 The first thing that we want to talk about is vector bundles that many of you will have heard about, but I want to recall some main properties of vector bundles that will be used when considering augmented bundles or decorated bundles, such as Higgs bundles. So how many of you have worked with Higgs bundles before, with vector bundles before? With vector bundles? So you probably know what degree and rank are, and the main thing that we're going to be looking at is stability conditions that we want to use for Higgs bundles. So an introduction to the properties that we want for vector bundles, given a vector bundle, and I should mention where we're going to be working on a Riemann surface. So I'm going to call sigma compact Riemann surface of genus G bigger equal than 2. You can put lower genus and you can put mark points on the Riemann surface, maybe we'll get to that part where we define parabolic structure on those mark points, but for now let's just stick with genus at least 2. So a vector bundle on sigma has slope, and the slope is the following, has slope equal to slope of E. So a vector bundle E here, slope of E is the degree of E over the rank of E. So if you haven't seen degrees and ranks before, what we're going to do in these lectures is something that can be do locally, and locally you can think of the rank as the dimension of your fibers over each point of the Riemann surface, you have a fiber, which is a vector space, it has a dimension, that's the rank, and the degree will be the number of zeros that a holomorphic section of the bundle has. So if I think of E over sigma, the section will be a map from sigma to E that will have zero sometimes, and the number of zeros will be my degree. So this is a slope, and we're going to say that a vector bundle is stable or semi-stable depending on the slope of the sub bundles. So we say, and you should tell me if the handwriting is too small, just shout and I'll make it big. So say we say that is stable or equivalently semi-stable if for every sub bundle that it has, so for every F inside E sub bundle, the slope of F is less than the slope of E. So the mu, I'm going to call this for making it concise, the mu of E, the slope of E. So mu of F is less than mu of E. And for semi-stability, we want less or equal. So this is a stability condition that we need to put in order to construct the modelized space of vector bundles. What we can do once we have this stability or semi-stability condition is we can look at isomorphism classes of vector bundles with fix, rank, and degree. And that's what we usually call N. So N, and we can put here the rank, so say rank R and degree D, the modelized space of rank and degree D vector bundles. So we need a stability condition of isomorphism classes of semi-stable bundles. Yes, yes, I mentioned the sections have to be holomorphic. So let me just put it here, holomorphic. So let's put it here, and I guess I put here the reference of holomorphic. And this is why I was telling you to think about the degree as the number of zeros of holomorphic sections that the bundle has. Thank you. So we can look at this modelized space, and we can ask what are the properties of this modelized space, and we can ask how its tangent space and cotangent space relate to Higgs bundles, and make sure that I have everything I want to say. So one of the things to make sure we remember, because it's going to affect the modelized space of Higgs bundles, is when is it smooth? So remark, when D and R are called prime, so for D, R called prime, then the modelized space is smooth. It's a smooth space, and the dimension so n square g minus one plus one. Look at this modelized space, but not just the modelized space, but its cotangent space. So I want to consider its cotangent space, which Higgs bundles from a different perspective than what those of you at Richard's talk read, which is from the perspective of the self-duality equations, just that you have a new direction to see them, and it's kind of a complementary direction to those appearing in the notes. So just that you don't get bored seeing the notes and seeing these, I want you to see that Higgs bundles can appear from many different ways. What's sorry? Sorry, the rank. No, it's R. Thank you. The rank is going to be R when I remember, and otherwise it's going to be N. Yes, thank you. Let's try to make sure that I, and g is the genus, so that bit is fine. Here I hadn't mentioned the rank. This is what we're going to usually call R, and this is what we usually call D. Okay, so what's the cotangent space? I need to get a representative of a point. So this is isomorphism classes of vector bundles in the modelized space. So we take a representative of an isomorphism class. Take V, a representative of a class, and so I want to look at the cotangent space at the representative of this modelized space. Let me drop the N and D, sorry, the R and D. Let's just call it N for now, since it's clear that we are fixing rank and degree. So what is this space? We can