 is a pair e5 for e a holomorphic vector bundle of rank. And on the Riemann surface, such that and the condition of the group that has determinant which is trivial is also going to be reflected here. So such that the determinant of the bundle so that the topics to your power of the bundle topics to your power of the bundle is trivial. So this is the line bundle. We're going to ask for it to be the trivial line bundle and the Higgs field which also has a trace as a matrix will have trace here. e from e to e 10 third with k holomorphic such that trace of the Higgs field is there. So if you have a special linear group this is what a Higgs bundle is. If you have a symplectic group what you get is a symplectic vector bundle so a bundle with a symplectic bilinear form and on the general bilinear form and the Higgs field will be compatible with the form so it will commute with the symplectic form. For orthogonals you have an orthogonal form. You can go back to this example and try to see can we see the structure here. Our vector bundle was the sum of a line bundle and its dual so when we take the topics to your power which is the second exterior power here for a rank two bundle and this is the product of the two line bundles and the product of a line bundle and its dual is trivial. So we can look here and notice that the top exterior power of e is indeed trivial. And we can look at our Higgs field and how we formed it and our Higgs field was indeed traceless. So this is not just rank two Higgs bundles but is SL2 Higgs bundles, SL2 C Higgs bundles. SL2 C, stable Higgs bundles. You can kind of see how from this perspective we have a notion of stability already for our old pairs and that's the stability that we can consider. For these groups, the stability condition that we define for the vector bundle expression of E phi will apply to principle bundles and so we're not going to go through any more stability. The important thing is that we can construct the modular space using that stability condition and I'm going to call it MGC. So let's put here the notation MGC, the moduli of isomorphism classes of, and we're always going to be talking in these lectures of A, B, C, D or the GL case of a semi-stable Gc Higgs bundles. In general, this space is not smooth but it has a hypercalor structure that we're going to be using. So it's hypercalor, which means it has a family of complex structures and of symplectic forms associated to them, which are compatible and which will give a lot of structure to the moduli space. Okay, so we can carry on defining more generalizations of Higgs bundles. You can look at real groups. So at the end of the lecture, we can take a real group and put the same idea into a real form of a complexly group. You can put mark points and define Higgs bundles which have structure on those mark points. That's parabolic structure with with filtrations of the bundle and weights for that filtration that satisfies some conditions with respect to stability and then you have a moduli space of Higgs bundles over this manifold with mark points. You can also drop the condition of holomorphicity and allow for Higgs bundles to have poles over certain mark points with which you would get wild Higgs bundles or tame Higgs bundles. You'll see these things at the end of the lecture. These are a few more generalizations that we can do to the moduli space of Higgs bundles and the Higgs bundles themselves which will carry a lot of interesting information. And before we do that, I want to mention a few things that are interesting. Yes, so actually not necessarily, I think. For now, we're just going to be ABCD type but I'm going to show you, when looking at correspondences between Higgs bundles that you could take a non reductivity group, yeah. So yeah, we're going to look at the cases, for instance, later on of SL2 cross SL2 and see how Higgs bundles can be put together. Probably, I don't know, sorry much. I can look for the references. I can't recall how they construct the moduli space. Yeah. Remind me, and I'll look for references for you. I think it's probably going back to Simpson, yeah, or faultings. There's a few papers that have some information of more general league groups. Moduli space of Higgs bundles is what some people call the dalbo moduli space. Let's keep this example here. So it'll be up. One of the interesting things appearing from Higgs bundles and let's just use the more general version that we just defined, the moduli space of principle GC Higgs bundles are correspondences between the space and other spaces of objects. So the first correspondence, or the couple of correspondence I want to mention, come from work of Simpson, Hitching, sorry, non-order Donaldson and Corlett, back in the late 80s, beginning of the 90s. And this is the moduli space of Higgs bundles, like we said, and this is through non-Abelian Hodge correspondence, this is the name that Simpson gave, non-Abelian Hodge correspondence that we sometimes write NACC, is in correspondence with the space of flat connections. So MDR, the dalbo moduli space of flat connections. So flat connections for our Riemann surface, our base Riemann surface that we are talking about. And through the Riemann-Hilbert correspondence, this moduli space is in correspondence with the space of representations of the fundamental group of your surface. So this is what we call the petty moduli space. So moduli MB, the petty moduli space of representations, reductive representations of the fundamental group of the Riemann surface into your group GC, quotiented by conjugations. So by studying the moduli space of Higgs bundles, this was of GC, Higgs bundles, you can