman. We have the following. I mean, this is a hard theorem. This is hardly non-trivial identity. Although me being a physicist, I'm going to take it basically as a definition for McDonald polynomials, in my case. So it's given by this following formula. q1 and q2 are the natural generators for the character lattice of this guy, of this torus, algebraic torus. This is the character, and these are the true functions. So as I said, this is a hard result. It has a complicated and hard proof. But this is, for us, the most natural definition for this. I mean, it's not a definition, but for the purpose of this talk is going to be a definition. All right, so it's very intriguing that indeed they appear, these functions appear in this particular formula. And I should say that the way they arrived at this formula was by counting. So what they did was to formulate the problem over finite fields, because the character variety is defined over finite fields. And they counted rational points. And what they did is that eventually using them, the real conjectures, they arrived at this formula. And they even proved a specialization of this formula, which I'm thinking sets z equals w inverse or something. So that is proven. That's a theorem. Just a second. If you can't write your final field, you can count on the weight, but you cannot distinguish between conmology. Exactly what? That's what they proved, the specialization of this formula. Yeah, specialization where z equals w inverse, I think. That's exactly what happens. A turn of the conmological degree. Exactly, yeah. And that's proven. But then based on some symmetries and some they conjectured this. All right, so now, as I said, me being a physicist, when I saw this formula, it looks very, very intriguing and very elegant. So I was thinking, how can I get this out of string theory? Is there any way of deriving such, not proving it, but deriving it or explaining it from string theory? And especially, how can I explain the occurrence of these guys, of the McDonald polynomials? And how McDonald polynomials are proved? Well, in their proof, what they get is a specialization of these McDonald polynomials. Because, you see, that's what I'm saying. They collapse with something much simpler. So only when you write the full conjecture, then you get this two-parameter family of functions. All right, so. OK, so the way this goes is, by first, employing another piece of mathematics which relates character varieties to the modulized space of Higgs bundles. And in this particular case, it's going to be Higgs bundles, parabolic Higgs bundles on the curve. It's not a piece of beautiful mathematics which goes back to work of Hitchin, Donaldson, Simpson, which tells us that there is a deep relation between character varieties and the modulized space of the Higgs bundles. And in this particular case, we get a special flavor. I mean, we get a flavor of Higgs bundle, which has parabolic structure. So I want to say a few words about to bring the Higgs bundles in the picture. And the p equals w conjecture. This is a conjecture formulated by Cataldo Miglorini. Oh, sorry, Halzel and Miglorini. But it really relies on previous work, as I said, of Hitchin, Donaldson, and Simpson, which tells us basically that as a real manifold, this character variety we had with defined, it's actually canonically identified to a modulized space of stable Higgs bundle. And in this particular case, parabolic Higgs bundles on C. So I'm going to call it H, let's call it H e lambda of C from Higgs. Hopefully it's not going to be too much confusion between this and this H and that H. Maybe I should call it Higgs to make sure that Higgs is the degree. I haven't told you the map between character variety that and Higgs bundle. But e is going to be just the degree of the bundle. So as I suspect, most people in the audience know a lot more about this than I do. Just to fix notation, what I'm talking about here is pairs e phi where e is a vector bundle on C and phi is a, so we're talking about pairs e phi where e is a vector bundle on C, phi is an endomorphism with values in a twist of Kc by the point P. And I'm just looking at a single point for simplicity. And what we need to know is that we have to specify, we specify a flag of type mu at P, sorry, in the fiber, in the fiber ep. And then we also have, we have some parabolic weights. Well, OK, I'm being a little bit sloppy here, but I just don't want to get bogged down in details. What I am going to do here, basically, is take a special, I'm going to take the lambda to be something like a diagonal of pure phases, e to the 2 pi i square root of minus 1 alpha 1 and so on. And the parabolic weights are going to be just these numbers that appear there. So it disqualifies as sufficiently generic according to the definition. I mean, I can make this sufficiently, I can make this sufficiently generic. And from that, you should also order some terms of partition. You're absolutely right. I'm being really sloppy here. Yeah, what you have to do is take the first multiplicity of the first guy, and that's the dimension of the first quotient, and so on. The fastest one. Yes, so I'm being sloppy. I could go through the details, but yeah, I should say the type ordered mu with the natural order given by the order of eigenvalues. And then maybe that's it. And then there is a notion of slope, parabolic slope, and the notion of slope stability. It's all standard. And you get, you have slope stability for these guys. And you get the smooth-modelized space, which is this Higgs-modelized space, e lambda c. So this is, again, it's a smooth quasi-projective variety. And as I said, it's, as a real manifold, it's identical to the correct variety. But the complex structures, the