 So, we had two lectures on essentially building up to understanding sheaves, their defamation theory and virtual cycles. If I lost you, which I'm sure I did, just take that now as a black box. There's this virtual cycle which plays the role of the fundamental class of the moduli space, has good properties, if you perturb things, it stays the same, it's defamation invariant. It's the correct fundamental class for this moduli space. Now we're going to apply it in Vaffa-Witton theory, but we had Lota's lecture, right? So maybe I just skip this and go on to the next lecture. No, okay, we'll go through it. So there's this paper hundreds of years ago by these guys, and Donaldson even gave it to me in my thesis during my PhD and told me I ought to think about making mathematical sense of it. It was around the time of Donaldson theory, well, it was around the time of Cyborg-Witton theory rewriting Donaldson theory. So at the time, there were all these gauge theories in physics which were leading to really exciting mathematical, rigorous mathematical invariants like Donaldson invariants. And the idea was, could you do something with this? And for 25 years, the answer was no, fundamentally because there's no natural way to compactify the moduli space here, and therefore get invariants. Okay, so very briefly, you're not meant to follow this, don't take notes. This is a theory given for a real four manifold, like a Riemannian four manifold. You pick auxiliary data, a bundle over it, and then you're supposed to count solutions of these gauge theory equations. And you're not meant to really understand what they are, but just roughly, you've got something here which is the self-dual part of the curvature, which is what's relevant to Donaldson theory, which is what's relevant on a Keiler surface to, via the Hitchin-Kobeashi correspondence, or Donaldson-Ullenbeck-Yau theorem. It's what's relevant to stable bundles. And then you've got some additional fields, which we'll incorporate in one field later on and call the Higgs field. And then you're supposed to count solutions of these equations. And that's the tricky bit, how to do that, but let's assume you could do that. Then there's this voodoo in physics which says that when you form a generating function of Fourier series or something of these invariants, these counts, then you should get modular forms. And so in particular, this sort of infinite collection of numbers should be determined by only finitely many numbers. And so in the Keiler case, so we still can't define Vapour-Whitney invariants, I should say. But there's been some very recent progress of some physicists defining them in other ways, using TQFTs. And now more progress on really using the equations and almost complex structures. So there's some hope now that maybe one day we'll be able to define these invariants. But what these lecturers are about are defining them in the algebraic case or the Keiler case. And so the problem is that the modular space is inherently non-compact, these fields can grow. And you'll see that better in the algebraic case in a minute. So from now on, I'm going to work on a Keiler surface or an algebraic surface from the next slide onwards. I'll call it S, so that's my four manifold. And then you can package these fields in a certain nice way using the splitting of two forms on a Keiler surface. And you end up with these equations. So this is the intergrability condition, which says that your connection defines a holomorphic structure on your bundle. And then this is a moment map equation, which tells you how to fix the metric on that bundle. And then this says that your Higgs field is holomorphic. So you end up with this data. Okay, so again, you're not really supposed to be paying too much attention yet. And then there's a hitching Kobayashi correspondence. So people have proved that the solution, at least in the algebraic case, that the solutions correspond to the following data. So this rather linearizes the problem and simplifies it. So this is where you have to start paying attention. So this is what the Waffa-Witton equations are for us. So they're the data of a vector bundle over our complex surface. And a Higgs field, there's this twisting by the canonical bundle of the surface. The Higgs field should be trace free and the determinant of the bundles should be fixed. And then there's a stability condition that ensures that you can solve this equation here. So you get rid of this nonlinear equation by seeing it as a moment map. This is annoying. So you see this middle equation here as a moment map equation and then you can solve it so long as this holomorphic data satisfies a stability condition. Okay, and the stability condition is the same slope stability that we used in the first lecture, except this Higgs field here modifies it so that instead of testing via all sub and quotient bundles of E, and you're testing their slopes as we did in the first lecture, you only take those invariant by phi. So you take phi invariant sub bundles of E and you check that the slope is less than the slope of the quotient. And that's the stability condition, okay? And yeah. The previous notation, the previous, yeah. Is it omega one one? I think this component on this, okay? Is it? Here? Is it omega one one? What would omega one one mean? One one comma one, this one. Yeah, what would that mean? So you decompose this, so originally this is kind of omega one one. So omega here is the scalar form. So it's already a one one form. So this wedge here is a two two form. Okay, so yeah, ignore everything I said. Here's what we're interested in algebraic geometry. Analysts have done a hard job for us and now we can forget about the equations. And we can reduce it to studying stable Higgs pairs. So there's the bundle field, endomorphism, twisted endomorphism, satisfying a stability condition. And one of the kind of insights that this work has produced is that you shouldn't, bizarrely, you shouldn't take slope stability. You should take Gizika stability because otherwise it's all a mess and you get the wrong answer. So that's the first thing we'll do is we'll replace slope stability by Gizika stability or semi stability. And the second thing is this means we can partially compactify things by allowing E to be a torsion free coherent sheaf instead of a vector bundle. So that will partially compactify the modular space, which gives us more of a chance of defining an invariant. But you'll still have non compactness because you can scale this phi. If you just scale phi, if it's non zero, then you'll get another solution. Okay, so exercise when the degree of the canonical bundle is negative, stability, that stability condition I told you forces the Higgs field to vanish. Okay, I mean very roughly it's a map from a vector bundle to a more negative vector bundle and so you, by taking kernels and co kernels and so on, you can prove this rather easily. So Vaffram Witten have a vanishing theorem that essentially when the curvature of the manifold is positive in some sense, then you get this vanishing theorem. All