 So this is my very first atelier. So I'm very excited to be participating in this new kind of event, even though I missed the first half. So what I'm going to be talking about, it's really about a pair of natural moduli spaces associated to a closed-oriented surface. That's the S and a league group G. In fact, it's really one space that has two different presentations. So with any topological space, maybe the first question you might be interested in is the number of components. And that's really what I'm going to be talking about. And we'll see that for these moduli spaces, this question about components has two aspects. What I've called here a mundane aspect, where there are obvious topological labels that are obviously constant on connected components. And so there's no surprises there. But there's also what I call an exotic contribution to this number of connected components, where you see components that are not detected by the obvious topological invariance. And that's really the subject matter. These moduli spaces, I'll say more about what they are in a minute, but one of the perspectives is from Higgs bundles. And that's what I'll be talking about here. So these will be moduli spaces of Higgs bundles that we'll be talking about. So really, I hope the high point of the talk will be discussion of a new result that's joined with Brian, Brian Collier, Oscar Garcia Prada, Peter Gothen, and Andre Oliveira, where we have detected some new exotic components in the moduli spaces associated to this group, SOPQ. And we'll see that part of the reason that this is interesting is because there are two well-known sources of exotic components for various classes of groups. And this group doesn't lie in either of those up till now, well-known sources. So there's something new going on here. But so before we can get to that, there's a certain amount of scene setting and background that we'll go through. So the plan is basically to tell you a bit about these moduli spaces. Say something about the ways that exotic components are known to arise, and then tell you something a little bit more about these new results. So that's the plan. So let me try and tell you a little bit just what we need to know about these moduli spaces. So the two of them, the one is representation variety. So homomorphisms from the fundamental group of the surface into the group G, up to conjugation. And the other one is a moduli space of Higgs bundles. So Higgs bundles I'll say more about in a minute, but these are holomorphic bundles with some extra structure. So this seems like a very different sort of beast. The relation between these two, I mean, this bridge is the result of big theorems. This is the non-Abelian Hodge correspondence. So this is the non-Abelian version of what Alex was talking about earlier today, the linear Hodge theory correspondence. So these spaces are not linear spaces. So that's sort of related to the non-Abelian. G, right. So S is a closed oriented surface of genus bigger than 1. And G is in principle any reductive Lie group. But for this talk, really, these are the ones we're going to see. So it's going to be SLNC, or special orthogonal group, or some of the real forms of those groups. But this does fit into a wider setting, but that won't appear much in this talk. You'll notice that in both of these modular spaces there's some adornment here. Here there's a plus, and here there's this word polystable. And I'm not going to say too much about either of those, except that so these are the conditions that you need in order to get nice modular spaces. I mean, both of these are parametrizing equivalence classes of objects. Here are the equivalence classes by action of conjugation of G on space of homomorphisms. And in general, if you just take the full set of equivalence classes, you don't get a nice geometric space. You have to impose some restriction. And this means reductive representation, and here the restriction is inspired by geometric invariant theory. So those are the two spaces. And we want to understand the connected components of both because they're homeomorphic. Sorry? Any n. Doesn't matter. We'll see that when n is 2, it's a little bit special, but any n. There was one other thing I was going to say. Oh, yeah. So notice here that all we have is the surface and the group. So this is purely topological data. Here we have to fix a complex structure on the surface, so turn it into a Riemann surface. And that's what allows us to talk about holomorphic objects over here. OK, so as I said, the focus for today is going to be on the connected components of pi 0. And we're going to be focusing mostly on this side of this correspondence. OK, so let me tell you a bit about what Higgs bundles are. So there's a notion of them for any g. It's somewhat easier to state for slnc. And then I'll say something about how you modify that for subgroups of slnc. So we've fixed the complex structure. So now we have a Riemann surface. And then a Higgs bundle on the Riemann surface is a pair consisting of a holomorphic bundle, so rank n holomorphic bundle. The determinant is fixed to be trivial. That's related to the s there. If that was a g, then you wouldn't fix the determinant. And the Higgs field, the extra