 Introduced today are brains. So subspaces of the modulate space of Higgs panels, which are going to be interesting to consider, partially because of their relation with other areas of maths and physics. So this is brains of Higgs panels. What we're going to do is we're going to fix the Riemann surface just like we were doing. So sigma is a compact Riemann surface, if seen as at least two. Partially, I emphasize that we're fixing our Riemann surface because Higgs panels we mentioned are related and in correspondence with representation groups. So surface group representations. For that, you only need a surface. You don't need the complex structure giving you the Riemann surface. But for Higgs panels, you do need the complex structure. So there's a difference there on which data we're going to need. I'm going to let Gc be the complex. The group and just like yesterday, I'm going to let G be a real form. So a real form of G fixed. So we said it could be thought as a fixed point set of an anti-holographic evolution. So that anti-holographic evolution, I'm going to call it sigma. So sigma acting on Gc. This is the data that we have fixed. And we mentioned we could construct the modulite space MGC. MGC, the moduli of Higgs panels of semi-stable. So it's isomorphism classes of semi-stable Gc Higgs panels. The properties that this modulite space has is that it's hypercaler. It's hypercaler. Those that haven't seen hypercaler spaces before, what you need to remember is that there's a sphere of complex structures and a sphere of associated symplectic forms. And we can choose from those compatible complex and symplectic structures. We can choose three in the sphere and they're going to satisfy quaternionic equations. And once we fix those three, we can ask questions about subspaces. Are these subspaces complex or manifolds with respect to someone? Are they Lagrangian subspaces with respect to someone? So I'm going to fix. I'm going to fix i, j and k such that they satisfy quaternionic equations. So I'm going to say i, j is equal to k and these are all complex structures. So i squared, j squared, k squared equals to minus one. I'm going to fix the corresponding compatible symplectic forms. What you are doing, so omega i, omega j, omega k. You could also fix the scalar forms associated. What you do is you take the Riemannian metric that the mod light space has and you put inside i, j, or k and you get a symplectic form associated. So a non-degenerate, excuse the metric to form, bilinear to form on the mod light space. I'm going to not give all the details of this construction, the notes. So we're doing chapter three of the notes. You'll get much more from there. You may wonder, so these are complex structures. Let me put it here. Complex structures. These are symplectic forms and they are corresponding to each of these structures. You may wonder, how am I going to choose these three structures from a sphere of structures that are available for us? Well, you have to remember that we start with some fixed objects, some fixed properties of our mod light and those objects carry structures. So our Riemann surface is a complex surface and our group is a complex group. So they both have some induced complex structure on our mod light space. So what we're doing here is we're taking from the structure we fixed first, the induced complex structure. So i is going to come from the Riemann surface. So the complex structure from the Riemann surface induces a complex structure on our mod light. The complex structure from the group will induce the one that we call j. So this is the convention that we usually have. The way that you can think about it also is if you go back to Hitching's equations, Hitching's self duality equations, equations for a connection on a Riemann surface and you look at the space of solutions and the manifold that it gives, if on that space of solutions you put complex structure i from the Riemann surface, you go to the mod light space of Higgs bundles. But if you put complex structure j, you go to the space of representations. So through the change of these complex structures, you switch from these mod light spaces. And once we fix those complex structures and those implicated forms, I want to ask how do subspaces of the mod light space of Higgs bundles behave? So given a subspace, subspace of MGC, how does it behave with respect to these structures? How does it behave with respect to the structures? What's its geometry? So is it related to representation theory in some fashion? So and what is its geometry and topology? Complex structures and three symplectic forms that we fixed, then we need to try and remind ourselves that we're working with all of those six objects. And to do that, we borrowed the terminology from physics of brains. So for those that have worked on brains, you'll spot that I'm borrowing part of the notation, not all of it, I'll make a comment about it. So we're going to define the following. We're going to say we say a Lagrangian sub manifold, sub manifold or subspace is a Lagrangian subspace with respect to one of the symplectic structures to a symplectic structure is an A brain. And we say a complex sub manifold with respect to a complex structure is a B brain. So let me just make a few comments about this terminology. If you've seen brains before, you might know that brains in modular space of Higgs bandos or more generally, brains are subspaces of your modular space which supports some sheaf on top. So some hyper holomorphic sheaf most often. So it's not just the subspace, the subspace is the support of the brain itself. It's already very hard to understand the supports of brains. So that's why I'm just going to focus on the support and just call brains by these names. The second thing to mention is that I've done this in this form. So I haven't put definition and I haven't put an A brain is this because there's other brains that are A brains that are not Lagrangians and we're not going to consider them today. It's already very interesting to find all the Lagrangians in our space. And if you haven't seen this notation before, the way that I remember is Lagrangian has an A and that's going to be your A brain. So, and it works