Quantization of Hamiltonian loop group spaces
Loizides, Yiannis
I will describe a map from `D-cycles' for the twisted K-homology of a compact, connected, simply connected Lie group to the Verlinde ring. The induced map on K-homology is inverse to the Freed-Hopkins-Teleman isomorphism. An application is to show that two options for `quantizing' a Hamiltonian loop group space are compatible with each other. This talk is partly based on joint work with Eckhard Meinrenken and Yanli Song.
 Thank you very much to the organizers for giving me the opportunity to speak at this interesting workshop. So my talk is mainly going to be based on an article that's not quite available on Archive yet, but I put a copy on my webpage in case you want to look at it. And it's very much inspired and motivated by earlier joint work with Akron Meinrank and Yanisang. Oh great, yeah. Okay, yeah, so my talk is mainly going to be based on the... Yeah, so the first thing listed here, but as I said, it's very much motivated by joint work with Yanisang and Akron Meinrank. Right. Ah, okay. Right, so my talk is related to the Freight Hopkins Tellment Theorem, so I'm going to start with a very brief introduction to that theorem. And part of the Freight Hopkins Tellment Theorem is a map going from representation theory of the loop group, LG, of some compact lead group, to twisted K theory, or twisted phenomenology of G. And so one thing I'm going to do in the talk is describe a map going in the opposite direction, going from twisted geomology to representation theory. And I'm going to be especially interested in what that map does to what are called d-cycles. This is the terminology of Baumkering-Wang. So d-cycles are sort of a nice geometric package that gives you cycles in twisted geomology. And then time permitting, I'll talk about sort of the original motivation that I had for thinking about this stuff, which has to do with Hamiltonian-LG space. Okay, so let me give you kind of a light mean overview of the loop group. So throughout my talk, G is going to be a compact, simply connected, simple loop group. And I'm going to fix the maximum torsity inside G. LG is going to be the loop group, so the maps from the circle into the lead group. And there are some subgroups of the loop group that are going to come up in my talk. So base loop groups, so these are loops which begin and end at the density element. And then sort of complementary to that, there's a copy of G sitting inside the loop group as constant loops. And another subgroup that will come up is the integral lattice. So you look at the kernel of the exponential map for this maximal torus. This is a lattice which we call it pi. And this naturally sits inside the loop group as the exponential loops formula that we did. And the loop group has a famous U1 central extension. And so I put the formula for the Lie algebra cosec of the central extension. And it involves this, V is the so-called basic inner product on the Lie algebra. It's an invariant inner product on the Lie algebra with a particularly nice normalization. And yeah, so one other thing I should mention about this central extension. It has a symmetry. So as one acts on the loop group by reprametrization, by rigid rotation of loops. And this symmetry lifts to the central extension. I am confused by this high definition. So that's the natural map from T, I guess it's going to, from the Lie algebra of T to T itself. Right. So how's the kernel of that inside omega G? So I'm identifying it with a subset of omega G. So if you have some elements in the kernel, you take, I guess this should be XT. Look at the loops that you get by exponentially. The one parameter subgroup generated by that element. So maybe instead of inclusion, I should put like an error, an inclusion error. Okay, and so let me tell you a little bit about the representations. So the group has an interesting family of projective representations called positive energy representations. And these are representations of the central extension with some conditions. So the representation should extend to an action of this semi-direct product group by this rotation circle. And the weights of the action of this rotation circle should be bounded below my zero. That's the positive energy condition. And then we say that one of these representations is at level K if the central circle. So the other circle acts with weight K. K is some integer. So these are positive energy representations. And they have a nice theory, very much parallel to the representation theory of G, the finite dimensionality group. So in particular, irreducible positive energy representations are parameterized by their highest weights. And another interesting thing that happens is that if you fix a level, so if you look at the fixed level K, it turns out there are only finitely many of these irreducible positive energy representations. And they're parameterized by the so-called level K weights. So my notation for this is pi K star. So pi star is the weight lattice, the dual of the integral lattice. And pi K star, you take the intersection of pi star with the K-fold inflation of the fundamental alpha. So I should also mention here and throughout my talk, I'm going to be using the basic inner product to identify G with its dual. So the alcove sits naturally inside the algebra of the maximum torus, but I'm thinking of it as a subset of the dual space. Right. And then so the verlimeter ring is sort of the analog of the representation ring for the loop group. So the level K verlimeter ring. So as a group, you can think of it as