piece of data, he has one description of it. It's a holomorphic map from the bundle to the bundle twisted by the canonical bundle. So this is the holomorphic cotangent bundle of the surface with a restriction on the trace. So this locally is just a matrix of one forms. The trace restriction here is also related to the s there. It's telling you that this lies in the Lie algebra of slnc. The important things we need to know about these Higgs bundles are, so here we have a holomorphic bundle, a Higgs field. If we add an extra piece of geometric data, a metric on the bundle, then that combined with the other defining data allows you to define connections. Here the d is the churn connection. So that's determined by the holomorphic structure and the metric, but then it's modified by the Higgs field. So this defines special metrics built on the Higgs bundle. The stability notion, which I mentioned as necessary for building the modular spaces, ensures that you can find a preferred metric, actually satisfying a harmonic condition, which has the consequence that this connection is flat. And so that establishes the connection between stable Higgs bundles and local systems and therefore representations of the fundamental group. So that's a key part of this non-Abelian Hodge bridge. You had a higher difference. Right, so then you can, I mean, this makes sense, but then you have to add an extra condition that the square of phi vanishes, phi which phi is 0. There's an extra condition. But I'm not thinking about that. Right, right. Right, right. But we're not going to go up in dimension. OK, so that's for SLNC. If you want to take a subgroup of SLNC, the way this description changes is you impose extra restrictions, which you can interpret as designed so that the holonomy of these flat connections lies in the group that you're interested in. And what that translates into, in some cases, is on this slide. So for SLNC, you have just a bundle with fixed determinant and a Higgs field with a trace condition. If you want holonomy in SONC, the bundle has to have an orthogonal structure. And the Higgs field has to be skew symmetric with respect to that orthogonal structure. And then the connections that you get from that will have the right holonomy. If you want to restrict to real forms of these complex groups, you have to impose extra restrictions on the Higgs bundles. In the case of SLNR, the rank n bundle again has to have an orthogonal structure. But the Higgs field now has to be symmetric, not skew symmetric. And in the case that is really the star of the talk, the SOPQs, the rank n bundle, I mean the rank p plus q bundle, has to be decomposed as a sum of a rank p and a rank q, both of which have orthogonal structures. So the orthogonal structure decomposes in this way. And the Higgs field has to be compatible with this decomposition in this, so it has to look like this. The t here means the transpose with respect to these orthogonal structures. Which objects? So phi is the Higgs field, so that's a map from v to v twisted by the canonical bundle. And q is the orthogonal structure that appears when the group that we're talking about is the orthogonal group. So q is a symmetric bilinear form that defines an orthogonal structure on the bundle. Orthogonal. No, there's no q here because this is the Higgs bundle for SLNC. So here we don't need any more data other than what I've described here in order for these connections to. Phi is a bundle map? Phi is a bundle map from v to v twisted by the canonical bundle. So v is over sigma, right? Phi is an endomorphism. If the k wasn't there, phi would just be an endomorphism of the bundle. Bundle endomorphism. Tough as the identity, right? Yeah, you can formulate those conditions in that way here. Right, right. So locally, this is a matrix of one forms. And those descriptions are related by SLNC transition functions for this case. Because you want this to take values in the Lie algebra of this group. That's related to where you want the holonomy of this thing to lie. Y is what? That's also related to the s here. It's a, I mean, you want, yeah. Well, no, yeah, g in this line is SLNC in this line. It's sO. Q is the new ingredient that you need to distinguish these Higgs bundles from those. And I'm advocating you interpret this as the condition you need so that the connections you build have holonomy not in SLNC, but in this subgroup of SLNC. OK. So these things have moduli spaces, whose components we want you to understand. The story is different for the case of g complex and g real, as we'll see. Let me start with the moduli spaces for complex g. Here's the classical semisimple Lie groups that are covered by this. So these spaces have many interesting properties. I've listed just some of them here, not all of which are going to be relevant in this talk. They, I mean, they're complex analytic spaces, but they actually have a hypercalous structure. They also have these two. Mg is the moduli space of polystable g-higgs bundles. The Riemann surface is fixed, and the topological type of the bundle is fixed. We'll see. We'll get to that. Right. From one point of view, we're varying the pair, the holomorphic structure, and the Higgs field. And we're looking at isomorphism