actually, so you should remember. So what types of brains can we have? This is linear algebra. We have a space with three symplectic forms that come from complex structure satisfying ij is equal to k. So if I want the space to be Lagrangian, if it's Lagrangian to one, it's going to be Lagrangian for another symplectic form because they are related. And if it's Lagrangian for three, they have to be degenerate. They can't be Lagrangian for three. So there are four types of brains. There are four types of brains. And the way that we're going to write these four times is by remembering the type, if it's A or B with respect to each of the three structures in that order. So we're going to say, we're going to be talking about A, B, A brains. We're going to be talking about B, A, A brains. A, A, B, B, B, B. So what do these notations mean? These means, for instance, is Lagrangian. Lagrangian for omega i and omega k. And just complex a manifold with respect to j. This one here is what we call a hyperholomorphic brain. So hyperholomorphic. In particular, it's quite hard to construct them in a non-trivial manner. If you want examples, the point is it's a space that it's a complex and manifold with respect to all of the complex structures. So it's a hypercalor subspace of your multiple space. A point is an example, or the whole space is an example. For Higgs bundles, if you take Higgs bundles for another complex group inside yours, then you get one. But understanding how to construct brains of any of these types is a non-trivial fashion. They're a non-trivial thing. The whole study of brains within Higgs bundles goes back to Kapustin and Witten. So if you've seen this paper, electromagnetic duality is like a 200-page paper. You go to chapter 12 in that paper. You see that that's back in 2006. You see that they are making some comments about brains in string theory, and in particular, as examples, brains in Higgs bundles. The cases that they consider were brains that are of type BAA, and that come from real Higgs bundles. So we want to enlarge that and ask, how can we construct brains of each of these types? What are the interesting properties that they carry? Do they give you real integrable systems? Do they give you any relation with representation theory? And what's a mirror symmetry for these brains? There are other things that you can do with brains that we're not going to do here, and I haven't mentioned much in the notes, but if you want to know more, you should ask me. One of those is, for instance, quantization. So you can do brain quantization, Gukov and Gukov and Witten have some nice papers about it. And the rough idea of what's going on is that quantization of a symplectic manifold, in this case, quantization of a BAA brain coming from real Higgs bundles can be understood in terms of homomorphisms between the brain that you start with, with some other canonical quiesotropic space, and your Hilbert space giving, the quantization will be coming from those morphisms. So because of that, it's also very interesting to understand brains, the global picture of how do they lie within the hitching vibration? How do they lie within the modified space? Any questions about this? Yes? Yes, that's a good point, thank you. So the grandeur of manifolds will be half-dimension. However, BBB brains don't necessarily have to be half-dimension. BBB brains can be a point or the whole space. It can be any dimension, and you can ask which ones are going to be half-dimension because those are actually also going to be interested. And yes? Yeah? Because your complex structures are coming, are satisfying the quaternionic equations, and once you build your symplectic forms, obtained through these complex structures, they're going to satisfy a similar relation, where one is going to be, so your symplectic forms and your scalar forms will be related as the imaginary and real part of a form, and once the symplectic form is zero, the other, the real and the imaginary will be zero, and with that you get a relation. These are complex Lagrangians. Yes, sorry, thank you, Vivek. I should mention these. So BBBs are things that are not Lagrangian, but all of these times, anytime you see an A, you should think of a B behind it. It's a complex sub-manifold. Yes, good point. Okay, so I want to construct examples of all of these. This is what we started a few years ago with David Baraglia, my collaborator. And later on, I started on working with other people. I'll mention them as we go along. We want to construct examples of all of these brains and see how they lie within the hitching vibration. How can we relate them to representation theory? The way that we're going to do it is by adding symmetries to our system. So the idea is add symmetries. Add symmetries to the theory and see what happens when you look at invariant points under these symmetries. I'm going to add symmetries and actions to all of these objects that we fixed here. That's why I kept them here. I'm going to consider these complex spaces, complex Riemann surface and complex Leroupe, and I'm going to add a real structure. That's the first natural thing you can do to a complex space. So I'm going to add here a real structure. A real structure and a Riemann surface means that you're adding an anti-holomorphic involution, F, that locally, so anti-holomorphic, that locally acts like conjugation, set, goes to Z bar. That's on the Riemann surface. On the Leroupe, we're going to add this real structure. So these real structures sigma that we mentioned here, real structure, and another thing that we can do on our Riemann surface is a finite group acting. So I want to add here a finite group gamma. Gamma. For those doing more representation theory side of things, you'll recognize that this is something that people care about as gamma-equivariant representations and their finite groups acting on a Riemann surface. So by looking at those things, I'm going to construct brains. I'm going to tell you first how to build them and which type of brains we get in each case. And then before we mention the geometry and topology, I want to give you an overview of how they feed globally in this picture. I find many times, and this is something