 So because it constrains the base, it tells us that the first invariant polynomial, remember that a basis of invariant polynomials was given by traces, for instance. So the first invariant polynomial is always zero. It means that our basis, when we think about it, is going to be the sum from i goes to two now, up to n of h zero sigma k to the i. So we're taking all of the differential forms, like we did for the general linear group, that we had all of them, but we're canceling the first one. Now what do we do about the fiber? The fiber is a condition that we want to get from the topic stereo power. So we want to understand, so we want to understand the condition, topic stereo power of E, isomorphic to the trivial line bundle in terms of a condition on the spectral cover, line bundles on top. So we want this in terms of a condition on the line bundle L over the spectral cover. So we need to try and reflect what is happening to the line bundle, which conditions do we have to put to the line bundle, such that when we push it down, we get trivial determinant. What does this mean? Because we want trivial determinant, we want O to be equal to the topic stereo power of E, but this is also the topic stereo power of the direct image of L. So how to understand the determinant bundle of the direct image of a line bundle? And for this, it's very useful to recall some properties of these vibrations and the covering. So let me just put the correct formulas here. There is a very nice formula that Bobilner, Assiman and Ramanan did use back in 89 when looking at the coverings of Riemann surfaces and direct images. So if you haven't seen this paper before, I really recommend it to you. So let me just write from Bobilner, Assiman and Ramanan. And once you use it a lot of times, you'll remember this formula by heart almost. The direct image of the determinant bundle of the direct image of a line bundle can be expressed in terms of the norm map. And I'll tell you what this is if you haven't seen it. The norm of the line bundle tensored with the canonical bundle to minus N, N minus one over two. So the norm map is a map that arises from your cover of the Riemann surface. We had here our cover S of the Riemann surface, sigma, and this was an N to one cover, right? So at the same time, when we think of line bundles on the cover of the Riemann surface, we can think of the line bundle, we can think of the divisor associated to the line bundle. So I want you to take a divisor here on S, so the divisor being the sum of p i's from i equals to one to some variable r. So this is a divisor on S. And I want you to look down at the image of those points under pi. So we're going to go down and look at the divisor that we get when we look at the image, the pi of p i. This is what the norm map does. So the norm map is sending a divisor down to its image on the map. If you have two points that coincide, we'll have something of multiplicity here. If otherwise, we have different points. And through this norm map, we can look at this formula. An interesting thing appears when you consider the kernel of this map. So the kernel of the norm map is a very interesting variety, which is called the print variety of the cover. So of S over sigma. So for a print variety, you always remind yourself what the cover is, what the base point is, and that information already tells you through the projection pi that already tells you what the norm map is. And the kernel would be all of the line bundles here that actually go to zero. So if you had a line bundle, for instance, of order two, when you projected to a two cover, you'd had the identity. But that's just one way in which you can get it. So the print variety will be relevant for us because we're trying to say this bundle has to be trivial, which means that this determinant bundle has to be trivial. So it means that we want all of this to be trivial. We want all of these to be trivial here. And a way to do it is by, instead of thinking of norm map of L tensored with k to the minus n, n minus one over two, is putting this k inside the norm map. When considering k and looking at the norm map, the norm map just multiplies it by n. So the canonical bundle gets multiplied by n. So if I want to put it inside, I need to take this n out because when I take it out, it's going to add an n. So this is equal to the norm map of L tensored with k to the minus n minus one over two. And what it's telling us is that this element is going to be trivial if we require, so this is our requirement and this is our requirement. So this determinant bundle is equivalent to this norm map and asking for it to be zero is equivalent to asking for this to be trivial. So what we have is that the direct image of L has trivial determinant, so the top exterior power is trivial if and only if the related line bundle, so L tensored with k minus n minus one over two is in the prime of s sigma. So the condition for the vector bundle to be of trivial determinant, this condition that we had here is actually a condition for the line bundle to have someone associated in the prime. So the prime variety appears here. What we're seeing is that the fiber over generic point here is the prime variety of s sigma. So we deduce how the conditions