Higgs fields, those Bs and gammas and whatever you vanish and you get reduced to the anti-self dual equation considered by Donaldson. And this is the complex analog of that. Okay, so in this case, the modular space is compact and you can do something. But otherwise it's tricky. Yeah. What do you mean by partially compactify? Partially compactify. A slightly bigger space, which is a bit closer to being compact but isn't compact, I don't mean anything, ignore it. See what the meaning of the zero is? Home ee, ee. Yeah, trace free. Trace of phi should vanish. But a trace is a section of phase. Yeah, that's right. Okay, so what we're going to consider actually is the moduli space. Ah, so yeah, so when this vanishing holds, then what we end up with is a moduli space just of stable sheaves. So this is what you get in Donaldson theory on a projective surface. With fixed determinant, maybe trivial determinant. And then what you find, as we'll see in the next lecture and as Lothar mentioned, is that the Vaffa-Whitney variant is some kind of oily characteristic or virtual or the characteristic of this moduli space. And there was a lot of work starting back in 1994 and going on about computing these numbers and seeing modular forms and so on. But really what this lecture's about is the other components when phi is non-zero, but we'll get to that. The triviality of a determinant bundle force from how do we consider SUR a bundle force? Yeah, but you don't have to do it. Later we'll fix the determinant to be just a line bundle. But it's sort of, this is to do with the Lie algebra of SUR and this is to do with the group SUR. So here you fix determinant and here you take trace free. Right, this is where I really want to start the lecture. So this is the cool thing. Those of you who know about Higgs bundles on curves or hitching systems or something will know this. The rest of you won't know this and you're really missing out. It's just a beautiful piece of mathematics. It explains what matrices are. So pay attention. Okay, so we have this data and here's what we're gonna do. We're gonna turn it into data on a bigger space on the canonical bundle of S, which we call X. Okay, so this is a Kalabiya three-fold. So this data is equivalent to a certain torsion sheaf. So a two-dimensional sheaf supported on a surface called the spectral surface, which is some kind of multi-covering of the surface itself. Okay, as I run out here with this picture, the vertical lines are the canonical bundle of S. And roughly speaking, this torsion sheaf is a line bundle on this spectral surface. Okay, so for now this is just the rough idea and then we'll do it rigorously. So what is this? Roughly speaking, the red points here, let's work on one fiber. Okay, so one point of S. So what do you have? You have this endomorphism, phi. So what you do is you plot its eigenvalues. This is the rank three case. You plot its three eigenvalues in KS. There's always this twist by KS. And what do you put over those eigenvalues? You put the eigenspace in E, okay? So there's an eigenspace in E, so that's some line in E at this point. Over there, there's one over there and there's one over there. And then you put it all together in a family and this gives you a line bundle over this surface, all right? That's somehow the generic situation. In reality, this surface might be some thickened version of S or it might just be S itself with a high-rank bundle on it instead of a line bundle. So it's more complicated, but that's the rough idea. So any questions about that? Okay, so next we're gonna do it rigorously. But first we're gonna do it over a point. So we're gonna understand a vector space and an endomorphism, all right? So we're gonna understand what matrices are. So we fix a vector space and an endomorphism. And so I'm always working over the complex numbers. So V is a C module. But now we've just made it into a CX module because X or a C phi module. We just let X act through phi. Phi commutes with itself, right? So this is a commutative thing, it's really a CX module. Okay, but we know what CX modules are. They're sheaves on the affine line on spec CX. And the fact that V is finite dimensional means that the sheaf has finite dimensional sections. So it's really, it's a torsion sheaf. It's only supported on a finite number of points. Otherwise it would be an infinite dimensional module. Okay, so we get a torsion sheaf supported. Maybe exercise, check you're happy with this. That it's supported on the eigenspaces of phi. Okay, that's why spec is called spec. It's spectrum, it's to do with eigenvalues. The eigenvalues, the points of spec CX are the eigenvalues of the operator X. Okay, so exercise, so this is really worth doing. This is really, you know, exercise about matrices. So here's a silly matrix, multiple of the identity. That corresponds under the spec correspondence. Check you're happy that it really corresponds to two copies of the structure sheaf of the point lambda. So the, of the eigenvalue. Okay, whereas if you take this Jordan normal form, what does this correspond to? So really check what module this gives you over CX and what sheaf that corresponds to. What you should find is it's very similar, but it's not the same. It's the structure sheaf of the thickened point two times lambda, the double point at lambda. It should actually be supported on eigenvalues of five, right? Not as in spaces. Oh, sorry. Eigenvalues, thank you. Yeah, I'll correct that, thank you. Okay, and more generally, another exercise. This explains the difference between the minimal polynomial and the characteristic polynomial. The modules actually supported on the zeros of the minimal polynomial, scheme theoretically. But it's divisor class is given by the characteristic polynomial. So I don't have type, so ask someone, if you don't know what the divisor class of a module is then ask someone. Or this is also called the fitting support or fitting ideal or yeah. But I can only go through so many things. So I'm not going to go over that. There's very not a pretty story. Someone could explain it to you very quickly. Okay, any questions? Yeah. What do you mean by generalized eigenspaces? Oh, the generalized, sorry. A generalized eigenvalue is something that, so is lambda such that five minus lambda to some power is zero. Not just five minus lambda is zero. And then the eigenspaces are the things that are killed by this five minus lambda. Did I say that right? No, I didn't say that right. I said that wrong. I thought you were waiting for me to get fixed. So lambda is a generalized eigenvalue and V is a generalized eigenvector. If five minus lambda times the identity V, but maybe you have to do that n times is zero. That's the condition. Okay, so now we want to do this rigorously and globally. So globally what we do is we take our sheaf E on S with its Higgs field phi, okay? And we make it into a pi star OX module. So we push down the functions down the fibers and we get this. This is just a polynomial, you know, always think over a point. This is just a polynomial ring in X. And you