the phyabilian group generated by these level K weights, generated by these irreducible positive energy representations at level K. And it also has some other descriptions. So it has a nice description as a quotient of the ordinary representation ring of G by the so-called verlimeter idea. And this shows you its ring structure. And it has another description that's going to be important for my talk. And yeah, so I apologize, the notation is a bit unearthly. So it has another description as certain formal characters for the torus, for the maximum torus. So formal infinite sums of irreducible characters of the torus with a certain anti-symmetry property under the action of the affine valgur. So to understand this, it's maybe helpful to think by analogy with the case of a compact loop group. So if you have irreducible representation of a compact loop group and you restrict it to the torus, you get a formula for it, the val character formula. And it's given by some numerator which is alternating under a certain action of the val group, divided by some universal denominator that doesn't depend on the representation. There's a similar story for the loop group. They also have characters given by something that looks like the val character formula. And again, it takes a form of a numerator which is alternating under a certain action of the affine val groups. This is sort of the loop group, analog of the val group, divided by some universal denominator that doesn't depend on the representation. So that's this isomorphism here. Okay, then I have a picture to kind of maybe give you a bit of a feel for what this affine valgur action looks like. By the way, this K plus H check has to do with, so this is describing what this action is. The H check that appears there is the dual coaxial number. And this is quite important in the free-throwing. So maybe I'll spend a minute to talk about it. By the way, this image I have to think accurate and David Lee Bland for letting use this image. So this is a picture for SU3 with K plus 3. So this level K is this 3. And the level K weights are the black dots inside this red triangle. So there's finally many from here. And so the green hyperplanes, this affine valgur action is generated by reflections and all of these green hyperplanes. So an element of this group of formal characters, it has a multiplicity function, which is a function from the weight lattice to the integers. And it should be alternating under reflections in all of these hyperplanes. So notice that the fundamental domain for this action, this reflection action, it's a bit bigger. So this is where the dual coaxial number comes in. So the fundamental domain for this action is a bit larger than the set of level K weights. So maybe you can see what this group is. Okay, so that's what I wanted to say about loop groups. The other side of the free talk in the theorem is twisted k-homology or k-theory. So for me, the twist of k-homology is going to be what's called the Dixie-Mute-Rede bundle. So it's a bundle of c-star algebras over your space x that you're interested in, with fibers equal to the compact operators on some Hilbert space. So locally trivial bundle, locally it just looks like an open set times compact operators, but globally there can be some interesting twisting. And so these bundles are classified up to isomorphism by something called the Dixie-Mute-Rede invariant, which is a certain k-homology class that has a component in degree three, which is maybe the most important, and then there can also be a component in degree one for some extra grading information. And so in the equivalent situation where we have a group acting, these are equivalent k-homology classes. So the compact operator is on a z-quadered Hilbert space here, right? Yes, yes. And... I'm going to say something else at the moment. Right, so I should mention, so different bundles will give you different groups. So we're talking about sort of very sparsible groups than you could get. So for me, I'm going to use the analytic definition of the twisted k-homology group. It's known to be equivalent to definitions of topology. So for me, the twisted k-theory of space x is going to be the k-homology, the analytic k-homology of the algebra of continuous sections of this bundle, vanishing at infinity if you're in a non-compact space. So again, I should emphasize, you get different groups maybe from different Dixme de Rady bundles, different C-style bundles. And just to remind you what the cycles in this group look like. So they look like triples h-row f, where h is some z-quadered Hilbert space. The row is a representation of your algebra on this space. And f is, you can think of as sort of an abstract, zeroth order, liptic operator. And what that really means is that if you look at these three operators, they should be compact. So this first condition maybe you can think of as sort of locality condition. And this says that f is close to being self-adjoint, and this is sort of a compact, resolvent type condition. So that's sort of an abstract definition. The fundamental example is coming from the Clifford-Algebra bundle. So if you take any even dimensional Riemannian metaphor, you can look at the Clifford-Algebra bundle. This is a finite dimensional Dixme-Druity bundle. This Dixme-Druity invariant is the third integral Stiefel-Wittney class. So this is the abstraction to the existence of a spin-z structure on M. And there's a class in this group that you can think of as being like the fundamental class of the manifold M. And it's basically the Durand-Durac operator