added to the Higgs field and to the vector bundle appear as conditions on the cover. So in what remains of the talk I want to tell you a little bit about how to see some other properties of the Higgs vibration. Let me before that make a couple of comments about it. So I want to leave the prime here, let's just put it there, that's a great question. So there's work by Simpson that explains how to think of them as shifts, so non torsion free shifts, shifts over the curve that's not going to be a smooth cover. For GLNC Higgs bundles, instead of Jacobians you're looking at compactified Jacobians obtained through this cover. For other groups, I think for many types of fibers is not really known, which geometry they carry. So when we look at the, let's follow on your question, we're looking at MGC over the Hitching Bay, so AGC and the Hitching Bay will have some non smooth points. So this will be what I'm going to call the singular or discriminant locus of the Hitching Fibration which are going to be those points corresponding to characteristic polynomials which don't give you a smooth curve. So corresponding to that phi minus lambda ID equals to zero, not smooth, not a smooth fiber. So you'll see, I tried to add some references in the lecture notes, if you are interested in particular details, there's some work done for SL and C, for instance. Work of Gothen and Oliveira showed that all of the fibers are connected, so even the singular fibers. But a question following what you said is what happens with the fibers here? Fiber over singular points. The fiber over the most singular points, the most singular point is when the Hicks field, when this curve is just zero. So when all of the coefficients are zero, this is over the zero. And the fiber here is what we call the Neil-Portin cone. And the Neil-Portin cone will encode information from many different types of Hicks bundles. If you remember yesterday, we mentioned when the group is compact, a real group is compact, your Hicks field is zero, so your Hicks bundle is just a vector bundle. So inside the Neil-Portin cone will be that case because when the Hicks field is zero, then the coefficients are zero, and we have a component which is the vector bundle, modernized space. However, there's going to be other people there. Hicks fields, which are Neil-Potin, are going to have the zero in the characteristic polynomials on time, and they're going to go over the zero. So understanding what the Neil-Potin cone is, in general, for other groups than GC, for GC there is quite a bit understood, for SL2 is understood, but understanding what the Neil-Potin cone is a very big question that would get a lot of attention if you could say something about it. Understanding the generic fibers is something that we can approach. Understanding the singular fibers, the whole of the singular fibers becomes complicated. What people tend to do is they understand more theory for the Hitching vibration through that C-star action that I told you about yesterday, and the fixed points of that action lie over the singular locus, but they don't really look at the whole fiber, they look at the fixed points over that singular locus. So what I'm going to be interested in the next two talks is to try and understand the singular fibers by not looking at the whole fiber, but looking at brains that completely lie over this blue part, so over the singular locus. For more information about these singular locus, you should take a look at the references by the end, the last section of chapter two of the notes, that's going to have some information about this blue part. So the next, the other thing that I wanted to tell you is about how to see some brains. We're going to do a bit more tomorrow, but I want to say, tell you how to see real Hicks bundles. Real Hicks bundles in the vibration. So we said real Hicks bundles can be thought as inside the modular space of complex Hicks bundles, so Mg is thought as inside MgC, which means we add conditions, so G Hicks bundle is Gc Hicks bundle plus conditions. So we're going one step further, we're adding some more conditions that makes it a real Hicks bundle, and hence those conditions should be reflected on the fibers on how they intersect, the fibers of the hitching vibration. So one of the examples we talked about yesterday, and see what happens, we're going to not need these. I want you to think about you Hicks bundles, so you need to have Hicks bundles with signature, or especially you need to have it with signature. So take, for example, take usupp, Hicks bundles. Hicks bundles. When you take real Hicks bundles, there's a lot of geometry that appears, and the geometry is going to be very interesting, sometimes involving quotient curves, or auxiliary spectral curves, and the spectral data that they will carry, sometimes it will be abelian, but many times it's not going to be abelian. So this is what we call non abelianization of Hicks bundles, in contrast with abelianization one, which is the situation that occurs when you take Hicks bundles for certain groups, for which there's no abelian data that can exist. But this one, for this case, it's abelian, and let me just remind you what these Hicks bundles are. So these Hicks bundles are our