classes of those objects. If you take just all isomorphism classes that doesn't have a nice geometric structure, but you impose a notion of stability throughout the unstable ones, and then what's left does form a nice moduli space. It's not always smooth. So when I say hypercalous structure, I mean on the smooth locus. In all cases, though, it has a very convenient C star action, which you get by acting just on the Higgs field, scaling that by non-zero complex numbers. It also has a very convenient function. You get by evaluating the L2 norm of the Higgs field. I mean, I put more function in quotes here, because when this isn't a smooth manifold, then you can't really be doing more theory. But you can extract topological information anyway from some of the properties of this function, and that will come up later in the talk. So that will play a role later in the talk. The other thing that we'll be using is this other feature of these moduli spaces, which is a vibration over a vector space, which turns out to be of half the dimension of these spaces, and such that the fibers of this vibration are generically abelian varieties. So generically means there are some singular fibers where that's not true, but away from the singular locus, there are abelian varieties. This map is defined by taking invariant polynomials and evaluating them on the Higgs field. And we'll see that this vibration has many wonderful properties, including being a convenient setting for understanding mirror symmetry. That's not something that we're going to say anything more about in this talk, but this structure will play a role in understanding some of the connected components. So these are all the interesting aspects of the space. The one aspect that is not so interesting for moduli spaces for complex groups is the set of connected components. In the case of the complex groups, there's really only obvious topological invariants that appear, and they do label connected components. So there's not much in the story for those moduli spaces. Oops, sorry. Right, so the critical points for this function are actually the fixed points for that C star action. I mean, it's a more spot function, I should say. I mean, the critical sets, the critical points are not isolated, but there are critical submanifolds. The main thing that we use about this is that it's a proper map, so it attains its minimum on every connected component. And that's the way that it's going to be used. And that statement is true whether the space is smooth or not. OK, so good. So yeah, please keep the questions coming. Slow me down if necessary. OK, so when the group is a real form of those complex groups that we just saw, the story about connected components gets considerably more interesting. A certain amount remains the same. I mean, this is still a moduli space of bundles with extra structure. But since there are principle G bundles underlying these Higgs bundles, those have topological invariance characteristic classes, and those must be constant on connected components. So those are the obvious invariance that partially label connected components of the moduli space. They decompose this space. So I hesitate to use the word component, because the whole point of this is that these labeled by characteristic classes are not necessarily connected. But it does give you this decomposition, where C is just here generically standing for whatever characteristic classes are appropriate for the G you're talking about. But what's new when the group is real is there are several phenomena that appear. On the one hand, in some cases, turns out that there are too many of these invariance. There are bounds that they have to satisfy in order for these components to be non-empty. So not all invariance can appear. The opposite phenomenon also occurs where there are too few of these. In other words, they fail to distinguish all the connected components. There's also a phenomenon that occurs for these real cases, where some of the components that appear have special geometric significance. We'll see in the example that I look at in a second, sort of the first and sort of prime example of this is that one of the components is a copy of Teichmuller space. But there are other geometric structures that are also detected by some of the special components that appear for special real forms. So that's sort of the overall general setting of the type of question that we want to look at lives in. So let me start by describing some of the phenomena in the case of SL2R. So let's look at this example. This is historically maybe the first one, but also we see essentially all the phenomena appearing in this example. So now we're looking at, from the one point of view, it's the modular space of SL2R Higgs bundles, which by this non-Abelian Hodge theory correspondence is the representation variety for the fundamental group. That should have been an S, not a sigma, because we don't need the holomorphic structure here into SL2R. There I've made it an S there. So this goes back to Worker Bill Goldman, who gave the count of the number of components for this representation variety, and here it is. So this is for a surface of genus G. So the number of components is here, this number, two to the two