acting on differential formulas. To make it exactly match this definition, you need to do something like apply a functional calculus to get a bounded operator. It's basically the Durand-Durac operator. You're assuming that was right. You're assuming that was right. Right. Yeah. I should have put it back. Right. So don't you need to assume this is spin-z structure in order to get a fundamental class of k-theta? So this is going to live in twisted k-theta. I guess that's part of the idea. So this won't live in the ordinary k-theta of M, but it will live in the a-theta of k-theta of M. So this is maybe one useful aspect of twisted k-theta. Okay. So now I'm going to describe a nice geometric package for generating cycles in this group. So these are called d-cycles or Baum-Carry-Wine-Covins d-cycles. And these are a twisted analog of Baum-Douglas cycles in geometric technology, which we've seen that before. So a d-cycle is a four-tuple consisting of a compact-oriented even dimensional Riemannian manifold. I'm just describing an even case. There's a similar version for the odd case. And then you have a vector bundle on M, a continuous map from your manifold into your space X. And the last piece of data is maybe the most interesting. So you have a marita bimodule linking a Clifford algebra bundle under your Riemannian manifold with the pullback of this twisting bundle, the pullback of this Dixme-Duriddy bundle. And so if A is trivial, if A is just the complex numbers, then this S is equivalent to a spin-C structure on M. In that way you recover the ordinary Baum-Douglas cycles. But yeah, so more generally, so if A is some general Dixme-Duriddy bundle, the existence of S implies that you have this relation. So the pullback of the Dixme-Duriddy invariant of A is this third integral C4-Widme class. And the way you go from a d-cycle to a cycle in this Tristic K-Emolegy group is fairly simple. So you take the fundamental class, you can compare with this vector bundle, and then the pair consisting of this map phi and the bimodule S you define a push-forward map from the Tristic K-Emolegy of M to the Tristic K-Emolegy of your space X. And so you can push forward to get a class, which shouldn't it be the search-geek-widney class of the normal bundle of M and X or something for this push-forward to work that way? So we should use the normal bundle. Not sure. I'm not sure. I'm not sure. So now I'll talk a bit about the free Hopkins-Tellerman theorem. I should say I'm only talking about a very special case of the theorem, a very general theorem for arbitrary compactly groups. I'm just going to be talking about the case where G is simply connected, simple and connected. So we take the space X to be the group itself, and so there's an action of G by conjugation. And so it's known in this case that third aquavirin homology is just isomorphic to the integers, and the lower homology is vanishing, which means that the Dixmeid-Durradi invariant is just an integer. So I'm going to use this notation A superscript L to just mean the Dixmeid-Durradi bundle with invariant L. And so the free Hopkins-Tellerman theorem is an isomorphism between the Rlindering and some positive-level K. And the twisted K homology, there's a version for K theory too, as stated by the Poincare duality, but here I've written it in terms of K homology. And the Dixmeid-Durradi invariant is shifted a little bit, so it's shifted by this dual coccidant number. And yeah, so more than this, so free Hopkins-Tellerman construct a very interesting map going from representations to twisted K theory. So given a positive energy representation, they explain how to construct a cycle for twisted K theory. So this is a very interesting part of their construction. But going in the opposite direction is maybe a little bit less clear. Given some cycle for twisted K homology, say D cycle, is there some nice description of the corresponding positive energy representation? So I'm going to be talking about this. So I'm going to bring things up into two main results. So first I'm going to describe a map. I'm going to construct a map for you from twisted K homology to the verlandering. And it'll be this particular avatar of the verlandering, so these formal alternating characters. And then the second thing is a more explicit cycle-level description of what this map does to D cycles. And it's going to turn out that this map is given by basically the index of a certain elliptic operator on a certain space. And what I mean by index here, so the operator will be on some non-compact manifold. So the kernel and the co-cernel will be infinite dimensional, but the multiplicities of the T-action, the tourist action on these Hilbert spaces will be finite. So it will make sense to talk about the index of this operator as a formal character of T. Okay, so I need to tell you briefly how to construct these bundles over a group using loop groups. So PG is going to, I should say this particular construction is not due to me, I think it's probably well known. So PG is going to denote the, for me it's going to denote the cosy periodic paths in G. So these are maps from the real line to G such that this product is some constant independent of S, S is the parameter in R. Is it still for any constant or is it fixed for that new PG? For any constant, the constant can be any element of G, but it's independent of S. Yeah, so different gammas will be sent to a different constant. Okay, yeah. Yeah, but and this gives you a map, so this gives you a map from PG to G. So you can take this cosy periodic path gamma and map it