 Thanks for coming back to the last lecture on the Advanced Topics and Higgs Bundles. What I want to do today is I want to tell you about some correspondences between Higgs Bundles and other Higgs Bundles or Higgs Bundles with other objects in maths physics. Before I get started, just a couple of comments about the question sheets. So you'll see that the question sheets are all related to each of the lectures, but they're not in any order, particular order. So they're not from easy to hard. You should definitely not start by the first one just because it's first. Take a look at them and start with the one that you feel most comfortable with. There are all problems that come from papers in the area. So there are all things that are not that trivial that will require some thought and that, ideally, it will take you to read those papers that are mentioned and where these things are done to see the methods a bit further. The methods are the ones that we described in the lectures, but they're a bit further developed in those papers. So don't feel upset if you can't solve something in half an hour, it's not supposed to. But if you want to solve things that are a bit easier, so there's some lecture notes from something I did in Singapore a few years back. So if you look at the archive, there'll be some lecture notes on Higgs Bundles and that wasn't advanced topics. That was a very, very introduction and there the exercises are much more simple and to do just in a few calculations. What I thought here since I was asked to do advanced topics was that I should challenge you just a bit more. But you should just, don't get stuck, don't get upset if you can't solve it and if you want to carry on thinking about these things later on, you can email me. So feel free, any of you, to email me. My email is in the website. Okay, so Higgs Bundles and Correspondences will be doing the things from the fourth chapter in the lecture notes, we'll see how far we go and also I'll have to go back to something that we skipped in lecture one since parable Higgs Bundles are related to what we want to do today. So we'll introduce parable Higgs Bundles and see some correspondences with Quivers. So let me just put here the plan. So we want to do Higgs Bundles and Correspondences and Correspondences. Yes, I left this for the last lecture since it's what I've been working on in the latest times. So I've been working with different collaborators, looking at correspondences, correspondences coming from a few perspectives. So I want to look at correspondences via three different directions and there's probably many more. I just want to start with these three. So the first one comes from dualities. So this could be Langland's duality or any other duality that you know between groups or between modelized spaces that you could apply to Higgs Bundles. Then I want to look at group homomorphisms, or group homomorphisms. The group homomorphisms or maps between groups that satisfy certain properties will induce maps on the modelized space of Higgs Bundles for each of those groups and it's interesting to ask what happens with the hitching vibration, what happens with brains inside this vibration, how do they transform with respect to these homomorphisms? And lately, I want to take, borrow some notation from graph theory, use quevers, which are graphs which carries some information on each of the vertices and arrows and look at correspondences between these objects and Higgs Bundles. So I'm going to be talking about work that I've done with David Baraglia, this paper published this year on SOPQ. I'm going to be talking about work that I've done with Bradlow and with Bradlow Branko. So this in terms of group of homomorphisms and I'm going to be talking about work with Steve Ryan involving quiver varieties and polygons and hyperpolygon spaces. So I want to start by looking at some of these. Let's talk about the dualities one first since we've talked a little bit already and we're not going to take it that much time today. So for Langlands duality, Langlands duality is a duality that we have between groups, they're Langlands dual groups. So if I have a complex league group, we're going to be working just like before, I haven't fixed them here, but just remember we used to have here in blue the things that we have fixed, complex league group, Riemann surface of genus at least two. Today we're going to just relax this condition but for now let's just fix them here. So given a complex league group, I can form its dual league group and the ones that you should keep in mind are the following. So if I take the general linear group GLNC, I'm going to come back to GLNC as the dual. If I take SLNC, I'm going to go to PGLNC, and if I go to SO2NC, SL dual SO2NC, and if I go to symplectic one, SP2NC, I'm going to go to the orthogonal with odd SO2N plus one C. So this is the correspondence that we're going to be considering and we want to understand what's happening with the hitching vibration in the modular space of Higgs panels for those groups. We mentioned just as a motivation in the last lectures that there was some correspondence and the correspondence is between the two modular spaces of Higgs panels for a group on its Langlands dual. What one can do is one can look at the hitching vibration that we defined before. Remember we're going to be taking a Higgs pair here so we're going to take a representative of a class here and what we're going to do is we're going to send it in particular for instance to the characteristic polynomial and its coefficients. So I'm going to send this to the coefficients of the characteristic polynomial of the Higgs field. So that phi minus lambda identity here. If I do the same taking Higgs panels for the Langlands dual group, I'm going to arrive to a modular space AGCL that one can show just because it's formed by the characteristic polynomials, invariant polynomials