 in points of order two. The intersection is in torsion two points. Intersection is in torsion two points, which means once you're looking at this vibration, you have the heat chain vibration, and you have the regular fibers lying over here. You have some singular fibers. And what you get from this BIA for these split real forms is some finite number of points in each fiber. It doesn't matter for which complex group, it's always torsion two points. One thing that you can ask, for instance then, is if I take a loop in the base of the vibration, so I take a loop here in the base of the heat chain vibration, what happens with the loop when I lift it? Where does it take my points? This is what's known as the monodromy action. Monodromy. And you're looking at the Gauss-Mannin connection for the vibration. There is a page or so in, what, sorry? I did, sorry, that's why the singular fiber was there. Yeah. You mean which points may collide here, these ones? Yeah, they don't collide actually. What do you mean by collide? I mean, you can come back to yourself. But I don't go through the fiber. I don't go through a singular fiber, I go around it. No, no, no, no, oh, this point, no, that's really bad. No. No, no, thank you. No, we're not going through the singular fiber, we're going around singular fibers. So we're looking at things around the singular fibers here. Yeah, thank you. So in that case, we're looking at how these points in the heat chain vibration go around. What, sorry? What is a split real form? A split real form, thank you. It's a form of maximal rank that complexifies to your group, and it's not compact. So for instance, there are some examples in this page 28, I think, in the titles, but you can think of SLR. So anytime you put an R instead of the C, that's going to be your split real form. So SPR, SP2 and R, SL and R for the orthogonal ones is SONN, or SONN plus one. Those are the split real forms. And for those ones, you can ask, what is this monotomy? In particular, the orbits of the monotomy will tell you about the connected components of the brain and hence the connected components of your representation space. So this is one of the things you can do. Since I mentioned the Riemann surface here and that you could fix from a surface, you could fix a complex structure, let me comment on an open problem that's available. That's quite hard, where you're looking at the space of complex structures for a given surface. So you're looking at Tecmoeller space. And on top of that, for each choice of Riemann surface, you have the modular space of Higgs bundles. So instead of looking at the hitching vibration, you're looking at the modular space of Higgs bundles over all of the types of Riemann surfaces available for a fixed surface. You could look at the monotomy there and see what happens. And that's not, I don't think that's understood in general for Higgs bundles. But we're looking at something else here. We're looking at monotomy for the hitching vibration and there's some comments there on what else you can do. Now, there's some brains that intersect with positive dimension. We mentioned yesterday a case. If you remember, we looked at UPP Higgs bundles and SUPP Higgs bundles. And we saw that those intersect in positive dimension, they intersect in Jacobian varieties or imprim varieties. But there's actually not much more. It's actually quasi-split-real forms. Quasi-split-real forms will have this intersection in abelian varieties, which are of positive dimension. And when they don't intersect, then it gets a bit more complicated because you can't use Hitching's construction back from 87. You can't use his papers on abelianization because those are for the regular fibers. So what happens when there's no intersection? When there's no intersection, the brain, so when we're going to look at this, when there's no intersection, the star, the brain in star lies completely over the singular locus. So it lies completely over this blue part in the vibration. Lies completely over discriminant of the base, A, G, C. So you have a brain completely inside the fibers over the singular locus and you can ask how do we describe them? Can we do, can we describe them with abelian data? With non-abelian data? Or can we describe them? Do we need both? Or do we need both? Abelian data will only appear when you're considering these quasi-split-forms. We can tell what we showed and this is a few years ago with Hitching. We showed that when you take groups like SU star, so this is SL with the quaternions as your coefficients or SO star or SP to P to P. So with the same signature, your modulate space of Hitch bundles for those groups and your brain itself lies within the vibration as a variety, which is a non-abelian variety. So the fibers, the fibers of the brain rank to vector bundle modulate spaces. So our spaces of rank two bundles on a curve. So on a curve S and a spectral curve that we define. So suddenly you have spaces which have ranked two objects on your spectral cover. So it becomes a bit more complicated. This is what we called the non-abelianization for these groups. Non-abelianization, and what we showed, this is work with Hitching. And what we showed with Baraglia in a paper that was published just this year is that in many cases you do need both. For instance, when you have orthogonal Hicks bundles with signatures, so P plus Q, P. So this is based on a lot of Lee theory and the compositions that you do to your groups seeing what the spectral data