write it down, actually, in terms of endomorphisms of V. So we can write it down as H1 over the Riemann surface of N, traceless endomorphisms of V. This is, you can think of them as the first cosmology, just like you see there, of endomorphisms of V over the Riemann surface. But now we want to use a duality to understand it as terms of sections, so H0. So we can use third duality to make it isomorphic to H0 and using third duality. We put the zero here and we look, what are we looking at? We're adding K, we're adding the cotangent space of the Riemann surface. So cotangent space of the Riemann surface and we're tensoring my end V. And are we remember to put the dual here? And I'm going to give a name to this space because we're going to come back a lot. So I'm going to get K. This is the cotangent space of the Riemann surface. It's also called the canonical bundle of the Riemann surface. The canonical bundle, when we consider off a Riemann surface, is a line bundle. So this is, if you haven't seen this object before, is a line bundle. Sorry, this is probably too small. Line bundle of degree 2g minus 2. So these are the properties we're going to use of the canonical bundle. Okay. Yes, thank you. And so I want to consider what this object is in terms of now sections. So these are sections over the Riemann surface of all of these. So this is a line bundle tensored with dendomorphisms of V dual. How can I think of that? Remembering that dendomorphisms of V can be thought as the product of bundles. This is the same as taking sections that are on sigma, which are traceless and go from V to V tensored with K. So from V to V tensored with K. So here we have K and this is a dendomorphism dual. We're thinking as maps from V to V. So these are maps. We can put them in name. These are holomorphic maps on the Riemann surface. There are sections on the Riemann surface that can be thought as traceless maps between vector bundles. And this takes us to Higgs bundles. So let me now tell you what Higgs bundle is. For those that haven't seen them before, a Higgs bundle, a Higgs bundle E phi on sigma is a pair, is a holomorphic vector bundle, is a holomorphic vector bundle. This is on the Riemann surface and phi is a map just like those that we have here. So it's a map from E to E tensored with K holomorphic. We mentioned that K is a line bundle and has degree 2g minus 2. If you've worked with line bundles before, you know we can take the square root of even degree bundles, line bundles in particular. So let's take the square root of K to construct a classical example of Higgs bundles. So you can see what we're talking about. So what's the example that we want to look at here? We want to consider E the sum of a square root K to the half plus the dual of the square root, the minus one half. So for these, we fix a choice of K to the half. There's 2 to a 2g choices of K to the half. We just fix one and then we let the vector bundle be that. Now our Higgs field, we're going to be doing a lot of linear algebra this week because as you can see we're talking about vector bundles which locally are vector spaces and maps between them will be locally matrices. So the Higgs field goes from E to E tensored with K. So K to the half plus K to the minus a half to K to the half plus K to the minus a half tensored with K. So this is a line bundle and a line bundle is a rank 2 bundle. So our Higgs field will be a 2 by 2 matrix. And the way that I want to do it, I want to have a 2 by 2 matrix which is of diagonal. And let's think about the entries. And what are the entries here? So this matrix when applied to this pair is going to go from K to the minus a half to K to the half to K. And so I'm going to put an omega here. And if we think just like we were doing here, alternating between vector bundles and sections of bundles and maps between bundles, a map which is from K to the minus a half to a half to K can be thought as a map omega which is a section on the Riemann surface. And what we do, just like we did in that side, we put this one, this degree in the opposite sign and we sum them to those ones. So K to the one half plus one half plus one plus two. So it's really a quadratic differential. So now we can look at the same thing for the other entry. This entry goes from K to the half to K to the minus a half to K. So they cancel out and it's going to be just the identity on K. So the identity. So what we have here is a family of Higgs bundles parameterized by quadratic differentials. So family of rank two Higgs bundles, rank two Higgs bundles parameterized by omega, quadratic differentials omega. This family we'll see later on that when considering an appropriate modulate space of Higgs bundles for some groups, this becomes a component. And this is what's known as the hitching component for rank two. So we'll come back to this example because it gives a lot of information later on in 92 hitching. For instance, when studying hitching components, he showed that it parameterized TecMuller space