is isomorphic to our original base. This isomorphism is not always the identity. It is going to be the identity for most groups but for instance for G2. So these correspondences also work for exceptional groups and when you look at G2, the isomorphism is not trivial. If you want to know more about that correspondence you should look at Hitching's paper from 2007. So what is the conjecture that appears when considering modular space of Higgs panels on their Langlands dual? Well, Ströminger, Jao and Zaslo, Jao and Zaslo conjectured that the duality between these modular spaces, the historic fibrations should be a duality between the fivers, duality of fivers. Which means we take the fiver over a regular point and we take the dual abelian variety in the other side and that should be what the fiver is in the dual side. So let's, quality of fivers and here we should emphasize this is regular, regular fivers. So when we were looking at the Hitching fivration we said in the Hitching base there's going to be regular fivers and there's going to be some space that's not regular. So there's going to be some discriminant locus and this duality is for all the fibers that are over the regular points. Remember over the regular points our fibers are abelian varieties but over the singular points there'll be spaces that are not necessarily nice or as nice and for those spaces this duality is still not well understood for the groups, for these groups and more generally. So back in 2002, so 2002 Housel and that is of proved that this is the case. So it proved that the duality between the fivers is indeed appearing for SL and PGL, SL and case. And soon after Donaghy and Pantheb and Kapustin and Witten around our six they completed the study of the regular fivers for these groups. If just as a little exercise for your head remember that for instance, for GL and C Hickspanos we mentioned that the regular fivers are Jacobian varieties. What's the dual space to that Jacobian variety? Any thoughts? Is the Jacobian itself? So for GL and C the regular fibers are Jacobians of curves and the dual space to a Jacobian. And the dual variety is again Jack. So we have a duality between the fibers, the fibers are Jacobians and the duals are also Jacobians and it agrees with our little table here that says GL and C goes to GL and C and they're like those duality. When you look at SL and SL and C then the regular fiber is the prem variety. You said we constructed the prem variety here in the lecture. And actually for the dual space the dual fiber is the prem variety of S sigma quotiented by torsion and points which agrees with the fiber in the dual side. So you can carry on and look at the duality in general for groups for any group that has a Langlands dual group or you can actually just do it for these classical groups and for that if you are curious about how to look at it I would suggest the work of Hitching O7 G2 curves and Langlands duality where he does very explicitly the correspondence between spectral data for these groups. The work of these other people is much more broad reaching but a bit more technical. The other thing that happens with Langlands duality and mirror symmetry is that once we have mirror symmetry between the fibers giving us the mirror symmetry of the two spaces we had also brains inside the mod light space. So brains brains here inside this mod light space and homological mirror symmetry also should give us a duality between brains so through conceivages work there should be some dual brain related which we talked already about when describing brains. So anytime that you study the mod light space of Hicks panels you should keep in mind that there is this duality there is this duality between fibers and there is a duality between subspaces and you should ask how does that fit into your research or your picture and in particular we can even ask how do those dualities fit with group homomorphisms or with graph theory and that's what we're going to do that's why I started with this one because I want to also mention some open questions that appear when you look at these two other correspondences together with the duality. Any questions about this? Yes, as dual, so when you construct a dual abelian variety looking at the exact sequence on the annihilator that's what we're getting, yeah. There's a bit more explanation I think in the notes about just more general. That's a great question also because when you look at brains inside the fiber so you look at the brain inside the fiber and you want to construct the dual object in the dual fiber you'll have to consider first the full fiber look at the dual space and look at the annihilator on the other side. Yes, yes. Because if you think about it what you're doing is you're taking your spectral curve you're pointing the base given by the zeros of this polynomial is your spectral curve which is generically smooth and it has the same linear system defining it. So you can calculate the genus that's a nice exercise to calculate what's the genus of this end cover of the Riemann surface and using that you get the Jacobian dimension, yeah. Any other questions? Okay, so let's take a look at the other dualities and see what happens. So I want to keep this Langlian's dual group here because the other correspondences the next correspondences the group homomorphism correspondences appear or I started studying them at least because I was wondering what else I mentioned some of these before what else can I put here which moduli space can I put here over the same base of the hitching vibration such that there is a correspondence which might not be the duality which shouldn't be the dualities there's a correspondence between these two vibrations these two moduli spaces and then there's a new correspondence between brains. So in particular from Langlian's duality we know that we saw yesterday that what we call BAA brains like