 Hello, everyone. Welcome back. Thank you for waking up early for this talk again. I want to tell you, carrying on from what we saw yesterday, is about some ways in which we can study Higgs bundles in their modelized space, some tools that Hitching introduced back in 87, and through which we can motivate a bit further the study of Higgs bundles. So this is called the Hitching Fibration, and that's what we're going to be doing today. So the Hitching Fibration. And this vibration was first introduced in 87 by Nigel Hitching when working with Higgs bundles on a Riemann surface. So just like yesterday, we're going to fix a Riemann surface here. Fix a Riemann surface, compact Riemann surface of genus, at least two. And we're going to be using a complex leaguer, just like we were doing yesterday. I'm going to tell you today and along the last two talks, when the Hitching Fibration has not been defined yet, when there is a chance to define it for these generalized Hitching systems. Let me remind you what Higgs bundles were for those that might not remember from yesterday. Recall that a Higgs bundle is a pair E phi for E holomorphic vector bundle on the Riemann surface, and phi a holomorphic section that goes from E to E tensored with a canonical bundle. So here phi is what we call the Higgs field. We saw yesterday, so Higgs field. And this cotangent bundle is the canonical bundle, which is the cotangent bundle of the Riemann surface. We formed yesterday when Higgs bundles for complex leak rooms. We said we'd take classical Higgs bundles and add some conditions to remind us of the structure of the grid. We'll see some more examples. But the interesting thing is that once you have your modular space of Higgs bundles, the one that we talked about yesterday, there is a natural vibration from the modular space of Higgs bundles for a complex group to a basis, and a fine space formed of differential forms on the Riemann surface. I'm going to give you first an overview of what we're going to do, and then we're going to define things properly and do some more proper descriptions. But I want you to have an idea of why we're doing it. So this vibration, this map is a proper map. The fibers are going to be abelian varieties, compact abelian varieties, which makes the whole space very nice. So you can describe the geometry of this space in terms of these abelian varieties. That's what Hitching called abelianization. This map, the Hitching map that we're going to define, it's called the Hitching Map, allows you to see many of the properties of the geometry of the modular space in terms of fibers and in terms of abelian varieties. Now, remember that we had here a relation with representation space. So we said non-abelian hodge correspondence and Riemann-Hilbert correspondence relates the modular space of Higgs bundles with different other areas of math. And in particular, we had representations of the fundamental group, reductive representations of the fundamental group of our surface, so pi1 of sigma into the complex league group. Now, we can also, we also mentioned that there are real Higgs bundles and the real Higgs bundles, we said they appear as some subset of the complex Higgs bundles. So if I put here the modular space of real Higgs bundles, so Mg inside the modular space of complex Higgs bundles, so here G is a real form of GC. This subset is the very nice subspace that we constructed yesterday and which we are going to be studying in the rest of the week. It lies inside the hitching vibration through this inclusion. So you can ask, what are those points? What are the points, the orange points that correspond to this space? And this space is just one example of what we know as brains. So a Lagrangian subspace. So this is an example of a brain that we're going to talk about in the next couple of days. Brains are subspaces that lie inside the modular space of Higgs bundles in particular, but inside other scalar and hypercaler spaces. And you can ask, how do those spaces lie within a vibration if there is a vibration? Now there's something else that we can look at. This vibration that we're going to define for a complex league group, you can ask what happens when I change the league group? Can I change the league group in some fashion that tells us something about the vibration? And one thing that you can do is you can consider the modular space for what's known as the Langlands dual group. So GCL. The Langlands dual group, if you haven't seen it before, you can take roots of a group and take the dual roots, the crow roots on the other side and construct a group. So just as an example, let me put it here. If you have, for instance, SP2NC, the dual Langlands group will be SO2N plus 1C. And you have a further list in the lecture notes and we'll come back when we do properly mirror symmetry. But one thing that happens here is that when you consider the hitching vibration for this other group, what you can realize is that it fibers over the same base. The base of the hitching vibration for the Langlands dual group is isomorphic to your original base. So you have a dual vibration, two toric vibrations over the same base. And what Sassler, Stromminger, and Yau conjecture is that mirror symmetry is a duality of these fibers of the vibration. So now suddenly you started with a modular space of Higgs-Mandels objects, which look rather simple. You constructed their modular space, you fiber them and now you have mirror symmetry appearing through these vibrations, through the hitching vibration. So there's a duality, a conjectural duality, Stromminger, Yau and Sassler as why is that conjecture that tells you the duality of the spaces and of the groups gets reflected on the duality of the fibers. But we also said that there was a subspace, that there was a subspace of Higgs-Mandels for real groups and in general of brains. So you can ask what happens with brains here? So brains, they're going to be dual brains that are also conjectured to exist for each brain that you take in one side. So the questions that we're going to ask are which kind of subspaces appear here as brains would as mirror symmetry tell you about the brains and the duality