make it into a module over this by using phi to the i. That's how you describe the action of this guy, this graded ring on E via this here. And obviously it's all commutative because phi commutes with phi to the i. Okay, so what you end up with is a pi star of OX module. And so what is that? Again, via the relative version of spec. That is a sheaf, in fact. Yeah, it's a sheaf over X. And it turns out the stability condition I told you about is equivalent to stability for this sheaf, okay? So we're interested in, I'll just say it. It's not so important. We're interested in Gizika stability for E phi. That's the way you look at the reduced Hilbert polynomial of phi invariant subsheaves. That's equivalent to the Gizika stability of this curly E phi, the spectral sheaf, which is to do with the reduced Hilbert polynomial of subsheaves on X of E phi. So stability matches up and I won't go into the details. All right, so exercise. I really advise doing this one. This is nice. So describe a Higgs pair on just the affine line. So this is really one dimension down. This is really sort of hitching theory. With spectral curve Y squared equals X. So you really want to see this branching. This non-trivial branching. And it's non-trivial. You see, what does that mean? The spectral curve Y squared equals X. So Y is the eigenvalue and it's got to be square root of X. Well, square root of X is not a well-defined function. It has monodromy, it's multiply valued. So you can't just write a diagonal matrix square root of X, square root of X, all right? You've got to work harder than that. And you know you can't diagonalize it because you can diagonalize it away from the branch point. But at the branch point, you're getting this double point vertically. And we know that that's supposed to correspond to a Jordan normal form. So you can't, yeah. We have a question on a biogena. Do you need some condition that support as affinate and flat morphism to S? No. It all comes out in the wash. There's no conditions here at all. This is an equivalence of categories. I mean, I haven't set up one of the categories, but it's absolutely fine. Everything just comes out in the wash. I have also a question. You assume that f-wedge f is equal to zero, like in NANDs. Phi wedge phi? Yeah, phi wedge phi. You don't have to because you don't have a one form here. Where was phi? Yeah, there, because... Because phi is already a two form, so there's no wedge. Yeah, that's right. In general, you would have to do that. You need some commutativity property. Okay, so you're going to get this branching. So the hint for this question is that a matrix is determined essentially by its trace and determinant, okay? So if you want eigenvalues square root of lambda and minus, sorry, square root of X and minus square root of X, you know what the trace and determinant are. And so now I think you can write down the right matrix, which is well-defined and is not multiply valued. Okay, and then exercise, do the same for Y squared equals X squared. So now there's no branching, but you can do two different things. You can describe one, you can describe two different spectral sheaves on that spectral surface. One of them corresponds to a diagonalizable Higgs field and one corresponds to a non-diagonalizable Higgs field with a non-trivial Jordan normal form. They're really great exercises. Okay, conversely, if I have one of these torsion sheaves supported over a surface, it'll be compactly supported. So that's relevant to the question online. What, this is an equivalence to compactly supported sheaves on X, okay? So if you have a compactly supported sheaf on X, then you recover a sheaf on S by pushing down, by taking sections. And then you can recover the Higgs field by the action of the sort of, the tautological endomorphism which you have on X. So, you know, at every point of X gives you a section of this line bundle. There's a tautological section of this line bundle, KS, or it's pullback, on KS, right? So it's zero here and it's one here and it's two here and so on, all right? If you multiply that by the identity, you get a sort of a, you get a canonical endomorphism which is zero on the zero section and one up here and so on, okay? That's the thing which gives you the Higgs field. So it turns out that Higgs pairs are entirely equivalent to compactly supported torsion sheaves on X, all right? So now we've reduced this Vafa Whitton thing to you can describe it either by Higgs pairs or you can describe it by torsion sheaves on a certain local Calabi-R three-fold, non-compact Calabi-R three-fold, X. And here you see the non-compactness, right? I mean, X is non-compact. You can scale the Higgs field, you can stretch everything in the KS directions and you can see it's inherently non-compact. You could try and compact projectively complete it or something, you'd lose the Calabi-R condition it's not clear whether that's a profitable thing to do. It probably is and maybe you could study it, yeah. You have a question, does the spectacostruction work more generally with an arbitrary line bundle instead? Yes, it does, yeah, there's nothing special about the canonical bundle here. Okay, and then what we're really interested in is two conditions. Remember we're dealing with SUR not UR and what that means is we wanna fix the determinant of E and take the trace of phi to be zero and what that corresponds to is that the determinant of the pushdown of E should be trivial so that's kind of not really a translation at all but the trace phi being zero is this condition that this spectral sheaf has sort of center of mass zero on each fiber. So where you would have to make sense of the center of mass, so in this case it would be sort of this point of the fiber plus this point of the fiber plus this point of the fiber but more generally you might have on the spectral surface might instead of being sort of three to one it might just be one to one and you have a rank three bundle on it instead of a line bundle on it so that's the case where your matrix is a multiple of the identity and in that case you have to weight those points by three. So you have to make sense of the center of mass the easiest way is to just say trace phi equals zero that's the best way of making sense of the center of mass. Okay, so that's so you could forget everything up until now and just say Vaffa-Witton theory is about these spectral sheaves these sheaves on X which satisfy this strange condition. Okay, they have this sort of center of mass zero on each canonical bundle fiber and there's a condition on the determinants of the push forward. Should we use it as some sort of other measurement? Yeah, yeah. Probably. Yeah, maybe. Yeah, if you take the divisor class yeah, yeah, yeah you could do that. There's probably a way of formalizing that. Okay, so we want a virtual cycle on this thing and then we've got to deal with non-compactness. Okay, so I already said this about stability we're going to use Gisica stability and the first thing we're going to do like in the last lecture is we're going to assume for simplicity