G plus one plus two G minus three. And so we'll understand that a little bit as we go down this slide. So the obvious invariant here is, I mean these representations and these Higgs bundles have associated to them an Euler number, so essentially an Euler number of an SL2R, of an SO2 bundle. So that's an integer. Turns out that there is a bound on this integer. So here we see the topological invariant the too many aspect. The bound is G minus one for the components where the invariant is not maximal, that those actually are labeled by that invariant, so those are connected, but for the maximal ones, there are two to the two G components all corresponding to that value of the invariant, each one of which is a copy of Teichmuller space. So that's what you see from the representation variety point of view, how does this look from the perspective of Higgs bundles, which is the perspective that we're going to extend when we look at other groups. So first of all, what does an SL2R Higgs bundle look like? So you probably don't remember from the table, but this is a rank two bundle, that's the two, but because it's an SL2R, it's a rank two bundle that has to have an orthogonal structure, and that can always be put in this standard form, a line bundle plus its dual. So the Higgs bundle is a line bundle plus its dual, the Higgs field has to be symmetric with respect to the orthogonal structure defined in this standard form, which turns out to mean that it has to be off-diagonal like this. Each one of these pieces in the Higgs field is a section of a line bundle, the line bundle determined by L in the canonical bundle. So that's the data that defines the Higgs bundles. There's topological invariant staring at you here in this line bundle, it's the degree of that line bundle. The bound on that is very accessible from the Higgs bundle point of view because the stability notion which I didn't define for you implies that one or other of these components cannot vanish. Which one depends on whether the degree is positive or negative, but the fact that it can't vanish puts a bound on the degree of the bundle of which it's a section because line bundles can only have sections if their degree is non-negative. So that's where the bound comes from. So you get a decomposition of the modular space of Higgs bundles into pieces labeled by the degree of this bundle L. And because like beta is a map from L inverse to L tensor K. Yeah, well no, it reinforces it. It's a map from L inverse to L tensor K. So it's a section of L squared K and the other one is of L minus two K. So when the parameter is maximal then the degree of one of these is zero and so to have a section it has to actually be the trivial bundle. So the structure of the bundle actually specializes. It's not determined just by its degree. It also has to be a square root of the canonical bundle. And that means that the description of the maximal components change is different from the other components. There's now one component for each choice of the square root of the canonical bundle and each one of them is determined just by the piece of information that hasn't been used yet, namely a section of this square of the canonical bundle. So by a quadratic differential. And so this identifies these components with a copy of the space of quadratic differentials. These are the Hitchin components. This is actually the identification of Teichmuller space with the space of quadratic differentials that Mike Wolf was also responsible for originally. But this is seen from the Higgs bundle point of view. And you see two to the two G copies of it corresponding to the choices of the canonical bundle. So there are many different phenomena at work here. And this is really a reflection of the fact that SL2R lies in, you know, it wears many different hats. There are many low dimensional isomorphisms at work here. SL2R is the same as SP2R and it's a double cover of SO12. It's also, you can also identify it with SU11, but that wasn't essential for this story, so I didn't put it here. These groups all lie in bigger families. This one is in the SP2NRs, this one SLNR. This one actually can be viewed, should be viewed as lying in two different families. In the one case, it's the SO2Qs, so keeping two fixed. And the other is SOPP plus one. Both of those are special cases of the group that we eventually want to talk about. So the groups that you see here live in, I mean, there's some special features that are the important things here. Some of them, the ones in blue here are like SLNR are the split real forms of their complex groups. While others, so you should look at this one and that one, this is in purple because it lies in both these families. But these groups are distinguished by, I mean, I've called them Hermitian. What it refers to is that the symmetric space you get from these quotient by their maximal compact is a Hermitian symmetric space. So those, these two features are the ones that are known to be responsible for special components. I'll say more about that in a minute. You'll notice that this one didn't get, wasn't either blue or red. So this is where the novelty is appearing here. Okay, so let me just say something about both of these mechanisms that are at work