algebraic varieties, a variety of structures are very different. And that leads to this highly non-trivial conjecture formulated by Cataldo, Hauser, and Migliani concerning what is the interpretation of the weight filtration on the Higgs side. So that's the issue. And again, without getting into too much detail, because this is really an embarrassing moment. I mean, most of the audience knows a lot more about perversives than I do. What they do is to define a perverse filtration on the homology of the modulate space of Higgs bundles using the decomposition theorem for the Hitchin map. So the story is that there is a Hitchin map to basically a vector space, which is obtained by taking the polynomial invariance of the Higgs field. So this maps to, by map H, to some vector space, which I'm not going to try to write in detail. And this is the map which associates the phi trace of phi to the k, a vector polynomial invariance for 1 smaller than k smaller than r. To e phi, but yeah, you just take the polynomial invariance of phi to the k. And in this complex structure, this is a holomorphic. It's an algebraic map. And using the decomposition theorem for this, we obtain a perverse, so-called perverse sheaf filtration on the homology of the Higgs and the modulate space of Higgs bundles. And this is going to be called P. And the conjecture is that, basically, Pi, if you set up your conventions correctly, which is a fairly subtle issue here, so set up your conventions, Pi equals W to I for all I. As I said, this was formulated by Cattadoe, Hauser, and McLean. And they proved it for rank two with no parabolic structure. So that's even there, which was a hard proof. So it's proven for rank two with no parabolic structure. OK, so these are the ingredients. So this is all a math story. So now what I want to do in the remaining, I don't know, most 15, 20 minutes maybe, is how this all ties together and all the pieces click together in the context of Motivic Donaldson-Thomas theory. So what is the plan here? Well, we want to have. Do they show that this conjecture implies the formula? The conjecture certainly implies the formula. No, no, no, no, no, no, no. No, I apologize. No, the conjecture doesn't imply the formula. The conjecture basically gives you another conjecture for where the dimensions of the W-graded pieces are replaced by the dimensions of the P-graded pieces. The conjecture doesn't imply the formula because this side of the formula came from, as I said, by counting points and then extrapolating from the point count. And on the Higgs side, it was checked for rank three by direct localization on the Higgs side, by just fixed point theorem on the Higgs side. It was checked. What was checked? What was checked was the prediction obtained from there for the Poincare polynomial. That can be checked. But for higher rank, it's getting really hard. So let me just, so what's the plan? We want to obtain some kind of derivation of the formula. Of the formula from, well, if you are a physicist from string theory, if you are a mathematician from Donaldson Thomas, or as you say, Motivic Donaldson Thomas theory. So Motivic DT theory. So I should say that from the outset, this is not going to be approved. But basically what I'm going to manage to do is reduce that formula to some sort of standard conjectures in DT theory. So I don't know how much time I have to explain, but that's going to be, so it's going to follow from some fairly standard conjectures in DT theory. But it's not, it gives you a framework for the proof. So basically it tells you when those are proven, then this is proven too. But OK. So all right. So now what's the first step? Well, it's just we're talking about Motivic Donaldson Thomas theory. So what we need is to find a collabial threefold lurking around here. So far, I've been on a curve. So everything was done on a curve. So where is the collabial threefold in this picture? So the first step would be construction. So what I want to do first, step one, we need to construct a collabial threefold x, for example, such that this modular space of Higgs bundles is going to be isomorphic to a modular space of Bridgeland stable, stable pure dimension one sheaves, pure dimension one sheaves on x. Now this looks kind of a loose statement, because I mean, how can you collabial? There are lots of collabial threefolds. But fortunately, there is here we can rely on some results obtained in the literature by, so this was born Greschenig, which basically tell us, give us a machinery of identifying parabolic bundles on C are identified with just ordinary vector bundles on a root stack, which I'm going to call C twiddle. So basically what's happening, informally, of course, you can immediately write down some rigorous formulas. You take this point P here, and you excise it. And what you glue in is a stack, basically, which looks locally like C mod mu L. You insert in a stack a point where L is just the length of the partition mu. How many parts you have in mu? So you insert an ugly form point. And what you get is a smooth ugly fold. It's a smooth little amount for stack, which furthermore projects naturally to C by some map row. And this is going to be a coarse-modulized space. I mean, C is a coarse-modulized space for C twiddle. And then there is some work I don't have time to review in detail, which allows me to reformulate the modulized problem for parabolic Higgs bundles on C as a modulized problem for just ordinary Higgs bundles on the root stack. And from there, the next step is to come up with a collabial three-fold. So coming up with a collabial three-fold, it's fairly easy at this point. It's kind of disappointingly easy. What you have to do is to take a fairly naive construction, which is to