between the objects in the other side? So these are questions that appear through the hitching vibration and once we put things in terms of this ability and variety. And finally, something that I've been interested in and I want you to think about for those that know a bit more about Lee theory and Lee groups and correspondences between modulite spaces, I'm interested in knowing whether there's other, there's other modulite spaces here that we can put of certain other Higgs-Mandels such that there's a duality, it won't be mirror symmetry and there's a duality between brains that won't be mirror symmetry. So I'll show you in the next couple of talks that a mirror symmetry proposes what dual types of brains should be in between these two modulite spaces and it's very strict, it tells you there's a brain of certain types so these MG we're going to see very soon is a type, something that we call BAA brain. I'll tell you soon what it means, it means that it's a Lagrangian with respect to some structures and it's dual brain here will be of type BBB and this is always the case. Anytime that you have certain type like BAA your dual will always be BBB. So I want to construct other modulite spaces and other correspondences which may be associated BBB to BBB instead of changing the type. So this is one example. So I'll tell you how you can solve this question in some cases but this is certainly very open. It's very open also to construct all the possible brains inside the modulite space of Higgs-Bandos and understand mirror symmetry. This will be telling you about objects in the Foucaille category for one space and in the draft category of coherent shifts on the other space and so understanding a basis for those categories for instance in terms of subspaces of modulite space of Higgs-Bandos is also still open. So there's tons of questions arising from the hitching vibration itself. So what we're going to do next is we're going to explicitly construct the hitching vibration. I want to tell you how to make it through Lee's theory and then how to get hands-on on seeing the abelian fibers and then I want to show you how these modulite spaces lie inside the hitching vibration as an example. So let's put a definition to define the hitching vibration. Consider the ring of invariant polynomials for the algebra of your group. So the ring of invariant polynomials for the algebra GC, for your group GC. We want to consider the ring and take a homogeneous basis of invariant polynomials. So and take, I want to take P1 to Pk, a homogeneous basis of invariant polynomials. Homo-genius basis for it of invariant polynomials. These invariant polynomials are homogeneous. They'll have some degree. And let me tell you that, so, of degree D1 to Dk. What we're going to do now is we're going to actually apply those invariant polynomials to the Higgs field. So the hitching vibration, the hitching vibration is defined, so is defined as a map H that goes from your modulite space of Higgs bundles and GC to certain base associated to each group. And the base is going to be a sum of differential forms and the forms will be of degrees Di, just like that. So it's a section over the Riemann surface of powers to Di of your canonical bundle. And it's defined by taking a representative of a Higgs bundle. So I'm going to always just consider the vector bundle expression with the additional constraints that we put for the group. And this is going to be sent to P1 of phi, Pk of phi. So just a word to understand, this is the delinearity perspective of the hitching vibration. Once we're applying polynomials, PI that have degree Di on a Higgs field. And the Higgs field was a section, so it was a map, an endomorphism of the vector bundle E with values on K. So you think of an endomorphism with values on the canonical bundle, you apply a polynomial of degree Di, you're going to give up something on the Di power of K. So you're applying the Higgs field again and again. A way of understanding these two is just thinking of a basis. So an example of basis, an example of basis, for instance, for most groups is trace of a matrix to a power. So traces of phi to the I, for instance, are examples. This is for groups, except, so for the A, B, C, D, but actually it doesn't work, not for S, O, 2, 1. See, where you need to take a path. So what happens with the hitching vibration? This vibration, the map H here, is what we call the hitching map, just like in that diagram. This is the hitching map. And what hitching showed, hitching map. So the hitching map is a proper map. So it's a proper map, in particular the inverse image of a point is compact and the generic fibers are compact abelian varieties. So generic fibers are compact abelian varieties. The dimension of the space and the dimension of the base are related, one is half of the other, which implies that with other conditions, implies that we have an integrable system. So it makes the moduli space into an integrable system, integrable system. These properties, interestingly, are independent of the choice of base. So independently of base. So you can take your favorite bases of invariant polynomials, apply them to your Higgs bundle and try to understand what the hitching vibration looks like and the properties should remain the same. Now this is the fact that the generic fibers are abelian varieties is what people know as the abelianization of Higgs bundles. What I want to do now is I want to tell you just some ideas of how to think about the hitching vibration and how to see those properties for some of the groups. And we'll come back to this diagram again, probably not today but tomorrow, to try and see which specific questions we can ask now that we know what the hitching vibration looks like. Along the way, you should stop me anytime you have some questions. So the way that the hitching vibration arises and the way that I want to think about it is through something that I mentioned here that you can take your favorite bases and look at the hitching vibration through that basis. So consider the basis of invariant polynomials that appears as coefficients of the characteristic polynomial of a matrix. So consider the basis