understand the information about the moduli spaces of flat connections and of representations. The properties that carry, so these correspondences are, these are the ophthalmophic correspondence as real manifolds and an analytic correspondence. They're not algebraic and so many properties that the cohomologies, for instance, of these spaces should have and that they should relate to each other are still not perfect. So if you've heard before of the work of, for instance, Tamash Housel and his collaborators on a P-col W conjecture, and there's people here working on that, I think maybe Vivek is working on that. There's people here that you can talk to about properties of these spaces and cohomology dimensions, for instance, of these spaces and how they relate to each other. But one thing that we can do, for instance, is understand the structure of representation. So can I deform a representation into another which would be translating to understanding the connected components of this moduli space in terms of the moduli space of Higgs bundles? If you want to see how this relation goes just from a rough perspective, let's say take a solution of the self-duality equations, that saw it on Friday, are equations for a connection on a Riemann surface. The connection will be a flat connection that has some holonomy and the holonomy will be the holonomy representation that will give you the representation point in this moduli space. And through the connection, you form the Higgs field that appears here. So there's going to be a relation that appears between these three spaces and we're always going to be asking the interpretation of objects in these three terms. So if I give you a Higgs bundle which is preserved by, say, a finite group acting on a Riemann surface, I'm going to ask what are the equivarian flat connections equivalent to in terms of Higgs bundles or what are the representations that are equivariant under that action? And we're going to understand the moduli spaces. Mostly people look at representation moduli spaces and understand them in terms of Higgs bundles. There are certain representations that become very interesting lately in the last year or so. For instance, Ghisharam Binhar proposed that there is something called Theta stable representations. And those Theta stable representations are not understood just yet in terms of Higgs bundles. So understanding who are these objects in terms of Higgs bundles and which connected components give us these Theta stable representations. Can we define a Theta stable Higgs bundle? This is something that's an open question. So a way of getting new open questions in the area is by looking at work done in the other two directions, so in plant connections and representation theory and trying to understand them here and be subversa. So in the last few minutes, I want to tell you one more description of Higgs bundles for a real group, since these correspondence with representation theory was later on extended. So at Prada, Lagarasiya, Prada, Gothen and Mundet and some of their collaborators, they extended the correspondence, the Riemann-Hilbert correspondence to representation into real groups and they correspond to real Higgs bundles. So for real Higgs bundles, we need to take a real group. So here we had a complex lig group. For now we fix our Riemann surface of genus at least two, a complex lig group and we can fix the real one. So G, real lig group. And many times we're going to consider real lig groups as real forms of the complex lig group. So recall that a real form, just recall, a real form of the lig group G C can be thought of as a real subgroup that complexifies to G C or as an anti-holomorphic involution that fixes G. So it's a real G such that the complexification of G is our complex lig group or equivalently an anti-holomorphic involution acting on the complex lig group such that it fixes or whose fixed points is the real form where the fixed point set G C fixed by sigma is a real subgroup. So we're going to take that, we're going to take a real lig group, take G, real lig group and take H it's maximal compact. And so that when we define Higgs bundles we can carry on looking at the constructions. Let's put here an example. So let's take SL say two P C as the complex lig group. So this is G C and let's take our real form to be S U P P. So special unitary matrices with signature P P. So matrices in SL two P C which preserve a Hermitian form with signature P P. We're going to do things with this, we're going to get the maximal compact. So what's the maximal compact in this case? It's S of U P cross U P. We're asking really, this should, let me just put S of U matrices with trivial determinant, special in the sense. Let me just write S as U P and S U P, okay? So, but the S is really outside. So you're not asking for matrices with trivial determinant, each of them you're asking for the product of those matrices to have trivial determinant. Does that make sense? Yeah, okay. So once we take a maximal compact, we can look at the Lie algebras. For Lie algebras using the maximal compact and it's Lie algebras, we can get the Cartan decomposition. So we've Cartan decomposition, the Cartan decomposition of the Lie algebra G of the group G as some M plus H. So M here is the orthogonal complement of H, okay? So here we have S U P cross S U P, we look at the Lie algebra. The Lie algebra will be of diagonal, sorry, the Lie algebra will be diagonal with something here, matrix here. Yeah, let's call it A and B. And when we look at the orthogonal complement, it'll be of diagonal, something here and something here. So by considering these Cartan decomposition, we can look into