 Sorry. Let's just call pi i, pi i the projection from the product as one plus s2 to s i. There's also the map from each of the spectral covers to the Riemann surface that we used to call pi. Sorry for the mix implementation, we're not going to call pi i that. So we're going to pull back by pi 1, the line bundle 1, and we're going to pull back by pi 2, the line bundle L2. What you can do is you can actually show that this is the line bundle that is going to give you the orthogonal Higgs bundle that appears here. So it's not any orthogonal Higgs bundle. When you look at the spectral data for this person, it is indeed this person here. In particular, this is already the normalized curve. So normalized. So the method through isogenes that allows you to recover a normalized curve already from where you are. Before moving on to the graph, part of the correspondence like me, just finish this part by giving you just a brief idea of what happens in the second case. This is in the second case, you don't have two spectral covers, but you do have one which you can take as five or product with itself. So if we call this one is i2, this is i3, the image of i3 has spectral data. The spectral data for this one, remember that's going to be an SO6, so it has a spectral data S6 and L6. Given by the following, you have to take the product S6 is going to be asymmetrication of some part of the fiber product. So I'm going to take S with fiber product with itself. I'm going to take away the diagonal component. So S with itself has a diagonal component. I'm going to take it away. So this means take diagonal component, take the diagonal component away, and then I'm going to symmetrize it. Once I do that, I have the six cover that I want, and then the line bundle L is going to be, just like we did before, is going to be the pullback under. So it's going to be the pullback under. Now there's two projections just like we did before, but it's of the same curve. So we're going to pull back the same line bundle. So pi1 pullback of L, pi2 pullback of L, and we don't want to take all of it. So all of it wouldn't give us orthogonal Higgs bundles. Remember the orthogonal Higgs bundles correspond to spectral data in a print variety. We said here there's a print variety. So it's anti-invariant objects under this involution sigma. So these are line bundles like this, and we're going to take the part of this line bundle that is anti-invariant under the involution. Anti-invariant part under the involution. Let's call it sigma hat. There's an involution here, sigma hat that switches the two curves, and this is what you have. So this one is a bit more work to show it, but you can actually prove that doing all these steps, so taking the diagonal out and symmetrizing then taking the line bundle and taking the anti-invariant part gives you the spectral data. All of this is done for the regular fibers of the Hitching Fibration. When you go to the singular fibers, we are completing some of the work with Cibralo and Lucas Branco, but there's still much to do in terms of these isogenes for singular curves and singular spectral data. And we're interested in singular covers and the fibers over singular points because when you look at the brains, these maps, so just a final remark. Both of these maps map BAA brains to BAA brains. When the form is split, when the form is split, we showed it. For other forms, we are in the singular fibers always. We're showing it for some other real forms, but there's many that are left yet to do. So you want to try and understand what happens with every part. Okay, so in the last 20 minutes, I want to mention a little bit about how graph theory can be used here and how do we get correspondences between Hicks bundles and quivers, which are types of graphs. The correspondences that we're going to get are in particular through parabolic Hicks bundles. So whilst we didn't cover them in the first class, I want to just make a note of what parabolic are quickly for you to have in the notes you're taking, and then we're going to compare with quivers. Any questions? We're okay? Okay, so we're going to finish off with these last things. For these last things, sometimes we're not going to be talking about the Riemann surface of genus two. Sometimes we're going to look at P1 and we can put groups, but let's just talk about GL for now. It's going to make things a bit more tractable. So let's define two things. We're going to have parabolic Hicks bundles, and once we have parabolic Hicks bundles, we're going to need quiver varieties to be able to go. So these are the graphs that we're going to use. For parabolic Hicks bundles, we're going to need a divisor fixed. We're going to follow Metan's Chaudry's work from the 80s, and we're going to fix a divisor on our Riemann surface. So fix an effective divisor D that's going to be P1 plus PN, a divisor on Sigma. These divisor, when we talk about polygons, I have polygons, we're going to need it to have different points for now we don't need. So what is a parabolic vector bundle? A parabolic vector bundle, vector bundle E on Sigma is a vector bundle of rank, say of rank R, is a vector bundle of rank R with parabolic structure. So is a vector bundle of rank. So vector bundle, we're still going to call it E with parabolic structure. And what is the parabolic structure? The parabolic structure we're going to use, or the way that one defines its objects, is by looking at a flag for the vector bundle and a set of weights that go with that flag. So we're going to have a flag. Let me put here a flag. And the flag is going to be a sequence of sub bundles, just like when you were doing Jordan Holder Filtrations. It's very similar for each point. So I want to keep the notation of the notes. So E, so over Pi I, for each Pi I, we're going to have Pi I is zero, included in E, Pi I, one, included dot, dot, dot, including E, Pi I, comma R, I. And this is going to be our E, Pi I is the flag is a sequence of sub bundles of the vector bundle and the weight vector. So the weight vector alpha is going to have alpha one to alpha. All right, so alpha. So it's going to have alphas for each I, it's going to have from one to our I. So let me just put it here. So some people go from one, the notation they use, the zero to our I, sometimes they put it back where it's once you're doing calculations, make sure that you know which notation they are using. That's why I'm trying to keep the same one that I used in the notes. So I'm going to go from one and this is what I'm going to call