brains for real hexpandos correspond to what we call hyperholomorphic brains, BBB brains. So I want to consider these new spaces these blue spaces here such that brains don't satisfy the pairing that appears from Langlian's duality in particular like BAA brains to correspond to BAA brains I want to keep the same type of brain if I can and that appears from group homomorphisms very naturally so let me tell you about it a little bit about what we've done with Steve Bradley and this is what actually how the lectures start in the notes here so I want to consider group homomorphisms group homomorphisms so these are maps between a group some G and some G prime I can look at morphisms between real groups I can look at morphisms between complex groups let's look at complex ones first and then we'll go to real ones once you have a group homomorphism between groups you can define a homomorphism between the representation space so this induces a map so I'm going to call it just the same between the representation space for your fixed Riemann surface and GC to representations of pi one again for sigma and GC prime and just by the non-abillion-hodge correspondence we talked before about we can then look at a map induced on Higgs bundles so induced map on Higgs bundles map on Higgs bundles from these finally what we have here is going to be the following so I'm going to take principal Higgs bundles so P let's recall that we're working with a complex group GC and P phi is going to go to so the same P GC but tensor through the morphism of groups with GC prime and the Higgs field will be a transformation of our original Higgs field and we're going to get through the action of our oh what's the name of this letter? Psi, okay thank you we're going to get through the derivative of Psi with phi so what are these objects? this is what we're going to call the new P GC prime and this is the derivative of so this is through the derivative of phi of Psi under the identity so at identity giving a map from the adjoint bundle of GC to adjoint of GC prime so once we have that we can ask what are Higgs bundles once we have particular group homomorphisms if you have your favorite group homomorphism we can check those ones and it's actually not been done very much so a few years ago we started doing this kind of thing with isogenes so we'll consider isogenes I'll put here a few so you know but if you can think of other morphisms that you're interested in or that you've seen and give you some nice property of groups then it would be very good to try and do it for the Higgs vibration it would suddenly be a new result I'll talk today about isogenes the other ones that I'm looking with Sebastian Schlöss so he's a student of Andi Naitzke actually we're looking at triality so how does triality influence the Higgs vibration? I won't talk about that today I'm going to focus on this isogenes isogenes are maps between groups of low rank that give you some unexpected equalities or or congruences but that are not there for higher ranked groups so the ones that I'm going to talk today about are going to be rank two and rank three so instead of putting all of them I want to just put the ones that we're going to use today so for example SL so I want to do SL2 C cross SL2C this is isogenes to SO4C and I want to do SL4C and SO6 so SL4C and SL, SO6C these are the two that we've considered both on the regular locus and on the singular locus I'll tell you about the regular locus today there are many other isogenes so if you look at page 36 you're going to see a long list of isogenes for instance SO5C and SP4C so another one that's here not mentioned which is what would arise from Langmuir's duality SO5C and SP4C so that's from Langmuir's duality the correspondence but these other two are not from Langmuir's duality and we want to understand what is the homomorphism for Higgs bundles what does it do to the hitching vibration and what does it do to brains in the hitching vibration the way that we're going to study this map is by considering the action of Higgs bundles will be coming directly from this map so if we apply it to Higgs bundles for each of those two groups then we get the following so for our particular cases let's put the cases one and two here that we're going to be considering so one and two for the cases one and two that are going to give us some of these blue correspondences so case one and case two we're going to apply the map to get from SL2 cross SL2 Higgs bundles and a natural SO4C Higgs bundle and the same for the second one so we need to say what we're starting with if we're starting with the SO... so the SOgene is going to send here so we're going to call it I2 the SOgene in rank two is going to send SL2C cross SL2C Higgs bundles into SO4C Higgs bundles which means we're going to apply it to a pair E1, phi1 and this is why I was telling you you can actually define Higgs bundles not necessarily for just groups of type A, B, C, D you can put other groups so like this product and we're going to get an SL2 cross SL2C Higgs bundle you can show that it's actually just two SL2 pairs so E2, phi2 and we're going to build the next Higgs bundle the SO4C Higgs bundle by doing the product of the bundles and doing the product of the Higgs fields with the identity so this is what we get we get E1 plus E2 and then for the Higgs field we get phi1 cross identity plus identity cross phi2 one can show that this pair actually belongs to the moduli space of SO4C Higgs bundles so it's not too difficult to check that the properties just carry through that the product of two Higgs bundles to do vector bundles from the SL2 group are going to give you a vector bundle that has an orthogonal structure and whose structure is compatible with that Higgs field now the I3 isogenic that takes SL4C Higgs bundles into SO6C Higgs bundles you might wonder how am I going to get a higher rank but we're just