is, trying to see how it lies within the Hitching vibration. You can ask, what is the dual brain? So we said these brains correspond to each other and they're mirror symmetry. There is a conjecture that we have on what the dual brain to the real Hicks bundles should be, but in terms of certain group that David Nadler studied a long time ago. But this is just a conjecture. It hasn't been proven. There's support for the conjecture coming from work of Hitching, of Lucas Branco, his former student, and a few other people, but there's no proof of what this conjecture. You can take a look in the notes for more details. But what I want to do in the next 15 minutes is tell you about the other brains because we've seen real Hicks bundles before. So I don't want you to get bored with that you've seen and I want to consider the other two brains. So we're going to forget about these VAAs and we're going to look at the other two brains. For the other two brains, the first one, let's take a look at the finite group. Oh, I raised more than I wanted. So let's keep here. I wanted to look at the BBBs. BBB coming from our action gamma. So gamma is a fix, a finite group, gamma acting on sigma, finite group, finite group acting on the Riemann surface. If it has fixed points, the fixed points are discrete. So it will have discrete number of fixed points. And you can study what there are gamma equivalent Hicks bundles in terms of the fixed points that this have. What we can show, so what the theorem says with Sebastian Heller, so with Heller, is that the equivalent Hicks bundles, the gamma equivalent Hicks bundles give BBB brain. And in the case of rank two Hicks bundles and for rank two Hicks bundles, some are abelian and I'll tell you how. Some are abelian. So there's a further question we want to ask following one of the things that you guys said, which is, when are these BBB brains meet dimensional? When are these BBB brains a bit more interesting? So when are the brains meet dimensional? In particular, because in those cases we can have Lagrangian fibers. So I want to mention the two cases where they can be meet dimensional. Let me tell you here. I don't want to forget any of those. Okay, so I think for more details on that brain you'll have to look at the last pages of these notes. What, sorry? Yes, I'll tell you. So it's actually a very open question in higher rank to understand this. So I would encourage you to try and study them. You'll see theorem 3.3 in the notes. They will tell you a bit more. We could only do it for rank two. So we can show for any rank that it's a BBB brain, but we're stuck when it comes to showing how it lies in the hitching vibration and which involutions give you meet dimensional. So for the case when we have rank two Hicks bundles then it's going to be meet dimensional when one of the two things happen, either so if option A is that gamma is that two acting fixed point three. The second option is when, oh sorry, oh you're good. When sigma is hyper elliptic of genus three and gamma is that two cross that two. And one involution, so one corresponds to four fixed points and this one corresponds to hyper elliptic involution. So this is the only two cases, rank two Hicks bundles of all finite group actions that you can have on a Riemann surface. There's only two cases that will give you a Lagrangian subspace. These meet dimensionals can we, we can show that they're Lagrangians and it's either we act like fixed point three or either we act by this involution on a hyper elliptic surface. Moreover, the brains in that case, so the two cases fall down to the same setting and these abelian spaces are prem varieties of the spectral curve and sigma, but quotiented by the induced, so there's an induced action of sigma, sorry, induced action of gamma and this is what we're going to take. So there's more details in here in theorem 3.2 and theorem 3.3 but this is the rough idea is that there's only two cases that will give you something that is mid-dimensional and you'll get prem varieties, so that's nice. Any other higher rank, still open, think about it. There's a lot of numerical calculations that you can do in order to check whether you could have something. So in some of the exercises that I put there, there are comments and I'm asking you to actually calculate dimensions of mod light spaces of representations for marked points or without marked points to see what the conditions are for mid-dimensional. Any questions? Yes? Yes, yeah, so wait. The BBB brain is for any group G. These ones are for SL2 but then we're ranked two, right? So SL2 is SP2, so, yeah, SL, which is SP, yeah. Yeah, okay. So, let's just finish off in the last 10 minutes of the talk with the last type of brain. So, the last type of brain that we have, so we're looking at ABA brains and this one. I should mention that these brains lie mostly over, they can lie over the whole vibration but some of them will lie over the non and over the singular locus and also you could ask what does mirror symmetry do to these objects? We're finishing some work with Heller and Biswas on a proposed dual for these brains using anti-invariant Higgs bundles. So, we have to find what anti-invariant Higgs bundles are and then we propose that that should be the dual, we have some support but this is going to be finished sometime soon. So, what happens with the ABA brains? These ABA brains and this, yeah, we have 10 minutes, so ABA brains from F. So, F was the involution here, F, this red F is an