that you might like too, it's very useful once you start some research program in one direction to keep track of how does your research and how do the results fit in a bigger picture, in a global picture of the area and related areas. So that when you're doing it, you know where the questions should come from. How should I relate this to this other thing? In our case, we are going to remember that our global picture is the modulite space of Higgs bundles over the Hitching base, AGC, and that we had a dual modulite space, M for GC, Langlund's dual, that we mentioned mirror symmetry, and S-Y-Z conjecture, relate them, and we're going to talk more about it tomorrow. But there is this relation and that we set here, there's a space of representations plus of pi1 of sigma to GC. So we're going to keep this in mind when we construct brains of each type, we want to feed them into this picture and try to ask questions about all of this. So what's going to happen? We're going to look at fixed points under the induced actions of these symmetries, yes, of sigma. So that's a good point. I should make a bit more clarifications of this. I'm going to give you more details once we see the geometry of the action, but really, we are taking a Riemann surface of at genus at least two, and you're looking at this involution F, and the involution F, if it has fixed points, you'll have circles as fixed points, and the circles, when you take them out, they will either separate your Riemann surface into two or not, and those are the two environments, and the environments for the involution will be the number of circles and whether it separates them, and the geometry and the environments of this involution will tell us what happens with the brain. So we will build brains, brains, from equivarian points. Equivarian points in the modular space, M, GC. You'll see that when we're defining real Higgs bundles, we mentioned there was the composition into carton subalgebra, and in the notes, there's a bit more explanation about how those compositions can be related to real groups. In particular, this is what you're doing. You're taking from the involution sigma acting on the group GC. You're going to get real Higgs bundles like we were doing before. We get G Higgs bundles, and the way you do it, this construction using involutions goes back to Hitchin's paper in 92, this Tecmuller paper, where he constructs Tecmuller component for split real Higgs bundles. And what you're doing is you're constructing an involution, say I1, that sends a pair E5 to a pair where you're going to compose sigma. Let me put it with color. So our sigma coming from the extra structure, we're also going to have another involution that I'm going to call rho E, and then minus sigma rho phi, where rho is the compact involution of the group, compact form. So the anti-holographic involution fixing the maximum compacts of group. This perspective and the appearance of rho is what gives you Hitchin's equations, and there's a bit more explanation of how to deduce that in a few pages from what we were doing. And the things that you get, so we get G Higgs bundles as fixed points of I1. So as fixed points of I1, this involution. And moreover, this involution gives the fixed points and gives BAA brain. So the first example of brains, so this BAA brain that we put here are these G Higgs bundles. So let me just erase this thing. And this BAA brain is of G Higgs bundles. And these are the Higgs bundles that Kapustin and Witten were looking at without thinking about involutions. But once you put the ideas of them together with Hitchin, you get involutions. And that's how the other three cases of involutions that we look at, in particular these ones, is what we did with David Baraglia from Australia, inspired by the first case. So what happens when you take other involutions with the other real forms? If you take the involution on the remand surface, so F, you can look at the fixed points by that. I'm going to call it I2, the fixed points of an involution that sends E phi to... And now we're going to also use the compact structure that's always going to be there. So the pullback by F of rho E minus pullback of F rho phi. When you do this, you get an ABA brain. So fixed by this involution. If you put the two involutions together, so if you put rho and F, so if you put F and sigma, you're going to get AAB brains. So there's a third involution here, I3, which is just a composition of I1 and I2. And it fixes AAB brains, fixed by this involution. And finally, with Sebastian Heller, what we considered was the finite group acting on the remand surface. So this is with Sebastian Heller from Tubingen. What we looked at is BBB brains as gamma-equivariant pairs. So you have gamma-equivariant pairs, you have fixed under different involutions, and all of those things fit into this diagram. I want to put them here. So let me put ABA, BBB, sorry, let me just do it like we were doing it there. So we have them in the same place. So AAB, this was BAA and BBB. And all of these come from these different structures that we're adding. This one comes from the involution F, this one comes from the involution sigma, this one comes from F and sigma at the same time, and this one comes from gamma. Gamma is a finite group acting, is the orange one here. Maybe I should put it with orange here so that you can remember. Now, yes, yes. Oh, so anytime you have a real form of a group, you can think of it as a fixed point set of an anti-holomorphic involution. Yeah, so if you have SL, say, you want an involution that fixes SU. For, so I don't want to write the list here, but if you take a look at page 28 of the notes, I put there two big tables of involution so you know what we're talking about of these things. Yeah, not necessarily. So I require it because it's more interesting because otherwise the brain is just vector bundles. So we mentioned before that if your real group is a compact group, then your Higgs bundles are just, your Higgs field is zero, and your vector bundle has to be a semi-stable bundle. Yes, it's a minus. Yeah, yeah, sorry, it's a minus, and you'll see in the notes how to deduce that. It's a minus that comes from how you're looking the