in the component story. Let's see, okay. So what is special about the split real forms? So here's a list of the real forms that fall into this category for the various classical groups. Here I've reminded you what the Higgs bundles for SL2R look like. In the, what the Higgs bundles that live in the special exotic components look like, I should say. There is some of the bundle is a sum of line bundles, but the line bundle is a square root of the canonical bundle. So some power of K and the Higgs field is constrained to not, I mean, the term that lies over there has to be nowhere zero and this one can be anything and that's where the parametrization comes from. The way this generalizes is, so there's a description in terms of the representation variety and in terms of the Higgs bundles. In this side, you get a generalization of the Teichmuller space. So these are the higher Teichmuller components. On this side, you get a generalization of the Hitchin components that were parametrized by quadratic differentials. And the way these appear just very briefly, the special thing about the split real forms is that there's an irreducible representation of SL2R that lies in each one of these. So if you take a representation corresponding to a point in Teichmuller space and compose it with the irreducible representation into the split real form, then that defines representations in here and the higher Teichmuller components are the components of representations which deform to one of these. On the Higgs bundle side, the Hitchin components that you get generalize the components which were parametrized by quadratic differentials. The quadratic differentials were actually the base of the vibration for the case of SL2C and the Hitchin component was parametrized by these by picking a section of this vibration and that construction, the existence of a section of this vibration which picks out a component in here is the way that these Hitchin components appear for the split real forms. So the one, okay, so don't panic. This is not as scary as it looks, but this is the example that is really the relevant one for our new SOPQ components. So this is what happens for the Hitchin components for the split real forms SOPP plus one. So these are according to this picture parametrized by the base of the Hitchin vibration for SO2P plus one and that contains not only quadratic differentials but differentials of all even powers from one to two P. The bundles in these Hitchin components just like these bundles are fixed, they're constrained to have a particular form, they have to be sums of powers of the canonical bundle and the Higgs field, I mean this is an SOPQ so it has to be compatible with the orthogonal structures on these bundles so it has to be of this diagonal form but the information in this sector that I've called eta, that's what goes where the star is, I've illustrated in this diagram. So what we have in this diagram are we have the summands in the two bundles distribute alternating from the summand of V to the summand of W and then the arrows are the non-zero components of the Higgs field. So the Higgs field is a map from W to V twisted by K. It has many components corresponding to this decomposition. These arrows indicate where they lie. Notice that these ones are all non-zero. These are essentially the identity. You might wonder how do you get an identity from the Pth power to the P minus first power and the reason is that these are Higgs fields so there's a twisting by the canonical bundle which isn't in this picture but if you put that in then that would be a map from K to P to K to the P. So this is the way the Higgs bundles in the Hitchin component for SOP P plus one look and quickly memorize that because this pattern will appear again in a couple of slides. Okay, so that's what happens for the spit real forms. For the real forms of Hermitian type so that was the other type of family that appeared in the case of SL2R and for which I said you get exotic components so there the relevant features are a little bit different so the example we saw SL2R everything is a sum of two bundles and the Higgs field is diagonal with respect to that that's a general feature for these real forms of Hermitian type. There's essentially integer valued invariant here it was just the degree of L in some of the other cases it's not exactly an integer it's a integer multiples of some constant but there's a bound which from the Higgs bundle point of view you can see as a consequence of the stability criterion when the obvious invariant hits its maximal value the structure of the bundle specializes in the way that in SL2R L had to be a square root of the canonical bundle there's something sort of analogous that always happens and so you get new invariants appearing because the structure has specialized so the maximal component decomposes into components labeled by a new invariant. Each piece labeled by this new invariant acquires a new interpretation which we'll see coming up as well it's like a modular space of Higgs bundles for a different group except now the Higgs field is not twisted by the canonical bundle but by the square of the canonical bundle so this is a K squared twisted G prime Higgs bundle for a companion