alpha I is zero. And this is always going to be bigger than alpha I, one, bigger dot, dot, dot, bigger than alpha I, R I, that is the last one that we have and this has to be bigger than zero, bigger or equal than zero. These are the weights and through the weights, we can find the parabolic degree. So I was very optimistic in space and we're going to define the parabolic degree of E, taking into account the degree of E, but also these weights. So it's the degree of E and now we're going to have the sum over all of the points and then over all of the weights. So it's going to be the sum from, I'm going to put J equals to one up to N and then we're going to have another sum from I equals to one up to R I and now I'm going to put the alpha J I, like we did before, but we're also going to consider the rank of the equations of these spaces. Remember when we're looking at stability, we're looking at degrees, we're using some ranks. So I'm going to put M I here J I and M J I is the dimension of E P J comma I over the dim, I'm sorry, over E P J comma I plus one. Once we have all that, we can actually define Higgs bundles. So the definition, and this is where I wanted to get with all these is a parabolic Higgs bundle, is a pair E phi, is a pair E phi for E parabolic vector bundle. So the parabolic vector bundle comes for you with the parabolic weight. So the weight vector and the flag, that's already given here on the Higgs field now is a section like we used to have a section over the Riemann surface of the endomorphisms of E, but instead of twisting it just with a canonical bundle, we're going to twist with the canonical bundle twisted by the divisor that you started with. And we have to have the extra condition that it preserves the flag. So this flag here needs to be preserved by the Higgs field, which means so such that the Higgs field when applied to one of these E E p i comma j has to come back to the same part. So it has to come back to E p i comma j tensored with k. If you had a sub bundle for each rank up to your original rank R, then you'd have a complete flag. Many people are interested in complete flags because they give you some more room to work and things become a bit easier. It's also sometimes good when you're, if you want to look at examples of parable Higgs bundles, just take rank two vector bundles so that the flag is just a line bundle and then calculations go very nice. And a remark here that we should make because we're going to come back to it is that we can think of parable Higgs bundles as Higgs bundles with tame singularities. So having these marked points and having the Higgs field having zeros over those points or having tame singularities will be equivalent. So we can think of parable Higgs bundles as tame what people call tame Higgs bundles. Higgs bundles with singularities of degree one over those points. With all of these, what I wanted to do is I want to have the space built. So once you have Higgs bundles and you have the parabolic degree, you can actually define parabolic stability and you can find the modulate space of parable Higgs bundles or tame Higgs bundles and that's the space we want to associate to some quiver varieties. So let me tell you a little bit in case you haven't seen quivers before about quivers. How are we doing? We have 10 minutes left. So this should be fine. So let me tell you about quivers. Any questions? Okay, so quivers and our goal. So whilst I do this, if you have seen quivers before but you haven't seen the correspondence with parable Higgs bundles, try to think how could you do it? How could you make that correspondence? If you've seen or read some of the advice from Athea to young mathematicians, he tells you even when you're bored just look at the lectures and try to prove the statements yourself better and in a cuter way. So try to do that if you get bored. Try to see how these correspondence works with quivers. So what is a quiver? A quiver, a quiver Q is a directed graph. It's a directed graph. So we can actually call it as a set of vertices, arrows and then I want the arrows to be directed. So I want heads and tails. So these are the vertices. These are the arrows. These are the head and tail maps. Head and tail maps. So what is it telling you? It's telling you for each arrow who is the tail and who is the head. So they are maps H and T are maps from the set of arrows to the set of vertices. Telling you who is the head and who is the tail. Quivers, what we can do to quivers is we can actually associate vector bundles to each point. So let me tell you here what do we want to do? Okay, so I want to keep this notation. I want to keep it on the board. An example to start with where I'm going to have a central node v zero and I'm going to make what we call star-shaped quivers. So star-shaped quivers are quivers that have a central node and arrows going from outside vertices to the central node here. So v one, v two, v three, v four. This is what we call a star-shaped quiver and you could have any number here of other arrows going star-shaped quiver. When you consider Higgs bundles and quivers, you should think that a Higgs bundle, so a remark, a Higgs bundle, remember, was a map, a Higgs field, a map from some e to e-tensor with k. So if you think of a vertex being e and another vertex being e-tensor with k and the map in the middle being phi, then you have a quiver, right? So just the easiest way of putting a Higgs field into this notation is also by doing this, associating two vertices and giving vector bundles here. So what we're doing here is we're going a step further. We're looking at quivers, but we're also putting vector bundles on each. So this is what it's called a representation of a quiver. A representation of a quiver will be an assignment of vector spaces or vector bundles to each point and a map between them. We can also write it in a different way as homomorphism. So let me give you both things. The space of representations, representations of a quiver, of the quiver is going to be the following. It's going to be a map. Sorry, it's going to be the space rep q, which is the sum of all of the morphisms. So here we have maps from the outer arrows to here. We are going to assign some value to v zero. We're going to assign a dimension vector that we usually call D. And this is going to give an assignment of a dimension for v zero. So I'm going to put R here and then dimensions for the outer vertices. And we usually take one to see the correspondence with Higgs bundles. So one, one, one, one, one. So