to the corresponding constant that appears on the right side of equals. And this makes, it's not too hard to see that this makes PG into a principal LG bundle over G. So you've got a principal bundle over G. And then you can build a Dixie 3D bundle by doing an associated bundle construction. You take a positive energy representation. So LG won't act on V, only some central extension of LG will act. But when you take compound operators, the action will descend to LG and so you can build this associated bundle. And yeah, so it's known that the Dixie 3D invariant of this is L. That's your start with a level L positive energy representation. Okay, right. So the next thing I want to this, it's not quite motivated yet, but I'll motivate it on the next slide. So this, yeah, so I sort of, I think of this original Dixie 3D bundle as being very, very big. The silvered spaces that you start with are very, very large. So what I want to describe is much smaller model for the Dixie 3D bundle over the maximum torus. It's just going to work over the maximum torus, but it's sort of much smaller and more manageable in some way. So I'm going to describe that. This is going to be an important part of the map. So, yeah, so remember the maximum torus and this lattice pi, they both sit inside LG and subgroups. So of course you can pull back the central extension and you get some central extension of this group. And then I'm going to take a certain representation of the central extension. So the way that you build it, I think this is kind of the nicest way to build it, you start with the regular representation of pi hat, the central extension of this lattice. And then you restrict to the sub space where the central circle acts with a certain weight. The weight should be minus L if you want to Dixie 3D bundle with, so if you want the numbers to work out, it should be weight minus L. So this is a certain sub space of this slightly larger Hilbert space. And yeah, so this carries a representation of this central extension. And then you can build a Dixie 3D bundle in the same way, using the associated bundle. So T here is the algebra of the maximum torus. And yeah, so we're building a Dixie 3D bundle in essentially the same way, using the associated bundle. And yeah, so it's not too hard to argue that when you take the Dixie 3D bundle over the whole group and restrict to the torus, these become rate isomorphic when you restrict to the torus. And this is essentially because you use the same central extension to build both of them. And you can write down, yeah, so the rate of morphism will be given by a similar associated bundle. So I like to think of this as sort of a small model for the bundle over the torus. OK, so now briefly tell you what this map is. The first couple of steps are basically restricting to the maximum torus. We restrict to some, so maybe one unfortunate aspect of this construction is we have to restrict to the neighborhood of the maximum torus. It's not ideal, but anyway. So the first step is we restrict to a tubular neighborhood, U, the maximum torus, and then you apply tomics and morphism to go to the maximum torus. Then we use this Morita isomorphism that I just talked about to switch to this smaller model for the DD bundle. And then we're going to use some tools from analytic k-homology. So it turns out, so one very nice feature of this DD bundle is that it has kind of an alternate description as a cross product algebra. You're already confused. Pi was the lattice. Yeah, Pi hat is the central expansion of the lattice that you get by. So it's not commutative anymore? No, it's not quite commutative anymore. Almost commutative. Right, yeah, so yeah, a nice feature of this small model for the DD bundle is when you look at this C star algebra, it has this alternate description as a cross product. And this is mixed available to you some tools from KK through. And in particular, you end up getting an element in certain aqua variant k-homology groups, k-homology just of a vector space, just of the algebra of t. We're now talking about a very simple space with respect to this group, so this central extension group that's playing a big role. And once you're in this group, there's another tool from k-homology called the analytic assembly map. This is kind of an integration over a T map, sort of a generalized index map going from this group into, so it goes into the k theory of the C star algebra of the group. And you can sort of think of this as being roughly like a representation ring for this group. Being something like a representation ring for this group. Yeah, so as I said, we use some, I guess, much loved tools in analytic k-homology as part of the construction. Right, and this is basically the description of the map. So the last thing I should mention is that with a little bit of extra effort, you can keep track of what happens to the val group symmetry when you apply this map. And actually all of the steps are equivalent under the normalizer of t, except for one. So this tom isomorphism used way up here, not quite equivalent, but has an anti-symmetry property under the normalizer. So if you keep track of the val symmetry, you can show that the range of this map is contained in the subgroup of this sort of generalized representation ring. That's isomorphic to formal characters with this anti-symmetry property. So going from here to here, we kind of are forgetting the pi action and just retaining the t action. So that's the description of the map. Right, and then, right, so for d-cycles, it gets somewhat more concrete. So if your