of invariant polynomials appearing as coefficients characteristic polynomial. Characteristic polynomial. So we can take the characteristic polynomial of our Higgs field and the coefficients will be giving us the basis of invariant polynomials. So take for instance, the general linear group classical Higgs bundles. What we do is we have a matrix, we express our Higgs bundles as a matrix so we're looking at the determinant of phi minus some variable lambda times the identity and this will be some polynomial that will have lambda to the d, so to the top of the degrees, dk plus a sub one lambda to the dk minus one, et cetera plus a sub k and this is the characteristic polynomial of the Higgs field. Here, probably we want to do it the other way around, sorry for the notation. Let's put here k minus one. Here, each of these a sub k minus a sub and sub i is inside the sections over the remand surface of k to the d i here. So the characteristic polynomial will give us some basis and these basis will be the points in the hitching base but the important thing to get from here is what you know from linear algebra about characteristic polynomials. Something that they tell you is about the eigenspaces and eigenvalues of the matrix you're taking. So the way to think about the hitching vibration and the way that I like it thinking about it and that we're going to do in the next two days is the following. So the idea is the following. The fibers are going to be defined through a point in the base so you need a point in the base and then the fibers on top. So the point in the base of the vibration, so the point in the base a sub gc is going to be given by the eigenvalue. So we're going to be given obtaining the point in the base by looking at these coefficients and these coefficients come from the characteristic polynomial that gives us eigenvalues. So it can be thought as the eigenvalues of phi. It has many eigenvalues over each point of the Riemann surface so really what it's giving us is giving us over the Riemann surface sigma it's giving us a curve of eigenvalues. So I'm going to have a curve here. I'm going to call it s. So curve of eigenvalues of eigenvalues. This is what people call the spectral curve of the Higgs field. So when you think about the hitching vibration and abelianization you think of a vibration where the basis, the point in the base defines an spectral curve, a curve of eigenvalues of a matrix. And then what you do to understand the fiber is you think about the eigenspaces. So the fiber will be given by eigenspaces of the Higgs field. The fiber will be eigenspaces. So let me put it in blue, in red. So eigenspaces and these eigenspaces will be a generically one-dimensional and there will be lines on this fiber. So you're going to be taking a spectral curve and lines on the fiber and the lines on the fiber will be the eigenspaces. So what do you get in the case of, for instance, gl and c-higgs bundles for gl and c-higgs bundles we're saying that the hitching vibration can be fully described by a curve that's a covering of the Riemann surface that sometimes will have ramifications and eigenspaces that are going to be lines here. So the data, the data giving the hitching vibration which we call usually spectral data. So let me put it here, this is what we call spectral is a spectral curve, a spectral curve S, a line bundle, which we're going to be able to show that can be thought of as in the Jacobian of the spectral curve. So the Jacobian is the set of line bundles that have degrees zero over our curve and although sometimes we're going to have other degrees so peak hard varieties with some fixed degree that size is more fixed to our Jacobian. And this is what we call abelianization. The Jacobian is an abelian variety that's going to be our fiber and we are hand seeing the mod light space of Higgs bundles in terms of abelian varieties. So how do we recover these spectral data? Yes, please, they are discrete but your Higgs field is over the whole Riemann surface and so over each point that you're perfectly right over each point of your Riemann surface you have a discrete count of eigenvalues and now I want you to take all of them at the same time, sometimes they will coincide and that's the ramification points. Yeah, that's a great point. Yes. You could think of it like that, yes. Yeah, you could think of it like that. So when people, if you look at the last few pages of Hitching's paper from 87 he'll give an example where he describes the geometry of a component of the mod light space, a component as a covering. So over the Riemann surface, a symmetric product of some degree giving you the spectral cover and then Jacobians. So you can put all of the details all together but what we usually do is we think of it as the Riemann surface, spectral covers and Jacobians on top for each point on the Hitching base. Yes, any other questions? I want to show you how from this data you can go back and forth from Higgs bundles. So what spectral data, how do you get spectral data from Higgs bundles and how do you come back? So the relation between, this is the relation between what we say, the pair S and L, the line bundle and the pair E and phi. So we started with a vector bundle on phi which was a section of an endomorphism of the vector bundle with values on K and now we're just saying you can forget about that and just remember a cover of the Riemann surface and a line on that, which is much a simpler data. So in one direction, when we are starting from Higgs bundles, so we start from a Higgs bundle, we just mentioned there how it arises the data. So we look at the eigenspaces and we look at the eigenvalues. So the curve S is, we usually just write as the zero set of our characteristic polynomial. So determinant of phi minus the variable times the identity, but to have a notion of where this curve lives to think about it as a cover of the Riemann surface, we need to give some meaning to these variables that we have here and so what we're doing to see it as a spectral cover is taking these as the tautological section. So it's a one form. Yesterday some of you were asking me how to think about eigenvalues and you were