this is the dimension of v zero, them for the vertex, the central vertex. And these are n dimensions one for ultra vector. So here we just have four. So it will be R and four ones. The dimension, so with that, we can form the space of representation summing up to whichever number in our star shape quiver we have. And we're going to do from one up to n. And we're going to look at the morphisms from dimension one to dimension R spaces. So homomorphisms from dimension, we're going to do complex numbers. So we're going to look at C, C R. What is this space? So we can think of these as just homomorphisms. Now we're doing from one to n of C to C R. We can think of it as C n to C R. And equivalently, we can just think of it in terms of products of bundles. So we can think of it as C to the N cross R. So this is one way of seeing the space of representations. But as we were mentioning, when putting Higgs bundles there, we can also think of it as an assignment of bundles, of vector spaces for each vertex and maps between them. So we can also think we can also express this as a set W composed of a collection of Ws, V for each vertex and phi sub A for each arrow, where W, V is going to be a vector final dimension vector space on the vertex Vi. And phi is a map that goes, so phi A is a map that goes from W with the head of A, from the tail of A to W with the head of A. Now, in this notation, I put it a bit more general. What we're doing here that we have this dimension vector with R and 111, this is telling us that in particular in our setting from that board, we're taking these final dimensional vector spaces as being of dimension R for V zero and one otherwise. But you could do it for any dimension. The interesting part comes by considering the space of representations and an action that it has through which we can quotient the space and get polygons and hyper-polygon spaces. So let me just make a quick comment of what's going on since we only have three minutes left. I want to consider these quivers. One thing that you can do to quivers is you can look at the double quiver. So we're following here work of Nakajima from the 90s, 96, 98. And the double quiver of a quiver is when we add arrows going the opposite direction here. So if we add these arrows going here, what we get is q bar, the double quiver. So this space of representations and this space of quivers can be defined, you can define a slope given a parameter. So there's a given a parameter, given alpha, a stability parameter, parameter, you can define slope for the space of representations. So for rep, q, and also for rep, q bar. And you can see more details there. And the other thing that you can do is you can consider a natural group acting on these star-shaped quivers, the group G that is going to be a group of rank R acting for these vertex and then S ones for the ones that have dimension one. So we can write it as the projectivization of U R cross S one and this S one is N times. So if we have an N quiver, N star-shaped quiver. So there is this group acting on the space of quivers, q and quivers, q bar and hence on the space of representations that we defined there. And let me just finish off by telling you, considering this space of representations that we have here, telling you two spaces that you can form by looking at the quotient space under this group action. So you can look at the space of representations and you can obtain the polling on space that people call p sub alpha for certain stability parameter alpha, p sub alpha is going to be the quotient of the space of representations of q under the action of G for that stability parameter. And if you look at the cotangent space of the space of representations, so if I look at the space of hyper-polygons, it's going to be some space that we usually denote by X. We usually denote by X. So here we need to remind ourselves we have the parameter alpha, we're looking at rank R and N points. So N R here is going to be N R alpha again and it's going to be the cotangent space of representations quotient by G for that parameter alpha and then zero for the cotangent direction. So this takes a few pages to define, but this is the rough idea. You can form these two spaces. When you consider these two spaces, you can look at points here X, Y and ask what do they correspond to? And I know I'm a minute late, but let me just have one more minute and tell you that there is a correspondence with parabolic Higgs panels. So what Ryan and Fisher showed a few years ago is the following and I'll just finish here. A Higgs field on P one. So there is a correspondence between parabolic Higgs bundles E phi on P one where E is equal to C R cross P one and phi is equal to, so phi of Z is equal to the sum for I equals to one up to N. So we're summing for each of these points. We're having a Higgs field which has a zero on the point of the star. So we're going to put phi I VZ Z and here we're going to put Z minus Z minus Z minus Z minus Z minus Z minus Z Pzu I and these phi I's correspond to having X I, Y I in your variables for the polygons. So there is a correspondence between these pairs and hyper polygons Y with divisor P one to P N. These correspondences between very particular parabolic bundles, so it's parabolic bundles which have this structure. What we can do now, what we're trying to extend Ryan and Fisher's work to have many more vertices in the star shape quiver which would correspond to the flag being longer and hence try to put the whole multiline space of parabolic Higgs bundles into this setting. I should stop here. Thank you very much. Any questions although we're a bit late. When you say any other, you mean like for other groups or for, I think you always have to consider some parabolic weight because mainly you're marking points in your remand surface and so you need to recall that information, not to make it obsolete. Why are you marking points? But if you have a favorite set of Higgs bundles which can recall that information of mark points without having parabolic structure that could work, you can do wild Higgs bundles. So that's something that we're doing. You can do further than P1 by adding loops. This is what work. If you add loops here in this quiver, then you can't use the setting of some of the nice properties of the Nakajima quiver varieties, but you can still define this parabolic or wild Higgs bundles for higher genus. Okay, maybe we should stop since you had to start again very soon. Thank you very much for all these four days listening to me. Thank you.