cycle is a d-cycle, so it's given by a four-tuple like this, so a manifold, mapping into your space, this marita bimodule. So it gets a lot more concrete. So first of all, what the corresponding analytics cycle looks like. So the Hilbert space looks like L2 sections of a certain Hilbert bundle over your manifold. And the Hilbert bundle has a Clifford module structure. And this operator F is sort of a bounded version of a direct type operator, acting on smooth sections of this bundle. So that's roughly what the cycle looks like. And, okay, so the first nice thing happens when you apply this marita morphism to the small model of the Schmidt-Riedi bundle. What happens, roughly speaking, is you replace these kind of big Hilbert spaces with these much smaller Hilbert spaces, which basically look like L2 of a lattice. And then there's, yeah, so it tends to be something finite dimensional. And then there's this nice correspondence between differential operators that act on the course of Hilbert spaces that look like L2 of a lattice, and operators on covering spaces, on ordinary finite dimensional manifolds. So in this step, when you pass to this smaller model, you end up getting something that looks like an ordinary direct type operator, so on a finite dimensional bundle, but on a covering space. Right, and then the last thing is you apply the analytic assembly map, and this ends up taking the index of the operator. Yeah, in this kind of generalized sense that I talked about. This is not quite immediate because the assembly map is defined in sort of a more abstract way, but if you do a little bit of work, you can show that it boils down to this in this special case. Okay, I'm not sure how much, probably, because I have a little bit of fun with that. So yeah, so with the remaining time, I want to tell you what my initial motivation was for thinking about these things. So initially I was motivated by an application to Hamiltonian loop group spaces. Yeah, so let me tell you very briefly what a Hamiltonian loop group space is. So the dual of the leogic of the loop group, we can think of it as a connection. So the algebra of value one forms on the circle, and we can think of those as connections on the trivial G bundle over the circle. And yeah, so the loop group acts on this space by gauge transformation. That's the action we want to consider. And then a Hamiltonian LG space is, so it's going to be an LG manifold, so smooth, barric manifold with an action of LG with a symplectic form and a proper moment map from M into LG star. And this moment map should be equivariated for the action that we talked about up here. So that's a Hamiltonian LG space. And so in this work with Yan Li Song and Akert, we were thinking about quantizing Hamiltonian LG spaces. How to quantize Hamiltonian LG spaces. And one maybe naive thing that you might try is to take some finite dimensional sub-manifold of the Hamiltonian LG space, try to quantize it with sort of more usual means, you do it with derap operators or something, and then get a positive energy representation by some kind of induction procedure. That's maybe sort of naively how you might go about it. So one space you might try is you might try to take inverse image of the Liegeber of the maximum torrents. So because this subspace is finite dimensional, the subset that you end up with, so if it's smooth, it will be some finite dimensional sub-manifold of M. But the problem, well one problem with this is that it could be singular. But we found a way around this, so we, some of this earlier work, we showed that you can always find a small thickening of this singular subset that's still finite dimensional, but it's smooth, and moreover we showed how to build the canonical spinacy structure. So our thinking was to look at the Dirac operator on this space. You can twist by a prequantum line bubble. And then essentially you look at the index of this operator, except not quite. So because we passed through a thickening of this space, you should, instead of looking at just the index, you should take an index pairing with a k-theory class that plays the role of a pointer-a dual to this x. Yeah, but there's some nice index pairing to study, and you prove that it gives you an, yeah, so when you take this index pairing, it gives you an element of this group. So again the kernel, core kernel, are infinite dimensional, but the multiplicities of the torus are finite, and in fact you get a character with this anti-symmetry property under the affine value. So you could define the quantization of this Hamiltonian loop group space as the unique element of the Rindring corresponding to this formal character. And one attractive feature of this approach is that you can do the wind deformation with it. This was really original motivation for doing this. And there's another approach that's a few years older to mostly to Eckert and also some collaborators. So you could, so this is another approach based more on, so it's based more on twisted k-homology. So you could, so the base loop group, this is sort of a, yeah, so I mentioned this is a subgroup of algae, so it turns out to act freely on n. So it acts freely on algae star and by equivalence it also acts freely on n. And you can take the quotient of m by this subgroup and you get a finite dimensional manifold, finite dimensional compact manifold called the quasi-Hematonian g-space. And this actually gives, there's actually a one to one correspondence between these things. If you put appropriate data on m, there's a one to one correspondence between these things. So