pairs V plus W, and a Hicks field phi, where V, W, are rank P, they're holomorphic bundles of rank P, and phi, it's an off diagonal Hicks field, beta and gamma, so beta going from W to V tensored with K, gamma going from V to W tensored with K, and the fact that we took the special unitary group means that the total space, so the topics to your power, which is the topics to your power, to the power of two P of V plus W is trivial, or equivalently we want the exterior power P of W to be equivalent to the exterior power P of V two. If you take the S, we get that condition, let's put the S condition in between brackets, and let's put this condition in between brackets. I want to understand first what the spectral data is, what the spectral curve is, and then I want to understand what the spectral fiber is, so how do these Hicks bundles lie within the hitching vibration? We have this hitching vibration, we have some smooth fibers in the hitching vibration, in this case we're not going to be interested in the singular fiber, because these guys lie over the non-singular ones. So we're going to have the non-singular ones, and we want to ask this real moduli space, this moduli space of UPP Hicks bundles, or yes UPP Hicks bundles, how do they lie here? Is there any variety? So what, what describes, what describes them G and over which points it describes it? So some points in the hitching base they will be describing this Hicks bundle. So we have 10 minutes or nine minutes, let's take a look at what we can see from here. The first thing we can notice is the structure of the Hicks field, the Hicks field is off diagonal, which means when we take traces as a basis of invariant polynomials, the trace of this one is there, the trace of phi square is not, but the trace of phi cubed is there, and so on. So every odd, every odd power of the Hicks field, so every power of two i plus one of the Hicks field will have trace equals to zero, which means that your spectral cover has only even degree polynomials. So this is a property that will carry to any basis that you have of homogeneous polynomials, and so we can express it as the polynomial only has even degrees. So our polynomial, which is the determinant of phi minus lambda times the identity, which was lambda to the two p plus, and now we know that it's only even degrees. So let me just pull them here. I'm going to put a sub one lambda to the two p minus two plus, plus a sub p, and this equal to zero defines our curve s, and here a sub i are sections over the Riemann surface of k to the two i, so our differential forms of degree two i. This curve, one thing to notice is that because it has only even degrees, there's an involution that preserves it, right? I can put lambda goes to minus lambda, I'm still preserved. So this has a natural involution sigma that sends lambda to minus lambda, preserving the spectral cover. Since I can preserve the spectral cover, I may as well just quotient it to see what happens. So I have my cover of the Riemann surface pi, and if I look at the quotient under sigma, I can define a new cover as bar, which is the quotient s by sigma. The quotient curve will be obtained through a two to one, so this is an end, sorry, this is a two p to one. We have a quotient, which is a quotient by a two one cover, and now this one projects as a p to one cover. This process of taking a quotient can be expressed in terms of equations and in terms of tautological sections. So remember that lambda was tautological section of pi, the pullback of k. I can take the quotient curve, so s bar, is going to be the zero set of, and now I need to put a new variable, so let me just call x i to the p plus a one x i to the p minus one plus a p equals to zero, equals to zero, and here the x i is the section of the pullback of k square. So this quotient curve here that we wrote here lives in the total space of k square, so it's not longer in the total space of k, it's in the total space of k square. Now we put the conditions on the hitching base, we know that it's only polynomials with even coefficients that are going to be here, so these ones will be just even coefficients, the points in the hitching base that will give us, so what's the fiber? What one can show is that the fiber for UPP Higgs bundles has to be obtained by considering line bundles, so for the fiber, we consider line bundles on the spectral curve s, which are preserved by the involution, so such that the sigma, the pullback of L, is isomorphic to L, so this is the condition that you want on here, and in a few minutes that we have left, I'll show you how you can see that, so what describes MG, this is the line bundles such that they are pulled back to themselves in the hitching, over the spectral cover. One thing that happens is that it's not as pretty to think about line bundles which are preserved, we started with a very nice variety, the Jacobian variety, a compact Abelian variety, which we know a lot about, and now we have a subset, line bundles preserved by an involution, the way that we can see that these line bundles have to be preserved, so the way that you see that the eigenspaces of these Higgs bundles are preserved by this involution is by considering how they behave with respect to the