that there's no strictly semi-stables so we're going to assume that anything that's semi-stable is actually stable that's just to start with. And then what you find is the deformation theory we've already done that. I've compactly supported coherent sheaves on the Clabia 3-fold is perfect. It's also got a certain symmetry. So it's the same as before there's deformations, obstructions they happen to be dual to the deformations. This gray piece is sort of for later that's when you keep track of the C-star action then if you know what this is this is when you keep track of the C-star action I'm just tensoring with the one-dimensional standard character of C-star there the standard representation weight one representation of C-star. So that comes in. The point is that this Clabia 3-fold has a holomorphic 3-form on it and the C-star action of scaling the fibres does not preserve that holomorphic 3-form it scales it by a factor or weight one or minus one. That's important. Assume the date is trivial the holomorphic 3-form is kind of trivial then the stream work is kind of controversial. Now that was the determinant of the bundle whereas this is the holomorphic 3-form on X. Okay. And there's no higher obstructions at least once you take trace 3 and I'm making Dennis nervous now but it can all be handled. Okay and therefore it inherits a virtual cycle and that's essentially what the last two lectures were about. And you know there's my picture remember all this. Okay so there's a way of describing the the moduli space by current issue models they give you these graphs and you make them vertical you get a cone in a vector bundle and you intersect that with the zero section. Okay. And so you get a zero dimensional virtual cycle. So this is essentially some kind of DT theory. Okay. But M is non-compact just as X is non-compact so really the virtual cycle vanishes it doesn't make much sense but we're going to use this C star action where we scale the fibers the canonical bundle fibers or equivalently we scale a Higgs field phi. You can work in on X or you can work on S with Higgs pairs it doesn't they're equivalent and you know at various points in these papers we switch from one to the other there's various things which are very convenient in one picture and various things are impossible in one picture and very convenient in the other. And now the C star fixed locus is compact and that's essentially because the fixed locus of C star on X is compact it's the original surface. Is the virtual cycle really zero? Yeah I think that's not quite true actually. Yeah we could talk about that after. I mean if it would be zero then the localize one would be all right. Yeah I think you're right. Yeah anyway. Yeah it's not a great thing to work with let's just say that. Okay. And in fact in the k-theoretic setup in my last lecture there's really no difference between the non-compact and the compact sir. But you have to use the C star action. Yeah so let's say it's uninteresting and one should use the equivariate version and then it becomes interesting. Anyway. Okay so the C star fixed locus is compact because the C star fixed locus is those sheaves which are really supported on S but only set theoretically on S. They can be supported on some scheme theoretics thickening of S. Okay we come to that soon. So we're getting sheaves supported on S and S is compact so you end up with a compact modular space of sheaves. And so because it's compact what we can do is we can just apply the virtual localization formula and we can integrate. So again this was something I thought long and hard about whether I was gonna discuss localization eventually I decided I didn't have time. And Lota already did it. So when you're doing integration, when you're dealing with coromology of a variety with a C star action, a circle action most of the information is contained in the fixed locus. You can localize integrals to the fixed locus. And so we do that here even though maybe the integral might not make sense on the non-compact thing we just pretend it's compact, apply the localization formula and we get an expression on the fixed locus and because this does make sense there's some kind of regularization of the integral on the non-compact thing. We just take this as our definition. All right. And because we've localized that involves inverting certain things like this. So we end up inverting integers. It turns out we end up with a rational number even though we're assuming that there's no strictly semi-stables we end up with a rational number. Don't you have to take some residue? Yeah that's, yeah this is kind of called a residue. Anyway, do you mean like a residue of a function of T like the coefficient of T inverse? No, it turns out because the virtual dimension's zero you're taking the constant coefficient and actually because the virtual dimension's zero this integral really is constant inside Q of T. Exercise. Yeah. By Dima, is this equivalent to the construction of virtual classes? No, this is a bit like the question yesterday is that can you define this via bare end functions and weighted Euler characteristics? So I refer, which Dima? Yeah, Zolkin. Ah, I referred Dima to the original papers where we defined two invariants one via bare end functions and weighted Euler characteristics and one in this way using virtual cycles. Because of non-compactness they're not the same and for a long time we worked with both and then we did computations which showed that to get the numbers that physicists get you have to take this definition, not the other one. And the co-section that he's referring to is much more closely related to the bare end function I think. So anyway, for now I say no. All right, so this is really just what's called a local DT invariant but this is really the UR Vaffa-Whitney invariant because I haven't fixed determinants and center of mass zero and trace zero and so on on this particular slide. And so what you find is this is kind of uninteresting. At the moment S, this is fine if the coromology of S is very simple. For instance when you have the vanishing theorem so if the canonical bundle of S is negative then this is a fine definition and it really is the Vaffa-Whitney invariant but in general for general S like general type S for instance this invariant's always zero. And that's to do with the action of the Jacobian of S and this extra piece in the obstruction theory of S. So I leave that as an exercise. That's a trivial exercise for people who know a bit about obstruction theories and duality and so on and it's an impossible exercise for most people so ignore it. Or ask someone and they can explain it to you. It's one of those things that's completely trivial when you see how to do it. But you would never see how to do it. So what we really want to do is define an S-U-R Vaffa-Whitney invariant that's going to have more chance of being non-zero. It's much more interesting and it's not just a DT invariant so it's something new. So what we do is we take this modulised space which I call M-PURP of those E with centromath zero and this