involution acting on your Riemann surface and the involution acting on your Riemann surface may have or not fixed points and the fixed points will be circles and the invariance that this involution has are two. So, F has invariance, F has invariance N and A, N is the number, if this is gamma I, a circle is the number of your gamma I's, how many fixed circles you have and A is either one or zero depending on whether when you take away those circles, sigma minus the union of those circles gamma I over I is connected or it has two components. So, you can't have more than two components even if you have more fixed circles, you wouldn't satisfy the condition of being an anti-holographic involution. This goes back, this classification goes back to Klein where he classified all of these involutions, he showed that given a surface, it always has a complex structure that makes it a Riemann surface for which there is a real form with those invariance. So, it's a one-to-one relation and these involutions have also been studied from the algebraic perspective, algebraic geometry perspective, Gross and Harris studied a lot real algebraic geometry in terms of Jacobian varieties so they studied real points in Jacobian varieties and that becomes very useful because we're looking at induced action on the modern life space and on the vibration are Jacobians or subspaces of Jacobians so we can put all that together and show the following. So, theorem in this case with Baradia is that the Higgs bundles given in this brain, so the ABA brain lies in the hitching vibration in the hitching vibration as a real integrable system so you have something new here, a real integrable system. It can be seen as the Lagrangian vibration so here where? The Lagrangian vibration, so this Lagrangian vibration can have singular fibers but the regular fibers has collection of tori as the regular fibers, collection of tori as regular fibers. And you may ask, what's this collection of tori? What do we mean? Is tori that appear when you look at the fixed points in the hitching vibration and the number of tori that appear here, the number of tori depend depend on those numbers that I wrote there. So the N and the A. So you have an N and an A invariant associated to your involution, these N and A will tell you the number of tori in your vibration. Moreover, if you were to consider this other brain coming from F together with a real thing, a real structure on the group that also satisfies the same theorem. So the same works for the AAB brain. The dual brain, when you ask, what happens with Langland's duality? Because there's a dual object here. We say it's self-dual. So there's a dual ABA inside the modular space of Higgs bundles for the Langland's dual group. And the natural thing that we can do, and that's how we propose the dual brain is, you take the same involution on the Langland's dual but you take the Langland's dual compact group. And then you have an ABA and that's what we conjecture is the Langland's dual. Now before we finish in the last five minutes, I want to tell you about some application that it has. So we mentioned the application with mirror symmetry, some application that it has with the space of representations. And the questions that we want to ask are inspired by knot theory. So those that have worked with knot some braids, I heard some nice talks yesterday in the women in math about people working with knots and braids. So when you look at knots and braids inside, so not braids, when you look at knots inside a sphere, inside a sphere, you take a tubular neighborhood and you take it out. You have a three manifold with boundary. Your boundary is a Riemann surface. It has genus one when you take a knot. And you can ask which representations extend from the boundary to the three manifold. So which rep? So which representations extend from pi one of the boundary of your three manifold to pi one of the three manifold itself? When you're looking at knots, you have this genus one space and the answer comes from the polynomial, the A polynomial of your knot. So you look at the zeros and that will choose some representations for you and that will be an ABA brain for that case. So one can show that you always get ABA brains when you ask this question, but actually forming those ABA brains and understanding their geometry, that's what's not been known. So with David Baraglia, we managed to find an answer to that for genus two surfaces. So this is like if you took a link or a graph from your sphere, you look at the complement of that in the sphere and you ask the same question, we can do it for certain ones, but for every other case is still an open question. So I want to answer that question for the following three manifold. So I want to consider three manifold. So you can put handle bodies in this way. I want to consider three manifold that you get from what some people call an iron triple. So I want to make the product of sigma times minus one one. And I want to quotient by an evolution that takes a point in sigma x and a point in the interval to f of x minus t. So our boundary M will be a Riemann surface sigma that we started with. When we ask that question, we want to see what is the space of representations that extend and the theorem here with Baraglia is the following. So the representations that extend extend correspond to key span law. So E phi in the ABA brain. So you have to be running the ABA brain, but this ABA brain had many components in the fibers. You may