set x is the total space of O C twiddle plus K C twiddle, which is, of course, I lied a little bit. This is not a smooth collabial three-fold. It's just a smooth ugly fold. This is going to be a collabial ugly fold of dimension three. And that's the setup. And then you can easily prove that. What you can prove, it's not quite that. What you can prove is that if you cross with A1, then the modulized space of Higgs bundles, it's isomorphic to the modulized space of dimension one sheath. But this is pretty harmless. I mean, this is pretty harmless there. All right, now, what we have is a geometric setup. So we lifted the problem from a curve to a collabial three-fold. So this is what physicists say, well, it just increased dimensions. So now what's the second step? Now, for physicists, I should say, once you get to this point, this modulized space of pure dimension one sheaths and so on, these are just d2, d0 bound states in this all-before background. So we managed to reformulate Higgs bundles to identify them with supersymmetric d2, d0 bound states. Now, the second step is to imply, now we're looking at the motivic dt theory of x, as I said, of Konsevich and Swabermann. And more specifically, we're going to look at specific objects in the derived category of x, which are the stable pairs considered by Pender, Rependi, and Thomas. So from Pender, Rependi, and Thomas, we know there is a counting theory for accounting theory for, let me write upwards, for stable pairs, which means our pairs o, x to f, where this guy is pure dimension one sheath, and s is generically surjective. I'm not going to, I don't have time to get in too much detail. The construction of Pender, Rependi, and Thomas produces a modulized space with symmetric virtual abstraction theory, so with symmetric, sorry, symmetric virtual cycle, and then you can get numerical invariance. But then using the machinery of Konsevich and Swabermann, we can actually get motivic invariance. We can get motivic invariance for such objects. And then from motivic invariance, I'm going to just specialize the polynomial invariance by taking this, so-called, a CERC polynomial. So this is, as I'm summarizing, a lot of mathematics in a few words, but I don't have time for saying more. I didn't want to erase this one, sorry. OK, whatever. So from Konsevich and Swabermann, what we get is we get a motivic stable pair invariance. And then you specialize the polynomial by taking the correct polynomial specialization, which is compatible with motivic decompositions. So we get polynomial invariance. And finally, we're getting some, we can get a partition function, which I can call Z, Pt. I should say that the polynomial invariance are called refined in physics. We also saw this refined BPS states occurred in Sergey's talk. Yeah, they are very much related. It's just the background. This is different. So we get some partition function like this. And yeah, so we get some generating function for such, for polynomial invariance. And OK, so now the main conjecture we formulate then. The main conjecture states the following, that this side equals this side of the HLRV formula up to a simple, a monomial change of variables. So it's not quite coming in W and Z. I have to take some combinations, but really easy combinations like Z squared and Z over W or something like that. Nothing complicated. So this is the main conjecture. This is one of the conjectures we have. Oh, man, I forgot. I should say this is all work with. We end Chuang, Ron Donaghi, and Tony Pantev. So this is pretty much, yeah, this is work done together with we end Ron and Tony. And one of the main conjectures we formulated is this one. It's to identify this side of this identity as a refined with the partition function. All right, now, what will this buy you? Well, if you combine this with the other identification of the modular space of Higgs bundles, what you realize then by staring at the formula is that, in fact, the whole HLRV formula, so what this buys you is that, so what we get from here is that if we get a diagram of our identities like this, this is the exponential of HLRV. And then we have a change of variables here and the change of variables here, the same change of variables, which identifies this side with a refined Donaldson-Thomas partition function here, or PT partition function. And this one, it's identified with a refined Gopal-Kumar-Vapha expansion. So this is refined Gopal-Kumar-Vapha expansion, which appeared in many places in physics. So this appeared in, well, in some ways, implicit in the original work of Gopal-Kumar-Vapha, Katz-Klem-Vapha, then it was written explicitly for Tori-Kulabi-Au-3-fold in Iqbal-Koskazin-Vapha, and more recently considered by Katz-Klem, and sorry, I forgot the full reference. So anyway, so it's considered in many places in physics. In this particular case, it's slightly exotic because we use this expansion for a non-Tori-Kulabi-fold. But anyway, the message here is that this reduces to, and this is one of the standard conjectures in Donaldson-Thomas theory. I should say that this is a, for mathematicians, this is a refinement of the strong rationality conjecture, conjecture of Pandare-Pandain-Thomas. So, OK, so it has a well, it has a good, it has its own place in Donaldson-Thomas theory, but to my knowledge, it's not proven, at least in the refined form. OK, so this is what you get from here, and then perhaps I'll finish in five minutes. The last thing I want to say is that what's the evidence for this conjecture we formulated? So there is two types. I mean, there are two things that physicists can do to get evidence for conjectures like this. One of them is the recomputations. And we were successful in doing some of the recomputations to