fixed determinant condition. Equivalently we take instead of all Higgs pairs we take those Higgs pairs or stable Higgs pairs with trace zero and fixed determinant. And you can relax this OS and fix just any old line bundle. That's fine. So I won't go through too much of this but it turns out you can change the deformation theory. You see the deformation theory of these sheaves on X these torsion sheaves was governed by this X group here and what you can do is you can prove there's a splitting. You can take out a piece. So this first piece the coromology of OS that governs the deformation theory of the determinant of the pushdown of E. So the determinant of the straight E on S. The deformation theory of that is the deformation theory of a line bundle so you get X from a line bundle to itself but that's just the coromology of the structure sheaf, right? This structure sheaf here is the endomorphism sheaf of the line bundle determinant of E. So this is what governs this first piece governs the deformation theory of that line bundle that I want to fix so I'm going to remove it. The second piece governs the deformation theory of the trace of the Higgs field or the center of mass of the spectral sheaf. And then the third piece is what's left over and you can prove there's a canonical direct sum decomposition and that is what governs the deformation theory of my SUR Vaffawitton Higgs pairs or sheaves with the center of mass zero. Okay, so if you wish again this is more for experts I think here's an exercise you can deduce this at least point-wise in the following way. There's a certain resolution of our spectral sheaf okay by you know remember this is an eigenspace and this is the vector space so this is just the map from the vector space projecting to the eigenspace by projecting out all the other eigenspaces but this is just the obvious junction this takes the sections of a sheaf to the sheaf evaluation of sections, okay that's the kernel well again intuitively this is kind of clear there is no kernel generically generically e phi zero it's a torsion sheaf but there's kernel when you're at a point where phi minus lambda or phi minus tor tors the eigen remember tors the sort of the tautological eigen function so where phi minus tor has some kernel or its rank drop so it has some kernel and co-kernel that kernel or co-kernel is the generalized eigenspace and so that gives you this resolution there should be an identity times that tor really so at the points where tor where how far you are up the canonical bundle is equal to an eigenvalue of phi then this operator here suddenly drops rank and it has some co-kernel and that is the eigenspace that's exactly what this resolution says so it's an exercise to prove it it's a pretty tricky exercise but it's in our paper the original paper with Tanaka okay and then maybe a more manageable exercise much easier is just to take x's from this sequence to e phi, curly e phi okay and then you get this use a junction and you get this long exact sequence here and this is what tells you how to relate the deformation theory of the Higgs pair to the deformation theory of the sheaf so in the middle here is the deformation theory of the sheaf on x sub is the deformation theory of the Higgs field so as I deform phi as I change phi I deform e phi and as I deform e phi by pushing down I deform e on s and not every deformation of e on s gives me an e phi because I might deform my e on s and not be able to deform the Higgs field with it there may be an obstruction to deforming the Higgs field and that's this last arrow here and this is the obstruction space for the Higgs fields okay and now using this you see you can see these pieces so the chronology of OS comes from this third term here so it comes from taking trace or identity it sits as a sum and in this x one of straight e straight e on s it sits as a sum and in there using the trace and identity maps and instead of talking about deformations of e it's talking about deformations of deti and it's said you'll as everything has this duality it's said you'll which is the chronology of chaos is sat in the first piece again as a sum and via trace and identity maps and it's sat there in terms of the Higgs field it's the trace part of the Higgs field and then the purpose what's left over yeah for x one could you view that the sum is a the one that comes from the similarity map here the blue lines here yeah the second sum this one for x one it's going to be h for x two so for the obstruction yeah it certainly has that flavor and I bet you could do it yeah yeah I think that's right yeah that's a good yeah that's a very good point yeah you could do it that way okay so I think I said this already alright so what you end up proving is that this m perp this s you are a Vaffawitton space also has a nice perfect obstruction theory virtual dimension zero it has this symmetry and you can make the same definition as before and you get a Vaffawitton invariant and okay here's a you know six month exercise if anyone wants to do it I'm not sure how valuable it would be but it's an alternative way of defining this perfect obstruction theory in the rank two case would be to take the fixed locus of dualizing and applying minus one on the fibers so you can dualize your e phi on its support I mean you can write that actually on X in terms of X to one instead of POM but anyway you can dualize that line bundle that spectral line bundle and apply minus one to the spectral surface and then the fixed points of that should give you this m perp and you know that Graeber and Panda told us how to pass from a perfect obstruction theory to a perfect obstruction theory for a fixed locus under a group action so that would give you a more immediate way of constructing this perfect obstruction theory yeah but you can surely adapt what they did the only reason I say this is because you know this slide is hiding like really 30 pages of hideous deformation theory I mean we should have found an easier way of doing it but we never did and this is one easier way of doing it in the rank 2 case it turns out to be really really hard to define this X perp perfect obstruction theory can we take some sort of uripe fiber yeah that's enough we also discussed that in the paper yeah if you we couldn't do that honestly because we didn't have the expertise but yeah if you're happy to say infinity derived algebraic geometry something then yeah you could do that hopefully it goes away in a second alright so just treat it as about black box there's this virtual cycle there's this we localize the fixed locus the C star action and we define an invariant alright and what these lecturers are about is essentially how you compute this and what the two contributions are there are two different types of contribution to this as Lothar mentioned there are two types of fixed locus one is where phi is zero that is obviously fixed under the C star action so that corresponds to sheaves on X which are scheme theoretically supported on S they're pushed forward from S so they're not line bundles on S they're