group G prime. So I want to try and work through the way that looks for this particular example because again this is the one that is relevant for the SOPQ so in this case so this is an example of a group of Hermitian type the Higgs bundle this is an SOPQ so the bundle has to be a sum of a rank two and a rank Q the rank two bundle is now an SO2 so again it has to decompose or it can be put in this form of a blind bundle plus it's dual here I've shown you what the orthogonal structure looks like with respect to that decomposition W is a rank Q orthogonal bundle so no special form for that and the part of the Higgs field that I've called eta is now a map from W to L plus L inverse so it has two pieces that are labeled there in a diagram similar to the one that we saw for the Hitchin components where we label the summands in the bundles and put in arrows for the components in the Higgs fields this is what the Higgs field looks like for an SO2 Q Higgs bundle so you get a quiver diagram like this in this case they're actually two invariants one of them is an integer it's the degree of L they're also invariants for the bundle W this is an orthogonal bundle so its degree is zero but it has Stiefel Whitney classes in principle it has two but in this example this little zero down there means that we've taken the connected component of the identity of this group and for that restriction that implies that the first Stiefel Whitney class of W is necessarily zero so we're left with just the second Stiefel Whitney class which lives in H2 with Z2 coefficients so they're actually two labels but this is the one that participates in the Hermitian symmetric story there's a bound on this which you can see in this case by looking at the composition of the map this part of the Higgs field and it's transposed with respect to the orthogonal structure that gives you a map from L to L inverse and at each stage remember we've twisted by the canonical bundle so this is a map from L to L inverse K squared and stability says that that can't be zero so you get a degree on the bound of L from that condition there it is so that's the bound in this case so now we get a primary decomposition of the modular space into sectors labeled by the degree of L and also this second Stiefel Whitney class that's also a legitimate label but we get something special happening when the invariant here is maximal has its maximal value and again it comes from the fact that when that happens this is really a section of a degree zero bundle so it says the bundle must be trivial and the section must be nowhere vanishing and that specializes the structure of the SO2Q Higgs bundle it fixes L to be essentially so it's got to be the canonical bundle twisted by a square root of the trivial bundle so the only thing that you allow to vary is I which has two to the two G possibilities point of order two in the Jacobian if you like also the rank Q bundle has to decompose into I plus a rank Q minus one orthogonal bundle so this picture has become this picture where W has split into I plus W naught and here are all the pieces of the Higgs field that you're allowed so now if you look at this diagram we can reinterpret the information that we have in there in a new way if you look at the information down in this part this piece beta naught is a map from the rank Q minus one orthogonal bundle to the square the point of order two twisted by the canonical bundle that's down there the information here is really I W naught and beta naught and that information apart from the fact that there's a square there defines a Higgs bundle for the group SO1 Q minus one so this part of the diagram defines a point in this space a K squared twisted Higgs bundle for the group SO1 Q minus one if you look at what's left the only thing that's okay so we've got the K is fixed and this I whatever I is A2 here is a map from I to I let's see did I write it correctly? Yeah you can write it as a map from I to I K twisted by K so that's a quadratic differential for part of the story it would be convenient to reinterpret it as a K squared twisted Higgs bundle for the group SO1 one but that's not so important for us here so what the bottom line is that the maximal component where the parameter tau has its maximal value decomposes into pieces labeled by the choices of the point of order two and for each choice the space that you get is a product of this one with that one this one is a moduli space of K squared twisted Higgs bundles for this group and this one is a power is a space of sections of quadratic differentials so there's the exotic components in this example so okay so let me let me with all that preparation now tell you about what we what we find for SOPQ so this I mean this includes SO2Q and it includes SOPP plus one or we'll see more relevantly SOP minus one P but it's those are just special cases of this and outside of those special cases this is not in either of those special classes of groups and so here's a here's a crude summary of the results so this is again let me just acknowledge my collaborators Collier, Gothen, Garcia Prada and Oliveira so what the result is the result says is that for any of these SOPQs there are extra components not detected by the topological