inside the modulate space of Higgs bundles. So I want to try and use the same conditions that we had for stability for vector bundles to try and construct the modulate space of Higgs bundles. Here we've added a condition. We've added the Higgs field. And so we should put some condition of stability which involves the Higgs field. And the more structure we add to a Higgs bundle, the more we have to add to the stability condition. So the stability condition to build the modulate space, we need stability conditions. We need stability. Define it following this definition here, but only asking for the condition for bundles that are preserved by the Higgs field. So not everyone matters for us now. We have a Higgs field. So it should only matter for those that are preserved. So we say a sap bundle inside E is preserved by phi is preserved by phi. Or, equivalently, we say that it's phi invariant. What do you expect happens? So if when applied to the sap bundle, we come back to it, tensor with F. So it's inside the image inside F tensored with K. Those are going to be phi invariant. And now we can do the stability definition. So we say that a Higgs bundle is stable. We say E phi is stable or semi-stable just like we did there. If for every sap bundle F inside E, which is phi invariant, so which is phi invariant, we have mu of F is less or less or equal for semi-stable than mu of E. So it's exactly the same condition, but we relaxed the number of vector bundles we're going to use. We're not going to use every vector bundle. We're only going to look at those that are preserved. So for instance, what happens if your Higgs field doesn't preserve anyone, then you're stable, automatically stable. And when you're trying to look at preserved bundles, remember the Higgs field is a matrix, really point-wise is a matrix. And it has a structure, and when you're trying to see which bundles are preserved or if anyone can be preserved, you can look at the matrix and see would it be preserving any of the sap bundles that you have. So we can come back to this example. Let me just erase. Erase in this case, we know that this is the quadratic differential, so we had omega quadratic differential. And you can ask, is there any bundle that is preserved? So from this matrix, can I come back if I apply this Higgs field to any sap bundle, in particular to k to the half and k to the minus the half, can I be preserved? So for omega different than zero, for our differential form, different than zero, no one is preserved because it's mixing them, the two bundles together. So no sap bundle preserved. So it's automatically stable. There's no one to check. But when omega is zero, who is preserved? Well, k to the minus the half, because it was sending me k to the minus the half, to k to the half to k, and that's going to be inside the image. So k to the minus the half is preserved. When we look at what is the degree and the slope of this, so what is the slope of k to the minus the half? Mu of k to the minus the half is the degree over the rank. The degree we said for k is 2g minus 2. So for k to the half is g minus one, but with negative. So it's minus g plus one, minus g plus one, and the rank is one, it's a line bundle. So the slope for this one is minus is negative because g is at least two. So this is negative. And what is the slope for our big Higgs field? So for our original pair, e phi, the degree of this bundle is a rank to a bundle who is the sum of a line bundle and its dual. So the degree is zero, it cancels out. So this is always less than zero, which is the slope of e. Hence our family of, this is of rank to stable Higgs bundles. It's not just Higgs bundles, it's stable Higgs bundles that we're looking at here. The same that we were talking about vector bundles when the rank and the degree are co-prime. Once we form the moduli space, it'll be smooth when they are co-prime. But there's one stability more that I want to tell you just in case it appears when looking at parable Higgs bundles, which is poly stability. So we say that it's poly stable if we can write it as a sum of Higgs bundles, which have the same slope as e. So if we can write e phi as a sum of say f1, phi1 plus dot dot dot plus fk, phik. So these are smaller Higgs bundles such that the slope of fi is equal to the slope of e. Yes, thank you. Yes, thank you. Okay, so we want to look at this moduli space. Now that we have the stability conditions, we're going to call it, so we're going to look at m. And so if the rank, I'm going to be switching between rank, I'm going to put n for rank. Sorry, people. Now let's just do it before. And d, the moduli space of isomorphism classes of semi stable Higgs bundles. Of fixed rank and degree. So we want to fix both things, the rank and the degree, just like we did before. So this moduli space, now we are not constructing it really. I'm telling you it can be constructed under these stability conditions. You can look at hypercalor equations to construct it from these pairs. And you can