you could try to define the quantization of the loop group space in terms of some appropriate quantization of m, whatever that should be. And Eckert and collaborators found a nice way of quantizing quasi-Hematonian spaces. So they noticed that these quasi-Hematonian spaces, they don't always have spincy structures, but they do always give rise to certain canonical d-segals, or certain canonical, yeah, so there's always this canonical rate of bimodular, this d-segal, at level the dual-coxeter number. So they called this a twisted spincy structure. And they're in their earlier papers. And yeah, so in a fairly natural way, if you have a pre-quantum line bundle in your loop group space, this also has sort of analog downstairs on the quasi-Hematonian space. So it takes the form of a marita morphism between a trivial line bundle and, sorry, trivial d-d bundle and the pullback of, yeah, so the corresponding d-d bundle on g. You start with a level k pre-quantum line bundle, you get this sort of marita morphism. And then, so Eckert gave a definition of the quantization of m as, well, basically the class of this d-segal, yeah, so you take the tensor product of this, so this is the analog of the spincy structure, and you take the tensor product with the analog of the pre-quantum line bundle, and this gives you a d-segal in this group. So you could, right, yeah, so you could then define the quantization of the loop group space as the corresponding element in the Verlind ring that Friedhawk can tell him if he gives you. I'm not sure I said that in the most clear way. Feel free to ask me about it. Okay, so now, right, so now I can, yeah, so the initial question that I wanted to understand was how these two pictures are related to each other. Initially, it seemed quite different to me, but I thought there should be some simple relationship between the two. So now I want to explain how that works. So it's almost an immediate consequence of what I've told you so far. So let X be this push forward of this d-segal. Yeah, so this is the quantization of the q-Hamiltonian space, the quasi-Hamiltonian space. And then, so according to the first theorem I told you, when I apply this map I, that's this complicated map that I told you about when you apply this map, you get some element, some formal character of t. And yeah, so this first theorem I told you about says that this element equals the image of X under the free Hopkins' element isomorphism. And then the second theorem I mentioned, so this was the explicit description in the case that you have a d-segal. So the second theorem tells you that I of X is given by the index of some first order elliptic operator on some space. And it turns out that this operator is exactly the operator that you get when you look at this index pair. So it's exactly the same operator. And then the last step, oh yeah, I didn't read it, but the last step is just when you apply the analytic assembly map, this is what connects the index of this operator to I of X, to the image of this under the free Hopkins' element isomorphism. Okay, yeah, and that's sort of the end of the proof that these two approaches give you the same thing. Take off. Thank you. Thank you. That's a good point. No, it's not. No, unfortunately, sorry, sorry. Yes, yeah, yeah, yeah, yeah, yeah, unfortunately I can't give an alternate proof. Yeah, at least not yet. I would love to be able to do this. So the free token cinema isomorphism is about GV covariance. Yeah, yeah, I was sort of implicitly using that, but I didn't say that. So everything in the description should be GAC overranged. So the manifold should have a G action. This vector model should be GAC overranged. The bimodule should have a G action. Yeah, and this map should be GAC overranged. Is it clear that like classes in GV covariance and K theory can always be realized versus the notion of insight? So I don't think it's known for sure yet. So the latest thing I heard was that, so maybe if Thomas Schick is here, we can answer. I don't know, but the latest thing I heard is that, so there are four authors including Thomas Schick have announced the proof that you can do this at least in the non-equivariant case. Yeah, I don't know about the equivalent case. So as far as I know, it's not, for my construction, I don't need it. So for my construction, I really just use it as a nice geometric package for producing cycle. It exists. Certainly for the groups I was talking about, it exists. I'm not sure about it in China. Is there any more questions? Let's give this to you again. Thank you.
 Thank you very much to the organizers for giving me the opportunity to speak at this interesting workshop.  So my talk is mainly going to be based on an article that's not quite available on Archive yet,  but I put a copy on my webpage in case you want to look at it.  And it's very much inspired and motivated by earlier joint work with Akron Meinrank and Yanisang.  Oh great, yeah.  Okay, yeah, so my talk is mainly going to be based on the...  Yeah, so the first thing listed here, but as I said, it's very much motivated by joint work with Yanisang and Akron Meinrank.  Right.  Ah, okay.  Right, so my talk is related to the Freight Hopkins Tellment Theorem, so I'm going to start with a very brief introduction to that theorem.  And part of the Freight Hopkins Tellment Theorem is a map going from representation theory of the loop group,  LG, of some compact lead group, to twisted K theory, or twisted phenomenology of G.  And so one thing I'm going to do in the talk is describe a map going in the opposite direction,  going from twisted geomology to representation theory.  