really rank R sheaves on S pushed forward to X so that's the most degenerate case where the Higgs field is zero and then the exercise is to show that you can also get fixed pairs from nilpetent phi so when you that's essentially down to the fact that when you scale a Jordan block with zeros and just a one in the corner when you scale that one to a lambda it's similar it can be it's what's the word I'm looking for it's similar to itself right if I put you know this matrix can be conjugated to this matrix under an endomorphism so what you need is an endomorphism of your E which is possible because E isn't strictly stable E is only stable when you pair it with the Higgs field so you can have an endomorphism of E which takes this to this so when you scale your Higgs field by an element of C star you end up with a similar Higgs field so you end up with an isomorphic Higgs pair it's easier to see the C star fixed low psi in the sheaf on X language rather than the Higgs pair language so on X what you're doing is you're taking sheaves which are supported on a thickening of the zero section you can see those could be C star embarrassed they're not all C star embarrassed the structure sheaf of a thickening of S for instance is C star embarrassed so we get a decomposition of the C star fixed locus into two pieces the so-called instant on branch so this is sheaves with fixed determinant on S this corresponds to solutions of the anti-self-dual equations by the Hitchin-Kobeashi correspondence yeah actually for what why do you need this kind of C star action here just to make things compact that everything's non-compact you can't define invariance because of that but by localizing to the fixed locus of the C star action you get a compact situation where you can define things and then what we call the monopole branch on which phi is nilpetent rather than zero so these are the sheaves supported scheme theoretically on S and these are the sheaves supported set theoretically on S but not scheme theoretically on S oh yeah I said that and I said that so the Waffa-Whitney invariance are some of these two contributions and they seem to be swapped in a certain sense by the S duality voodoo predictions of physics or this is some non-Abelian version of electromagnetic duality those are just words that you know that's some kind of jargon I heard and it Eugene is asking a question yeah if phi is equal to zero and the model space of sheaves on S is as smooth as it is easy to describe the virtual cycle yeah we're going to do that next and Loter did it yesterday okay there's my answer now we're going to do that next okay so I'm going to focus on the monopole locus because the other locus the instanton branch is where people made lots of progress over 25 years and the latest progress is by Loter and Martin Kuhl and a lot is known about that and mainly I'm going to focus on the monopole locus but to start with I will talk about this instanton locus very briefly and I refer you to Loter's lecture for more information so let's look at moduli of sheaves on S so it's just sheaves straight E with zero Higgs field that has its own virtual cycle based on the deformation obstruction theory of sheaves on S okay if you hate all this deformation theory just consider the case where it's already smooth of the correct dimension where this X2 here vanishes so yeah you can already form this moduli space of sheaves on S we know how to handle it in many different ways but this is the most general case so it has its own virtual cycle but considered in this three fold theory instead of a surface theory the obstruction it has extra deformations and extra obstructions okay so you can start to deform the Higgs pair that lives in here so you can start to deform the Higgs field that lives in here so that's an extra deformation and it's dual to the obstruction space and it has extra obstructions here the obstructions to deform in the Higgs field and they're dual to the original deformations and you know an exercise you can really work that out this is easier than the previous exercise because now my sheaves supported on S so it's much easier to resolve it and this is how you do it and you really just see that the deformations this is in the sheaf language so here's your curly E it's just the push forward of straight E when you take its deformation theory you get the original deformation theory on S but then you get this other piece which is essentially the dual of that up to a shift and a character of C-stop okay so in fancier language this says that the when considered on X you're taking a minus one shifted cotangent bundle as the derived scheme of sheaves on S so that doesn't help anyone I appreciate it but what it's saying is that the the virtual tangent bundle is the old virtual tangent bundle just of the surface theory plus it's dual which gets shifted and it picks up a character and that becomes the virtual normal bundle so that's the directions in which you deform the Higgs field that's normal to this monopole locus okay and from that you can compute use the localization formula you can compute what is the Euler class of this guy and you can see it's very close to being just the Euler class of the virtual tangent bundle in fact it's dual and so what you find from that is that when you take this definition by localization of the Vaffa-Whitney invariant or at least the contribution to the Vaffa-Whitney invariant from the instant on locus what you get is this signed virtual or their characteristic and I use a different sign from low term and this has been heavily studied over 25 years and the really the state of the art in recent years is Gertrude and Kuhl which you heard about yesterday and physicists in particular Peerlene and Mancho and other people and there's this vanishing theorem when the degree of the canonical bundle is negative that's everything so that kind of justifies studying this locus intensively but I'm interested in the case where this vanishing theorem doesn't hold and where you get monopole contributions and then we'll come back later to the interaction between these two contributions and how incredibly this estuality appears to swap the two and so that means you can make predictions on one side or the other sometimes you can do calculations more easily on one side sometimes on the other and then the estuality switches them and it's a very powerful tool as Lothar demonstrated yesterday okay so I'm really interested in this monopole locus so for instance here's what happens in rank two so an exercise is tensor as sheath supported on two times the zero locus so on the thickening in the zero locus tensor it by this obvious exact sequence to describe it in terms of rank one sheaths on S so this piece will give you a rank one sheath on S this piece will give you a rank one sheath on S and there'll be a map between them up to tensoring with KS and both rank one sheaths will let's assume everything's torsion free they'll give you ideal sheaths and the map between them will give you a nesting of the two ideal sheaths I will do that in more detail in