involution that sends L to L, but the involution that sends the tautological section to minus. Remember that we were talking about the tautological section giving us a Higgs field, so the tautological section, when we push it down, gives us the Higgs field, and now we have an involution that takes a Higgs field to minus a Higgs field, so the line bundle for an eigenvalue and the line bundle for the minus eigenvalue should coincide, and that's what's telling you here, that the two eigenspaces for the two different eigenvalues minus an eigenvalue, an eigenvalue itself, they should coincide, and this is the isomorphism. But we don't like, as a description, it's not telling us whether it's an Abelian variety, it's not telling us how many connected components that yellow part, that orange part here has, we're interested in knowing, are there many components here that describe these Higgs bundles? Are there just one? What's the geometry and what's the topology? So we can understand, or the aim here, aim is to get a geometric description of the spectral data for G-higgs bundles. So that condition is not enough for us to understand it, but one thing that we can do is we can figure out what it is in terms of this cover. So a line bundle that is preserved can be thought in terms of invariant and anti-invariant sections. So note, and this will be just the beginning, I don't think I'll have time to tell you much about it, but you can look at the notes, the extra notes, but what you have to note is that since the sigma L is brought back to itself, then when we look at the direct image on the quotient curve, so if I look at the direct image under this map, say, rho here, rho direct image of L, as well as the direct image under pi of L, they can both decompose in terms of invariant anti-invariant sections. Remember that we mentioned that to get direct image, we look at sections of the line bundle and we push down the sections. You're now thinking, when I push down a section and I apply the sigma, does it take me back to the section or does it take me back to minus the section? That will separate the sections into two sets, the invariant and the anti-invariant, and those two sets will separate the sections of each of these direct images and hence will separate the bundles themselves and you'll get the structure of a V and a W, coming one from the invariant sections and one from the anti-invariant sections. Moreover, these invariant and anti-invariant sections might be related. So for instance, when you're trying to see whether you have the condition on the determinant bundles, this red condition that we put here, you have to account for that in terms of invariant and anti-invariant sections and what you'll see is that when you take these fibers, in fact, you can describe without the S condition, so just for UPP, you can describe the fiber as the Jacobian of the quotient curve, which means you started with a line bundle on your curve, you push it down to rho star of L, this is on S bar, and these will separate into invariant and anti-invariant sections. Those two are related, one can be expressed in terms of the other, so we only need one and that gives you a Jacobian. If you were to put the extra condition here and fix some degree, you'd get in many cases the prime variety of S bar over C. So by adding extra conditions and the details of these are in references in the lecture notes and there's a bit more explanation in the lecture notes. The way that we do is we first look at the conditions, the overall conditions that give us structure, so that being invariant, so that we get the separation of the two vector bundles, and then we see how can we express that condition in terms of a variety that we know, like a Jacobian variety or a prime variety. What we're going to do tomorrow is we're going to look back at that diagram that we put here as the motivation and we're going to see which properties now that we know how the hitching vibration works and we know how generically we can get for complex groups or for some real groups, what happens with the duality with the mirror symmetry, what happens with other correspondences and how can I define sub-spaces which are interesting inside the monolid space of hitch bundles and which have interesting correspondences. So I think that's all for today, thank you. Are there any more questions? Yes, yes, yes, so that's a great point and that will lead us to tomorrow. As we mentioned, the monolid space of hitch bundles is hypercaler, so you have a sphere of structures. The triple means that I fixed three structures that I chose with some convention and for those three structures, my manifold or my sub-space is a complex sub-manifold which is a B or a Lagrangian, which is an A and that will remind me if it's a BAA or a BBB. These ones that we're seeing here, for UPP, these are BAA brains. So they're Lagrangian sub-varieties with respect to some of the structures and they're complex sub-varieties with respect to the other one. Yes, yeah, and we're going to see how duality could work for those objects. But for those that haven't seen this hypercaler structure before, we'll do that again tomorrow. Yeah, any other comments, questions? No, otherwise I'll see you tomorrow. Thank you.