check this for rational curves with a single marked point. So that was done by localization using, well, using further conjecture, further a k-theoretic index of Nekrosov and Okunkov. So that's one thing, so we can get evidence by direct computations. But we can do something perhaps more conceptual, at least from the physical point of view, which is try to use string theory dualities. So we can train this particular case, M type 2A duality. And this is a sequence of dualities which is now in the physics literature of Geometric Engineering. It's Geometric Engineering. So there is a long history in physics, which I don't have time to get into. But there is a mathematical formulation of this sequence of dualities which occurred in the recent paper of Nekrosov and Okunkov. I think, to my understanding, it's still conjectural. Part of the paper is still conjectural. But there is a mathematical framework proposed by Nikita and Andrei concerning about this duality for this duality. So how it goes is as follows. I'm just going to sketch it in a few minutes. I'm not going to give you the punchline. And the good news for us is that, although perhaps it's not completely rigorously constructed, it does explain the occurrence of McDonald polynomials. So it goes like this. So this is, as I said, using recent work of Nekrosov and Okunkov. So I'm going to state it in this particular case. Day work is more general. But I only need it in this particular case, which now we take a five-fold that's going to look like this. OC twiddle plus OC twiddle plus OC twiddle plus KC twiddle over the root stack C twiddle. And now this is a five-fold. And what they do is they conjecture the existence of a counting problem for membrane quantum states in M theory, which is a counting problem formulated on the five-fold. So there is going to be some counting, so-called M2 counting problem, which is in full rigorosity is going to be a very hard theory, I believe. But let's accept that that exists. So there is some counting problem. And what now they do, what we can do using this, we notice that there is two torus actions here. There is a C star 1 and C star 2, both of them anti-diagonal. So there is two torus actions. And what you can do when you have torus actions like this, you start with this complicated counting problem on the five-fold. And using localization, you can reduce it to a three-fold problem. And if you use localization for the first torus, what you get is the dT slash pT theory of our x. It's a long story, but I'm being schematic here. And now if we do localization for the second torus, well, what we get is something like pT theory on for a non-colabial three-fold, which is just C twiddle cross C2. So you get a pT. It's all equivariant. It's a long story. So pT theory for this non-colabial three-fold. And now if you analyze this theory to employing some physics tricks along the way, from here, with a lot of work, you can actually show that the partition function you get here, you get from this theory here, it's basically, again, given by this side of the formula up to change of variables. And the way this is done is by it goes, this arrow here goes exactly through to Heyman's construction for McDonald's polynomial. So from here, you get very naturally McDonald's polynomials using Heyman's results. But I really don't have any time to go into detail here. It's a fairly long computation. And it's quite remarkable that at the end of the day, going through this five-fold theory, that's kind of mysterious. Everything gets all parts clicked together. And I think you're convincing evidence for this conjecture. And I think it's a good time to stop. And thank you very much. And I want to wish Maxine a happy birthday. And the question, where is the character of a right? It's the last part of your talk. It's the one about Higgs bundles. Yeah, so it got replaced by Higgs bundles. And the W filtration got replaced by the perverse filtration. The filter level high. You're assuming Pw. I'm assuming. I'm sorry. I should have stressed it. Yeah, I'm assuming Pw. You're using that, actually. You know, I found another one. And how you use it? Well, when I. Your filtration has. Well, it's, yeah. Well, you see, there is further. OK. Yes, it's part of this refined Gopal Kumagwafa expansion. Because there is, even in 2000, when the Gopal Kumagwafa appeared, there was work by Hosono, Saito, and Takahashi saying that the GPS, the GV, BPS invariance should be identified with the dimensions of the graded pieces in the perverse filtration on the homology of Higgs. Well, so this is a natural interpretation. In this side, it's a natural interpretation for GV invariance. Of course, their claim is true, basically, for smooth spaces. But under the circumstances, this is a smooth space. I mean, the modernized space is a stable Higgs bundle. It's a smooth. So in this case, we can actually employ their definition and think of this as a mathematical definition for GV BPS invariance. It's very much in agreement with the intuitive picture of Gopal Kumagwafa. Because if you think how you compute the graded pieces here, you have this left-shits action by the vertical polarization. And that's really what it's in sync with physics. When you use necrosis, they have a very unpopularization. How are you comparing them? No, that's a further step. Now, at the end of the paper, they also talk about refined invariance. You have to upgrade. Yes, I gloss over a lot of stuff. I agree. But yes, it's kind of you have to, in particular, they have this definition of refined invariance as a case-erratic index. So it's not defined. It's identified with some case-erratic index. I think you have two more questions. All right, so that works. Thank you, Manuel. Thank you.