correctly saying we should think about it as one forms. So this is the one form that's the tautological, we call it tautological section of phi of the pullback of K to itself. So if you think about K as the line bundle over the Riemann surface here with some projection pi, anytime you have a line bundle on your Riemann surface, you can pull back that line bundle to itself and you can look at the evaluation maps to evaluate the pullback into the bundle that you started with. That's what we call tautological section and that's what lambda is going to be, which means that this curve here is inside the total space of the canonical bundle of your Riemann surface. So this is a spectral curve, it's a cover that we can, because it's inside the total space, we can call it pi here, the projection and we have this cover and the line bundle L is defined as the co-kernel of the Higgs field for that point and we can just think of it at the eigen space of phi. Go to the other direction, something is the interesting part of how do we get back a Higgs bundle. So let's keep this drawing here to remember what we're doing. We're taking a line bundle on top of a spectral cover. So we start as S being the zero set of some polynomial. So this is for the general linear group and for the general linear group, you have lambda to the N for GL and C. You have lambda to the N plus A1 lambda to the N minus one plus plus AN where each of the AIs is in H zero sigma K to the I. So we start with a curve like this and we start with a line bundle L that is in the Jacobian variety. How do we recover objects that we want? This cover here is a cover of the Riemann surface. So it's on top of the Riemann surface through pi and when you look at it, it's an N to one cover. So this polynomial is a degree N. This is an N to one cover. And moreover, you can think of it as a linear system in the total space of the canonical bundle. This is a linear system where you can check what are the base points? What are the points where the cover would be singular? And you can use theorems like Bertini's theorem to show that it's generically smooth. So this is a generically smooth curve. So this generically smooth curve is on the Riemann surface and if I take on top of the curve a line bundle L, so if I take a line bundle L here, I can look at the direct image of this line bundle. So I can do the direct image here through pi, so direct image of L and what you're doing if you haven't seen direct images before, you're taking sections of your line bundle and you're pushing those sections down to your Riemann surface. So you do it over open sets and then you extend it to the whole manifold, to the whole Riemann surface. This direct image over an N to one cover for a generically smooth cover is a rank N vector bundle. So we can actually just call it E rank N vector bundle, which is good, we wanted to get a rank N vector bundle and now we need a Higgs field. So how do we do the Higgs field? To get the Higgs field, we remember what we said that lambda was. Lambda was a tautological section of the pullback of K. So if I take L and I look at the map that goes just by multiplication by lambda, so I'm going to multiply by lambda, that means that I'm going to go, I'm going to arrive to L tensor with the pullback of K. That's just multiplication by our variable, the variable that we just took here. So we haven't introduced anything new and now I'm going to push it down to the Riemann surface. So I want to push that down and pushing that down is going to tell me that I'm going to the direct image of L to the direct image of L and because this is the pullback of the canonical bundle, this is actually just K. So this is the map that we wanted. This is the Higgs field here that appears as the direct image of the tautological section. So this is basically what you're going to do every time you're going to take a cover of your Riemann surface, you're going to take a line bundle on it, you push that line bundle down, you push the tautological section down and you get the Higgs field and the Higgs field and the extra bundle. Any questions about pushing down these things? So you might wonder, what happens with other groups? So we mentioned that for other groups, we have to put extra data on the pair, on the Higgs bundle, to reflect the nature of the group. So what would happen with respect to the spectral data? So the spectral data might need some other conditions. For what we just said here, is that we're taking line bundles and we push them down to rank n vector bundles and the line bundle tensor with the line bundle, sorry, the line bundle tensor with K gives us the arrival for the Higgs field when we push down the tautological section. So for GC, we get that the vibration MGC, sorry, for, I don't want just any GC, I want GLN for GLNC over A GLNC because of the fiber is the Jacobian variety. Jacobian variety of S is the fiber on top of a point defining S. So the base point will define a spectral cover and the fiber will be the Jacobian variety. Now, if you took another type of Higgs bundles, so take for instance, take GC equals to SLNC. We saw yesterday that we had to add some extra conditions to our Higgs bundle. So a Higgs bundle is, and not just any Higgs bundle, right? A GC Higgs bundle here is E phi for E rank. And with trivial determinant, we said, so the topics here, power of E is trivial and the Higgs field is as before satisfies the trace being zero. What does this mean in terms of our spectral curve and our spectral data? The first thing that we have is that the first trace, so the first form that we're taking in these cases A1, A1 is zero. So the implications in spectral data will be reflected from these extra conditions we put here. So to understand the vibration for SLNC and see what is it going to A, SL, and C, we have to look at the conditions on the Higgs field and on the bundle. Remember that the Higgs field tells us about the coefficients of the characteristic polynomial and the bundle tells us about the eigen spaces, the line bundles. So the Higgs field will constrain the base, so this person here will constrain the base and this person here will constrain the fiber.