invariance whose union is isomorphic to this space so this these pieces look like what we saw for the SO2Q and the SOPP minus one except here instead of being twisted by K squared it's twisted by K to the P and here instead of just having quadratic differentials we have a sum of all the even powers up to two P minus two so let me tell you what I want to do in the next 10 minutes or so is give you some idea of why why there was sort of some expectation that these components might be there or what the evidence was that there ought to be some exotic components in this case and then just say a couple words about sort of the precise thing that we actually proved but before we get there right I was going to make this a little bit more precise so let me, sorry I forgot about this so here the result said that the exotic components the union all looks like this space the invariance, the obvious topological invariance in this example SOPQ are Stiefel-Whitney classes for the orthogonal bundle V and the orthogonal bundle W so in principle four invariance two first Stiefel-Whitneys and two second Stiefel-Whitneys but because of the S over there these first Stiefel-Whitney classes are linked so there's really three invariance one first Stiefel-Whitney class and two second Stiefel-Whitney classes and so there's a component for each of those which is detected just by those topological invariance and the exotic ones don't occur for all possible values of this in the cases where the second Stiefel-Whitney class of V is not zero you only get the component that is labeled by the obvious invariance the exotic ones occur in these sectors where the second Stiefel-Whitney class of V is zero so these, this component that I've labeled with a zero are the ones which are labeled just by the topological invariance and another way of characterizing those is to say that those are the Higgs bundles where you can deform the Higgs field to vanish and you're left with just orthogonal bundles in the others, you can't do that in the others, so in the exotic components the structure of the Higgs bundles all is captured in this diagram so like the SOP P plus one example there's a part that is determined just by powers of the canonical bundle that fully describes V W has a part like that and a rank so this is a rank P minus one piece this is a rank P piece so this has to be a rank Q minus P plus one and that's the analog of what we got in the SO2 Q example so this is the structure of the Higgs bundle of the bundle and the arrows indicate the structure of the Higgs field for every point in these exotic components and you'll notice that some of these arrows have ones on them so they can never vanish so you can see that you can't deform this away for the Higgs field to vanish completely okay so let me just look a little bit more closely at this diagram and this is the way the Higgs bundles look in this exotic component if you look at the part at the top of the diagram that's exactly what we saw for SOP P plus one except now this is SOP minus one P but so this part of the diagram parameterizes I mean represents a point in that modular space and the rest of the diagram the part in red here encodes data for an SO1 this is the rank of W naught Higgs bundle but now this is a map from W naught to this power of the canonical bundle with an extra K because everything is twisted by K so that's why there's a K to the P over there so that's where the two pieces of the, in the description of the component come from or how they're encoded in this diagram okay so now I can tell you about some of the evidence that there was for why there ought to be these components so the first is evidence from the function that I mentioned at the beginning of the talk that serves as a Morse function on the modular space of Higgs bundles so even though these spaces aren't necessarily smooth the fact that this is a proper map means that it attains its local min on all connected components so you get information about the connected components by identifying the local minima okay so in the case of SOPQ this was, I've written this for SLN here's what it looks like for SOPQ the function is really the ultronorm of this part of the Higgs field so this function either in this case or that case clearly it has a global minimum at zero and that's attained on the components that are labeled by zero before where the Higgs field can be deformed to vanish but so if there are other local minima then that holds out the possibility that there could be other components and this was detected by Marta Royo in her PhD thesis in 2009 she was a student of Oscar Garcia Prada and she found local minima for this function on SOPQ where the Higgs field did not vanish and they had this form so this is the diagram that we've had up a few times but with only these arrows all the other arrows have been set to zero so I mean this doesn't prove that there are other connected components but it allows for that possibility so this was the first piece of evidence the second piece of evidence was from Brian's results for these split real forms SOP P plus one so this was a generalization of results for the special case SO2-3 which SO2-3 is almost SP4-R those groups are isogenous but SP4-R is a double