wonder which components do we need to take? And I have a couple of minutes. So let me tell you which components you take. So we're taking here our three manifold really bad drawing of a three manifold that's bounding our Riemann surface. And one thing you will have because we have this action that has some fixed circles. So we'll have any number of circles. There'll be some relations between the circles and the genus that you can have. Because we have those circles, we have the Higgs bundle here E phi. And we're looking for the brain which are invariant Higgs bundles. So invariant under the induced action here of F. The induced action will keep them invariant the isomorphism class, but in particular over each of these circles here, the vector bundle when I apply the involution. So when pulled back by this, it's going to be coming back to itself. It's a fixed circle. So we're fixed in the same point. And so for each of these and for each dimension, so for each dimension, of E over each circle. So E has some rank. That's the dimension of the vector space over each circle. There is a choice. There is a plus or minus one choice. Do I send E to itself or do I send it to E to all? So that's the two options we have with this involution acting over the fixed circles. This is over each of these gamma I. So this information actually encodes something of the vector bundle that appears because we have an action on the Riemann surface. And this is what we call the equivalent class of E. So this is the class of E, the k-theoretic class on the equivalent set two theory of the Riemann surface. This is the rough idea of how you can explain this topological class in terms of the actions that you have. And so you can come back here and say the E phi in the ABA brain for which the class of E in the twisted k-theory of the Riemann surface is trivial. And now we know which representations extend. If you wanted to carry on working on this kind of thing, then for instance, there is work of Satoshi Nahuata where he took a graph, a theta graph. So just the genus two, theta graph out of the sphere, looked at the complement and which representations extend and he showed that in terms of a generalized A polynomial. But, and that should correspond to an ABA brain coming like this, but that's not been shown. And for graphs and links of higher, giving higher genus compliments, that's not known from either of the perspectives from the perspective of generalized A polynomials and the matching with the ones that we've done. So I should stop here and tomorrow we're going to see about more correspondences like mirror symmetry and with other modular spaces. Thank you. Any questions? Yes? Yes, certainly and that would be very interesting. So there's not much being done in that direction. There is something that we have put in some remarks I can point you towards them. A particularly interesting thing that comes from what you're saying is that once you put two equivalent objects, you're looking at the intersection of two brains, right? Because you're asking for equivalence with respect to two things. And the intersection of brains is what gives you the maps in the Foucaille category. So you're having objects in the Foucaille category looking at the morphisms and that will come from intersection of brains. So you can have many more applications if you actually manage to give some examples of these objects, yeah. Any other questions? Yes? Can you maybe say a bit more about the superior context? Can these four types of brains be interpreted as deep brains? I think so. I haven't seen it mentioned many times. We put some comments in the non-abillion paper. We have some comments about how when brains intersect, usually you have a one-dimensional space. But when they collide in a higher dimensional space, that's what we're getting here, this non-abillionization. These BAA brains that are non-abillion should appear from the coalition of brains in higher dimensional spaces. There are some other types of brains, T brains. It's what we've looked at with my physics collaborators. These are brains which lie over the Neil Purden current. So they're brains coming from Neil Purden Higgs fields. And there's nothing like we've considered here. These brains all lie in the discriminant locus, but mostly outside the zero. Yeah. Yes, Vivek? What, sorry? Where do they meet? No, I think they will meet. This links up to his question. They will intersect in other places. It's just not been studied much, but he's generalized hitching. Yeah, I think they're going to be BAA brains. I haven't seen them expressed as involuntary. They could, because they are BAA brains that come equivalently to the hitching section. And the hitching section is one of the BAA examples we have here. I don't think they've looked at that. Yeah, I don't think they've looked at that. In fact, that's better if they aren't, because it would be great to have examples of constructions of families. So here, maybe I should make a point, emphasize that we like these four types of brains coming from involutions because they're all related to each other. We're taking involutions that when we put them together, we get the other types of things. If you could find another way of constructed families, maybe call yours one, or belong to some four types of actions that you can do that would give you the four types of brains, that would be really ideal. There's no other way for now of doing that for brains. Yeah. OK, I'll see you tomorrow. Thank you.