And I'm going to be especially interested in what that map does to what are called d-cycles.  This is the terminology of Baumkering-Wang.  So d-cycles are sort of a nice geometric package that gives you cycles in twisted geomology.  And then time permitting, I'll talk about sort of the original motivation that I had for thinking about this stuff,  which has to do with Hamiltonian-LG space.  Okay, so let me give you kind of a light mean overview of the loop group.  So throughout my talk, G is going to be a compact, simply connected, simple loop group.  And I'm going to fix the maximum torsity inside G.  LG is going to be the loop group, so the maps from the circle into the lead group.  And there are some subgroups of the loop group that are going to come up in my talk.  So base loop groups, so these are loops which begin and end at the density element.  And then sort of complementary to that, there's a copy of G sitting inside the loop group as constant loops.  And another subgroup that will come up is the integral lattice.  So you look at the kernel of the exponential map for this maximal torus.  This is a lattice which we call it pi.  And this naturally sits inside the loop group as the exponential loops formula that we did.  And the loop group has a famous U1 central extension.  And so I put the formula for the Lie algebra cosec of the central extension.  And it involves this, V is the so-called basic inner product on the Lie algebra.  It's an invariant inner product on the Lie algebra with a particularly nice normalization.  And yeah, so one other thing I should mention about this central extension.  It has a symmetry.  So as one acts on the loop group by reprametrization, by rigid rotation of loops.  And this symmetry lifts to the central extension.  I am confused by this high definition.  So that's the natural map from T, I guess it's going to, from the Lie algebra of T to T itself.  Right.  So how's the kernel of that inside omega G?  So I'm identifying it with a subset of omega G.  So if you have some elements in the kernel, you take, I guess this should be XT.  Look at the loops that you get by exponentially.  The one parameter subgroup generated by that element.  So maybe instead of inclusion, I should put like an error, an inclusion error.  Okay, and so let me tell you a little bit about the representations.  So the group has an interesting family of projective representations called positive energy representations.  And these are representations of the central extension with some conditions.  So the representation should extend to an action of this semi-direct product group by this rotation circle.  And the weights of the action of this rotation circle should be bounded below my zero.  That's the positive energy condition.  And then we say that one of these representations is at level K if the central circle.  So the other circle acts with weight K. K is some integer.  So these are positive energy representations.  And they have a nice theory, very much parallel to the representation theory of G, the finite dimensionality group.  So in particular, irreducible positive energy representations are parameterized by their highest weights.  And another interesting thing that happens is that if you fix a level, so if you look at the fixed level K,  it turns out there are only finitely many of these irreducible positive energy representations.  And they're parameterized by the so-called level K weights.  So my notation for this is pi K star.  So pi star is the weight lattice, the dual of the integral lattice.  And pi K star, you take the intersection of pi star with the K-fold inflation of the fundamental alpha.  So I should also mention here and throughout my talk, I'm going to be using the basic inner product to identify G with its dual.  So the alcove sits naturally inside the algebra of the maximum torus, but I'm thinking of it as a subset of the dual space.  Right. And then so the verlimeter ring is sort of the analog of the representation ring for the loop group.  So the level K verlimeter ring.  So as a group, you can think of it as the phyabilian group generated by these level K weights,  generated by these irreducible positive energy representations at level K.  And it also has some other descriptions.  So it has a nice description as a quotient of the ordinary representation ring of G by the so-called verlimeter idea.  And this shows you its ring structure.  And it has another description that's going to be important for my talk.  And yeah, so I apologize, the notation is a bit unearthly.  So it has another description as certain formal characters for the torus, for the maximum torus.  So formal infinite sums of irreducible characters of the torus with a certain anti-symmetry property under the action of the affine valgur.  So to understand this, it's maybe helpful to think by analogy with the case of a compact loop group.  So if you have irreducible representation of a compact loop group and you restrict it to the torus,  you get a formula for it, the val character formula.  And it's given by some numerator which is alternating under a certain action of the val group,  divided by some universal denominator that doesn't depend on the representation.  There's a similar story for the loop group.  They also have characters given by something that looks like the val character formula.  And again, it takes a form of a numerator which is alternating under a certain action of the affine val groups.  