hitching equations. You're asking for your connection to have holonomy in the real group, and that's what you need to prove. Yeah, there's always a minus here. So you're always composing two anti-holomorphic things. You're getting something holomorphic, which acts on the mod-less space, a holomorphic space, and your subspace that it fixes is a complex sub-manifold. Before we look at the geometry of all of these space that are going to be really interesting, I want to convince you that they're interesting and convince you that if we could find some new ones that could also be great. So maybe you worked with Lagrangians before and you have your favorite way of constructing Lagrangian sub-manifolds, then we can try and apply them here. In order to see their importance, I want to put them in our little world map and see where they fit. One thing that we mentioned here is this blue arrow, the mirror-symmetry correspondence between the hitching vibration. What we know from string theory, and you can go back to the paper of Kapustin Witt in this chapter 12 thing, is that there's a correspondence between brains, brain types. So not between brains themselves, but in terms of specific objects, but between the types. And the types that are dual are, ABA is a self-dual, so let me just put it here. This blue arrow, the mirror-symmetry arrow, will tell us that ABA type is self-dual type. It'll tell us that BAA brains are dual to BBB brains. So if I have a BAA brain in MGC, I will have a BBB brain in the dual-modelized space corresponding to it. And it tells us that AAB brains are self-dual. So anytime you construct a brain, you're going to ask, what should the dual object be? We'll see tomorrow how far mirror-symmetry has gone in this setting and what is missing. And so you'll see that in particular, it's not completely known what mirror-symmetry is for singular fibers, for instance. So if your brains line the singular fibers, you'll need to develop some new methods to study mirror-symmetry for those brains. The other thing that we were thinking about was representation theory. So we knew from non-Abilian Hodge correspondence that we had correspondence with representation theory and Riemann-Hilbert representation to complex groups. And we know that that's also the case for G-higgs bundles. There's a representation, it goes to representations here, rep for the group G. So representations in the group G. The interesting thing that happens for these brain, ABA brain coming from the real structure on the Riemann surface, is that we can actually put the objects in that brain in terms of representations of three manifolds. So until now, we were looking at Higgs bundles in terms of representations of Riemann surfaces, so two-dimensional spaces. And now we can actually express representations of three manifolds in terms of Higgs bundles. So this is rep of three manifolds. Now tell you how this relates back to here. So in what is left in the half an hour that we have left, I want to tell you about properties of all of these brains and the geometry that they have. Again, there's much more if you're interested. There is going to be much more in these notes. And in particular, there is a bit of a separation that I made so that you can see which types of objects you have of each case. What, sorry? Yes, yes. So a fundamental group of three manifolds, fundamental group, so here I should have put, not be lazy and just put rep pi one of sigma into G. And this is three manifolds, is really pi one of a three manifold M into GC. Yes, thank you. Okay, so let's get ready to see the properties of all of these brains. I'm going to leave that one there so you remember which brain is which. We want to remember which one is which. We want to see which types of brains we have in each of these cases. So we want to study these brains inside the hitching vibration so that then we can apply some kind of freedom of kite transform to see what the dual objects should be. And in particular, also we want to see whether there's an avalanization or not available for our brains. So remember the mod light space of Higgs bundles through this hitching vibration age was avalanized. So the first question we want to ask is how do these brains lie within the hitching vibration? So how do the brains in the hitching vibration? I'm going to first think about the real one. So I can give you some overall details of that brain and then we'll move to the other ones that are a bit more interesting. And we're going to probably out this one. This is what sometimes is known in the literature when you actually calculate the points. Things like these appear sometimes of see the real Higgs bundles and you can do the same for see the real vector bundles. And the properties of that brain will be very similar to the ABA. So I'll mention when these things coincide. So let's take a look at the BAA of real Higgs bundles. So it's a mid-dimensional space, a Lagrangian that lies inside the mod light space of Higgs bundles, giving you real Higgs bundles. And you ask how does this brain lie within the hitching vibration? And I want to consider three cases. I want to consider when they intersect the regular fibers. So first one is intersect regular fibers in zero dimension. The second one is intersect the real Higgs bundles. In zero dimension. The second one is intersect regular fibers in positive dimension. And the third one is when they don't intersect the regular fibers. When they completely over the discriminant locus. So no intersection with regular fibers. The first case when they intersect in zero dimensional, the study of BAA brains, when I was doing my thesis back almost 10 years ago, I was thinking of Higgs bundles, not in terms of brains or string theory or anything like that. I was just looking at the structure of Higgs bundles for real groups. Now looking back, I can realize that all that work actually fits in this much nicer global picture of a world where we can look for different objects of different types which are related with each other. And the case where brains lie in regular fibers that I mentioned goes back to that thesis work where I show that if G is a split real form, lit real form, the brain intersects.