more generality later what that corresponds to in Higgs language is that you can decompose the Higgs pair using the C star action so you can decompose your vector bundle E in terms of a weight zero piece which is fixed by the C star action a weight minus one piece which is not and the Higgs what you find is the Higgs field what diagonal is this Jordan block the Higgs field maps the E zero to the E one stability forces the E is to be torsion free so up to tensoring with a line bundle they're ideal sheaths and what it turns out is that this phi ends up being a map between two ideal sheaths for of sub schemes of S so the green bit was doing it in sheath language on X the black bit is doing it in Higgs language on S and so these monopole contributions have something to do with nested Hilbert schemes and they're essentially computing with these things is the subject of the rest of the course and also eliminating other components you know in higher rank there's other components where you get more than where these aren't both rank one this could be rank two and this could be rank one and then the rest of the course is about showing why they don't contribute okay that's just some history so we did some computations a few years ago just in very low degrees before we had general methods for computation that I'm going to talk about in this lecture and those computations are very hard work and Martin Kuhl observed that our answers which we thought were rather disappointing turn out to give the first few terms of modular forms predicted by Vaffer and Witten hundreds of years ago so here's what it was Gertrude and Kuhl were already seeing these terms this is cut out of Vaffer Wittenpaper in 1994 Gertrude and Kuhl were already seeing these terms but they weren't seeing this first line and what we were seeing with this term the first four terms that we managed to calculate Martin observed were exactly this term the first four terms of this this is some generating series so we were calculating I think with fixed determinant equal to the canonical bundle for convenience if you calculate with fixed determinant equal to zero you'll get this term and Larakar has done that computation since so that was what told us that this is the right definition that I've been giving to you the other possible definitions that have been discussed are incorrect so this is saying that the bare-ended weighted definition that some of you will be aware of gives the wrong answers okay so I should let me just do this slide okay so more generally something in the monopole locus is fixed by C star you can lift that being fixed by C star to prove using stability you can you can show that what that means is that actually you can make C star act on the sheaf you can make the sheaf be C star and then you can decompose it using that equivalence into C star weight spaces and you end up with a picture like this so you split your bundle into pieces these are just the these just correspond to the Jordan blocks so you should think of this in the sheaf on X language this is the bit supported scheme theoretically on S this is the next bit supported on the first order thickening of S and so on up to the kth order thickening k minus 1 and phi which has weight 1 because it's being scaled by the C star action always maps from EI to EI plus 1 so you get this sort of chain of maps and so this defines components of the fixed locus labelled by integers later will show that those integers have to be decreasing otherwise it contributes 0 so you can think of this actually as a young diagram or a partition so you get these components of the modularized space where these are the ranks of these pieces so these are the ranks of the sheaf you've got the sheaf on this thickening of S and it sort of has on its support it has ranks on different thickenings of S you can look at what its rank is and those are these RIs and the most important components as Lothar explained yesterday are the two extremes where the whole thing is supported on S so everything is in E0 that's the instant on locus as we discussed and I'm not going to talk about much and then there's the other extreme where there is spread out as much as possible and they're all rank 1 and that's these nested Hilbert schemes so that's where the profile looks like this he called this 1 to the R and what we'll see later is that when your surface has a 2 form by something called co-sectional localization which I'll discuss the only non-zero contributions come from constant profiles so where all these ranks are the same and where actually these fies are generically isomorphisms generically over the surface so we'll discuss this later but in particular what it means is and Lothar mentioned it when the rank is prime of course there's no way of splitting up into constants like this except for the two extremes so when the rank is prime so I tell you what a prime number is here only the nested Hilbert schemes contribute and the instant on focus so all I'm doing here is setting up for next time explaining why nested Hilbert schemes are important they're really almost everything in Valfa Witten theory or they're everything we know until now and then this question was asked twice since succession yesterday what about the first the case not covered by this so that's rank 4 I'm going to turn it to 2 okay and I believe Sheshmani yeah I've been thinking about this the issue is that these individual E's needn't be stable so when they are stable Andre has done a great deal of work studying you know correspondences between modular spaces of sort of E0 and E1 set up by maps between them and so on the problem is that in general they needn't be stable because the stability condition is more complicated so I think the idea of Sheshmani is that you should try and wall cross through some space of stability conditions to the situation where you have stable sheaves where the two EIs are both stable and if you could do that then you could start to study this problem but at the moment no one knows how to do that okay so next time here with the semi-stable case so far I've always assumed that my sheaves are stable and that any semi-stable sheaves are actually strictly stable next time I'll show you how to deal with a semi-stable case any question comment don't be shy just it's off is it the monopole equation like Bogomolni or Hitchin sorry any other monopole equation related to this Rafa with him I mean the Hitchin equation is essentially this equation one dimension down and there again you have a C star action and you can localize to compact fixed loci and you know Hitchin does that that is not the question the monopole means where you say monopole is it oh yeah honestly I have no idea that's probably a bad so in the original Wafa Witten paper they very briefly in one section mention that they do exist solutions which aren't just push for they aren't just instantons and so they're interested in the rank two case so what they end up with is two rank one sheaves so essentially what we call in the nested Hilbert scheme but in their case they don't allow