also consider the appearance through Hitching's equation. So self-duality equations on Riemann surfaces. And what Hitching was able to show is that the existence of a solution corresponded to having this moduli space. So a point in this moduli space. I want to say a few properties that this moduli space has. And then we're going to come back to the example. One of the properties that it has, and that is a remark that is very useful. We are not going to use it very much, but many authors in the area actually use this action. So I want to make sure that you know that it exists. So if we have a stable Higgs bundle, so if I is stable, then if I multiply by a non-zero complex number, the Higgs field, then I remain stable. So then e lambda phi for lambda and non-zero complex number is stable. I can also look at an automorphism of e and pull the Higgs field back by it. And that's also stable. So e and then let's call it alpha pullback of phi is, so for alpha automorphism of e is stable. So it has to be holomorphic. Remember, we're looking at holomorphic bundle, so holomorphic. And this is what's known as the c star action of the moduli space. So c star action of m. So once we fix rank and degree, we just call it m. This moduli space has a few properties that are going to be useful. It's a quasi-projective scheme and we've said it's smooth when rank and degree are called prime. And also it has a hypercalor structure that we're going to come back, so we're not going to talk much today, but it has a lot of geometry appearing. So let me just keep the Riemann surface since that's the one thing that we have fixed for now. And let me tell you a bit more about other Higgs bundles that we can define. Are there any questions? These are just the definitions. You have to forgive me that I have to go through the definitions at the beginning, but it's going to get interesting once we start looking at the geometry. And I want to give you a few, yes, yes, it's for the whole moduli space. So people will look at isomorphism classes of semi-stable bundles and they'll have a c star action. And this c star action will allow you to look at things like more theory for the moduli space. Yes, please. Yes, it should still be there. So whenever I talk about Higgs bundles, you should be considering it traceless. Yes, it wasn't. No, actually it wasn't in definition, so that's true. I told you when I was constructing it from vector bundles, we used traceless. I'll show you how it appears in traceless condition. So for the definition, we don't need it to be traceless. And what we're actually getting is what we call GL and C, so general linear group Higgs bundles. Once we start putting other groups, we'll come back. That's a great point. So let's do that. Let's take a look at some generalizations of Higgs bundles. And I want to before, let's do one generalization. And before we go into more, let me try to convince you that it's an interesting subject. So let me do that. And then I'll give you a map of interesting questions that we'll be touching during the other lectures, just to make sure I keep you interested. So the same way that you can generalize a vector bundle to a principal bundle by asking for the fibers to have the structure group of a complex lig group, for instance, we can do the same for Higgs bundles. So we can put a structure of lig groups. So let me put GC, a complex lig group. You might want to put some extra conditions to this, but later we'll see that we can define them for almost every or all of complex lig groups. So let me just put GC, a complex lig group. Let's try to define what Higgs bundles are for a complex lig group. So we'll come back to the space. So definition. We're going to do the same that we do for vector bundles, but for Higgs bundles in terms of principal GC bundles. So GC Higgs bundle, P Phi is a pair for which, so it's formed a principal GC bundle and Phi, a holomorphic section of that joint bundle to P tensor with K. So Phi, a holomorphic section of at P tensor with K. So we're using the joint representation of P to do that. And we're going to take that as the definition for principal Higgs bundles, but you could use other representations. This definition comes also from 87, from Hitching Second Paper, where he was looking at different groups. And the most interesting part from our perspective in this week is that even though this might seem a bit too lithoretic, because now we're getting into principal bundles, for classical, so a remark, for groups GC, which are GL, or groups of type A, B, C, D. So for SL, for SP, and for SO of any rank, this definition can be put in terms of our all definition of vector bundles. So GC Higgs bundle can be expressed if I, as before, plus conditions reflecting the nature of the group, the nature of the group. What do we mean by this? Let's put an example here. If you take SL, and then we'll get to your question about the trace. So for example, example, if we take GC to be SL and C, then we have then an SL and C Higgs bundle.