the isotropy representation for the group. So let me just make sure I don't forget anything here. So let me define the isotropy representation or what people call isotropy representation. It's obtained through the joint action of G restricted to H on, so we're going to say is from H to GL and it's obtained on GLM for M that orthogonal complement. And what we're doing is we're taking a G and we're making it act. So a G in this here, we're making it act on a point X. So it's going to act on X as G on X. And through these isotropy representation is that we can form, we can look at the complexified one that goes, so IC, complexified from HC to GLMC. And through these, we can form vector bundles whose structure is in the fibers is MC. So with these, we consider bundles, vector bundles. Well, they are principle bundles, right? So let me just put vector bundles. Since we're going to see them as vector bundles, E, which are going to be, we're going to write it as E of MC and it's really just a natural vector bundle E with a probe through the isotropy representation MC. So MC is given the structure to the fibers of E. And the definition we can do now using that is that a G expands, so a G. So now it's not complex, it's the real group. A G expands is a pair E phi for E holomorphic HC bundle. So the complication of the maximal compactives, the structure to the vector bundle, and phi, it's going to be a holomorphic section, but now it's a holomorphic section over the Riemann surface of this bundle that we constructed here. The MC principle bundle answered with K. The same happens as when we were doing Higgs bundles for complexly groups. We can think of them as pairs E phi where we're adding some extra conditions which reflect the nature of the group. And we can see them just as we were doing before in terms of matrices. I have four minutes left. It's an hour, right? Is it? Okay, so since we started with this, let me just do a couple more steps to show you what this is happening here. So what do we have here? If we were taking, if you don't like the S, the S here, we can start looking at UPP and see what happens for UPP. Here, our H was here. The little H, the algebra complexified that we want. Let me put it here, the complexified the algebra. We have GLPC plus GLPC. So our complexified the algebra and then complexified orthogonal complement will be matrices A and B here. And when we look at UPP, Higgs bundles, or for that respect, we could just put a Q. There's no need of having the same thing here. We can just have a different rank. We just need to make sure that we remember and we put it in the complex one if we want P plus Q. And so a UPP or a UPQ, UPQ, Higgs bundles is a pair, E5, where what happens? Now E was coming, the structure of E comes from HC bundle. So H is UP cross UP. So it's going to be a decomposition of bundles of E into two UP cross UQ. So for E equal to some V plus W, where V and W, VW holomorphic bundles of rank P and Q each. And the Higgs field structure comes from the structure of the orthogonal complement. And here the orthogonal complement we said was the off diagonal matrices, the GL. So and phi is going to be a matrix. Can you see if I write down here? Yeah, phi is a matrix. So phi here is a matrix, which is going to be off diagonal. So I'm going to put a beta and a gamma here and where this beta and gamma, beta goes. So we just do it in terms of this bundle. So beta goes from W to V tensored with K holomorphic and gamma goes from V to W tensored with K holomorphic. So just by studying the nature of the group, the algebra, the cartonic composition and its isotropy representation, we can deduce the structure of these Higgs bundles. And this pretty much tells you also is example how to do, how to deal with Higgs bundles with signature. Anytime you have signature, you're going to have to take a Hermitian form that's preserved and you're going to have the maximal compacts decompose. One more thing to mention is that, and then I'll stop, is that the structure group, as we said, comes for the Higgs field from the orthogonal complement of the maximal compact. So what happens if our form G is itself H? So if G is H, then the maximal compact is the whole thing, there's no orthogonal complement, right? It's zero. And indeed, whenever G, so remark, and then we'll finish, remark, if G is compact, so G compact implies that the Higgs field is zero. So when G is compact, you recover pairs E phi for phi zero, but when you look at the modulate space and you ask for stability, if phi is zero, you're looking at every sub-bundle. So really the modulate space of Higgs bundles for G compact is the modulate space of vector bundles. And this is how you can see vector bundles inside, a way of seeing vector bundles inside Higgs bundles. What we're going to do tomorrow is we're going to study these modulate spaces through an integrable system, and we're going to use that integrable system to motivate a bit further beyond these three correspondences. I want to introduce a few more directions through which we can ask questions and which you should keep in mind when considering research projects. Thank you very much. Yes. So once I look at the composition of M and G, of G in terms of M and H, you mean? So that's a good point. So what you need to remember is which representation you were using. So we've been using standard representations for things. In principle, you could write UPQ Higgs bundles in terms of more complicated things if you looked at other representations. But if you fix it from the beginning, yes, it's a unique object that you get. Up to isomorphism classes. Yeah. Any other question? Thank you. Thank you.