actually going to apply the same correspondence that we wrote there for the psi we're going to start with a pair now it doesn't have any number it's just E5 in SL4C and we're going to do is we're going to look at the Higgs pair twisted with itself so we're going to use the bundle will be the exterior power of E so this is a rank 4 vector bundle from SL4C and when you take the watch 2 you're going to get a rank 6 bundle and here the Higgs field will be very similar to that one but just itself taking both times so we're going to put just phi identity plus identity times phi but what we're interesting most on is not just these maps these maps just carry on naturally from what we explained here but we're interested in these correspondence with the hitching vibration so as I mentioned before I want to not only see the map or the new mod light space over the same space so here what we're showing is that we have SL2 cross SL2 Higgs bundles when you look at the bases for these two groups and for these two groups they're the same so we can put one here and one here and we have these correspondence between the bases but we need to understand what the correspondence is between the fibers so I want to mention in just a few minutes what the correspondence between the hitching fibers are since these fiber product construction could possibly have some other applications in other areas so what we're going to do to understand to understand the correspondence between hitching fibers so the hitching fibers for SL2 cross SL2 Higgs bundles corresponding to SO4 and to SO4 corresponding to SO6 we need to consider fiber products of curves so we need to consider fiber product of curves so fiber products are products that we do over the Riemann surface for instance if we have for one curve so we're going to have the curves given by eigenvalues over each point of the Riemann surface and the fiber product of the curve with itself it's all of the combinations of those eigenvalues repeated so we have a higher rank cover so the fiber product we're going to be looking at in the first case so in the first case we're going to be looking at the two cases that we have there in the first one we're going to have two spectral covers because we have two Higgs bundles E1, phi 1 and E2, phi 2 and these two Higgs bundles have spectral data we saw already spectral data for SL Higgs bundles so we know that spectral data here is a curve S and a line bundle in the print variety so L1 and here there's an S2 and a line bundle 2 in the print variety in the regular fibers so for regular fibers we saw then the same for the spectral data of SL4 we know that there's a curve and a line bundle in the print variety so we want to do something to these curves and these line bundles in order to arrive to data for SL4C one thing to note is the following so let me put a remark here because this will be something interesting remark you might have seen it in the problems if you got to do something about orthogonal Higgs bundles when considering SL2NC Higgs bundles your characteristic polynomial so Higgs bundles define a curve S a curve S through a characteristic polynomial which is the zero of our lambda to the 2N plus a1 lambda to the 2N minus 2 plus a2 lambda to the 2N minus 4 plus our last coefficient which is not really going to be an A and N but so this is going to be A sub N where A sub N here so A sub N remember this is a section of K to the 2N and it's not just any section it's the square of a Fafnian so it's a Fafnian square so now your linear system has base points the base points are the zeroes of these differential because those zeroes appear twice so your curve is always singular so whenever you look at Higgs bundles for SL2N you are always on a singular curve so this is always singular what's that? of a Fafnian yes of a Fafnian sorry so always singular and what you have to do to define spectral data is you have to separate those points and define the normalization so you have some normalization S hat normalized and then it's the prime variety of this S hat that you're going to be looking at so what you're going to be doing for the spectral data in this curve is you're going to look at the map that's a little involution that sends lambda to minus lambda in the original curve has fixed points the fixed points are those base points but when we normalize and we separate them our new cover S hat doesn't have any ramification points and our new involution doesn't have any fixed points and the data, the spectral data in this case is going to be coming from a prime of S hat and S hat quotiented by sigma so this is a very very quick explanation of something that took hitching many pages in his paper to describe so you shouldn't worry if you can't deduce it right away I just want to give you the gist of what's going on for orthogonal heat expansions and in particular the fact that spectral data has to be coming or comes from a normalization of a spectral curve so if there's any way in which you could recover the normalization automatically that would be great because you don't have to normalize the curve and that's what happens when we do our method of fiber products so through fiber products we can show the following so the image so the image let me put it here the image of I2 has spectral data I'm going to call it just trying to follow the notation from the notes so you can go back to them S4 hat and L4 for S4 being S4 hat being the fiber product of S1 fiber over the Riemann surface with S2 and these two spectral covers are given over the Riemann surface so there's a pi1 and there's a pi2 from each of these two covers and L4 the line bundle for the rank 4 orthogonal bundle is just going to be the pullback of of each of these line bundles on each of the curves so let me write it like this L4 is going to be if pi1 is the projection from the first one to the first factor so let's not call these ones