cover of SO2-3 and in both those cases so in SP4-R it was known that there were extra exotic components first detected by Peter Gothen and Brian's analysis of the case SOP P plus one found analogous exotic components for NEP for these split real forms so these are components I mean we know that since this is a split real form there are hitching components but these are components, I mean others not the hitching components so in fact there are components labeled by an integer D that, let's see did I put it somewhere up here? Yeah, yeah, it can range between zero and P times two G minus two at the maximum value these are the hitching components but Brian actually described not only detected them but described them for the other values of D as well so this is for every value of D if you take all the components all the even D's and all the odd D's then the union of those actually gives you the special case of the KP twisted SO1Q minus P plus one Higgs bundles that we saw in the SOPQ example so this was a special case I mean now we can say it's a special case but it was an indication of extra components in SOPQs there was one other stream of evidence and I know I'm gonna gobble this explanation but so you can talk to my collaborator Brian afterwards if you want a more coherent explanation but so this was evidence from the other side of the non-Abelian hodge correspondence so what I described before the more theory evidence and the SOPQ plus one came out of Higgs bundle analysis but so Livier Gichard and Anna Wienhardt analyzed the I mean had a prediction for the representation variety the corresponding representation variety for SOPQ which came out of their work analyzing what so what the I mean what these things have in common the Hitchin representations and the maximal representations for the symmetric case those are the two cases where exotic components are known in both of those cases the representations in these exotic components all have the special property known as the you know that there are Nossov representations as defined by Laverie's concept of a Nossov what Olivia and Anna showed was that there's actually a perhaps a more subtle property at work here which relies on a positivity notion for a group and actually a group and a parabolic subgroup of it which both in both of these cases you have but they showed that this property can occur more generally than in these two cases and they were first of all they they conjectured that when the group has this positivity property then there ought to be additional exotic components and it turns out that the only case not included by these two classes of groups okay except for some exceptional groups but the only other case is SOPQ so this suggests that there ought to be exotic components of SOPQ which have this property that they identified let's see I should stop in two minutes so all right so I'll just quickly summarize what we actually did so I mean once we had finally sort of understood what the components ought to look like then showing that you have components like that amounts to finding a map from what the space that they ought to be into the modular space where you think it should live the one slight wrinkle here is that these are modular spaces of isomorphism classes of objects what I've been writing down has mostly been just objects not isomorphism classes in fact the description of the map from data in the space that describes the components into the big modular space is really a map on objects not isomorphism classes so what we showed was first of all that this actually does define a map on between the modular spaces and then we showed that the map is has the right properties to actually define a component so it's closed and it's open and so okay so the first one is showing that that you actually have a map you more or less do just by direct observation direct computation and observation to show that the map is closed the Hitchin vibration actually plays an important role there because you wanna you know you want to show that the map is closed one way to do that is to show that if you have a divergent sequence then the image under the map is also divergent and if the sequence is divergent using the Hitchin vibration really the only way you can detect when a sequence is divergent because it diverges in the base of the Hitchin vibration and that's a sort of the convenient tool for showing that the map is closed showing that the map is open we actually could show really a bit more than that I mean we could show that the derivative of the map is a local isomorphism so that locally this is a diffeomorphism so it's open and but for that you need a good model of local neighborhoods of the modular space and that's really where sort of the work goes so okay that's all I'll say about the proof so you know one obvious question that has yet to be answered is whether these components that we've detected are actually the same as the components predicted by the representation variety picture in the Gishard-Vinhard conjecture so in other words are the representations in the components corresponding to our exotic components are they the ones that have the correct positivity property and stay tuned we'll tell you at the next atelier okay thanks