for singularities they have two line bundles and then they have this Hicks pair which is therefore a section of a line bundle the homes from one line bundle to the other so they end up with an equation for a section of a line bundle and this equation looks very much like the Cyberg Witten monopole equation and they call it monopole and then it's become known as the monopole locus I'm sure that's a dreadful name but I'm sure Hicks is a dreadful name as well so yeah any more questions how badly does the getting this capability composition work here oh yeah I mean what's sort of relevant is the the virtual or derived version of that and then so my understanding is um Davesh has announced this localization formula right I think that's basically a derived version of this BB theorem and he hasn't written it right and he never will okay exercise yeah I'm not an expert on this um yeah I'm a bit nervous about saying anything because the versions I know are in the language essentially of the bare end function and so they're not entirely relevant to what I'm doing here but there are different versions so there's different refinements of DT theory where you take you replace the bare end function by perverse sheaf of vanishing cycles and then there's other versions using K theory which I'm going to talk about but they all have localization formulae and Davesh has proved a localization formula which unfortunately hasn't written down but it's very much like a derived version of this BB formula or decomposition but let me not say more than that because I don't you know I'm starting to say things that I don't really know what I'm talking about if you were to work with compact kalabia 34 are there any way to put the conditions here like here trace 0 and fix determinants in the intrinsic language of yeah that's a great question sometimes but not usually but usually that it's done already for you um I'll come and talk to you about that later yeah um often it's done for you the very fact that you're deforming inside this kalabia kind of stops you seeing it already looks like an SUR theory rather than a UR theory essentially because the kalabia you probably assume has no Jacobian right and no H2 zero yeah so then physicists since extended these estuality conjectures from Vuffer-Witton theory to the kalabia case they predict that when you calculate with two-dimensional torsion sheaves in a kalabia 3 fold you should get modular forms without putting extra without putting extra conditions yeah there's no problem there just the same as for the UR Vuffer-Witton equations that's really just the local DT theory again there should be estuality for that it's just often a duality between zero and zero so these are not the nicest modular forms they're mock modular forms in general and they're vector-valued modular forms so they're hard things to deal with but nonetheless so according to what I said yesterday Gromov-Witton theory might be determined by some mock modular forms some vector-valued modular forms but I'm not convinced actually that that's a useful statement at all because the length of the vector gets bigger and bigger each time you try and pin something down and make a statement that you have to make the vector longer and then you know you want to use modularity to say well there's only a finite dimensional space of modular forms of this type therefore once I know 10 coefficients I know everything or something like that but you often find that once you compute 10 you find actually you need more because the length of the vector is longer than you thought and I don't know when you compute some more and you try and use our theory between DT theory and Gromov-Witton theory you find some parameter has to get bigger in order to follow our theorem through and when that parameter gets bigger suddenly the length of the vector gets longer it's not entirely clear yet that it's a useful theorem I think in the Waffa-Witton theory absolutely where should it come from? actually geometric mathematical? it's an artifact to formulate at the moment but it's not some graphic theorem I've often thought about it it'd be nice wouldn't it at the moment no I'm going to explain how you're supposed to go between instants on, so sheaves on S and these nested Hilbert schemes the easiest way to see any kind of link is at nested Hilbert schemes as I will describe by degeneracy loci you can express in terms of just Hilbert schemes so that's one way that reduces the theory of Hilbert schemes which gives you modular forms, vertex operators instantons you can do something similar you can do this mochizuki wall crossing where you take instantons plus another field and then you start wall crossing and then you get the ability conditions for them and at one end you get instantons at the other end you get Hilbert schemes again you manage to decompose your bundle into sums of rank one bundles and there's another route to Hilbert schemes and then maybe astuality is just relating these two statements about Hilbert schemes but it's awfully non-direct is the integrant in the Hilbert scheme points do they even look similar to see this? not on the face of it and you have to use lots of results of necrosoph and people integrable systems people and so on lots of combinatorics to eventually see that the generating functions are related at all yeah I think this is really a mystery I mean people should really are you reducing this astuality at the level of instantons and some kind of others like monocles it's more general than that is it not? as it seems that astuality is like a kind of duality between the instantons on hand and the other hand monocles it's not well that's just one element of the astuality group there's others so even if you just form the instanton generating series or just the monopole generating series those also have modular behaviour so it's more general I mean it's not something we understand we have a question how does this astuality relate to the astuality for one dimension one sheaths and a compact three people dimension one sheaths co-dimension one sheaths yeah it's the same thing yeah there's a generalization of astuality to that setting which when you restrict to these non-compact calabiows it becomes the astuality I'm talking about I have a question you said that using the Beren function gives the wrong answer it simply means that there is no physics counterpart to that numbers yeah that's right it gives very uninteresting answers it also gives modular phones but they're all just the eta function and they're not the numbers that physicists predicted yeah so if you take an arbitrary line bundle instead of ks do you expect some sort of partial modularity? oh yeah I don't know you need something like that line bundle to have a degree lower than the canonical bundle otherwise you'll get x3's and stuff you won't get a perfect obstruction theory but then you could ask that that's a great question I have no idea I don't think anyone's done a single computation so if you're a gay or student that's what you'll be I think we can thanks Richard thank you