This is sort of the loop group, analog of the val group, divided by some universal denominator that doesn't depend on the representation.  So that's this isomorphism here.  Okay, then I have a picture to kind of maybe give you a bit of a feel for what this affine valgur action looks like.  By the way, this K plus H check has to do with, so this is describing what this action is.  The H check that appears there is the dual coaxial number.  And this is quite important in the free-throwing.  So maybe I'll spend a minute to talk about it.  By the way, this image I have to think accurate and David Lee Bland for letting use this image.  So this is a picture for SU3 with K plus 3.  So this level K is this 3.  And the level K weights are the black dots inside this red triangle.  So there's finally many from here.  And so the green hyperplanes, this affine valgur action is generated by reflections and all of these green hyperplanes.  So an element of this group of formal characters, it has a multiplicity function, which is a function from the weight lattice to the integers.  And it should be alternating under reflections in all of these hyperplanes.  So notice that the fundamental domain for this action, this reflection action, it's a bit bigger.  So this is where the dual coaxial number comes in.  So the fundamental domain for this action is a bit larger than the set of level K weights.  So maybe you can see what this group is.  Okay, so that's what I wanted to say about loop groups.  The other side of the free talk in the theorem is twisted k-homology or k-theory.  So for me, the twist of k-homology is going to be what's called the Dixie-Mute-Rede bundle.  So it's a bundle of c-star algebras over your space x that you're interested in,  with fibers equal to the compact operators on some Hilbert space.  So locally trivial bundle, locally it just looks like an open set times compact operators,  but globally there can be some interesting twisting.  And so these bundles are classified up to isomorphism by something called the Dixie-Mute-Rede invariant,  which is a certain k-homology class that has a component in degree three, which is maybe the most important,  and then there can also be a component in degree one for some extra grading information.  And so in the equivalent situation where we have a group acting, these are equivalent k-homology classes.  So the compact operator is on a z-quadered Hilbert space here, right?  Yes, yes.  And...  I'm going to say something else at the moment.  Right, so I should mention, so different bundles will give you different groups.  So we're talking about sort of very sparsible groups than you could get.  So for me, I'm going to use the analytic definition of the twisted k-homology group.  It's known to be equivalent to definitions of topology.  So for me, the twisted k-theory of space x is going to be the k-homology, the analytic k-homology  of the algebra of continuous sections of this bundle,  vanishing at infinity if you're in a non-compact space.  So again, I should emphasize, you get different groups maybe from different Dixme de Rady bundles,  different C-style bundles.  And just to remind you what the cycles in this group look like.  So they look like triples h-row f, where h is some z-quadered Hilbert space.  The row is a representation of your algebra on this space.  And f is, you can think of as sort of an abstract, zeroth order, liptic operator.  And what that really means is that if you look at these three operators, they should be compact.  So this first condition maybe you can think of as sort of locality condition.  And this says that f is close to being self-adjoint, and this is sort of a compact, resolvent type condition.  So that's sort of an abstract definition.  The fundamental example is coming from the Clifford-Algebra bundle.  So if you take any even dimensional Riemannian metaphor, you can look at the Clifford-Algebra bundle.  This is a finite dimensional Dixme-Druity bundle.  This Dixme-Druity invariant is the third integral Stiefel-Wittney class.  So this is the abstraction to the existence of a spin-z structure on M.  And there's a class in this group that you can think of as being like the fundamental class of the manifold M.  And it's basically the Durand-Durac operator acting on differential formulas.  To make it exactly match this definition, you need to do something like apply a functional calculus to get a bounded operator.  It's basically the Durand-Durac operator.  You're assuming that was right.  You're assuming that was right.  Right. Yeah.  I should have put it back.  Right. So don't you need to assume this is spin-z structure in order to get a fundamental class of k-theta?  So this is going to live in twisted k-theta.  I guess that's part of the idea.  So this won't live in the ordinary k-theta of M, but it will live in the a-theta of k-theta of M.  So this is maybe one useful aspect of twisted k-theta.  Okay. So now I'm going to describe a nice geometric package for generating cycles in this group.  So these are called d-cycles or Baum-Carry-Wine-Covins d-cycles.  And these are a twisted analog of Baum-Douglas cycles in geometric technology, which we've seen that before.  So a d-cycle is a four-tuple consisting of a compact-oriented even dimensional Riemannian manifold.  I'm just describing an even case. There's a similar version for the odd case.  And then you have a vector bundle on M, a continuous map from your manifold into your space X.  And the last piece of data is maybe the most interesting.  So you have a marita bimodule linking a Clifford algebra bundle under your Riemannian manifold with the pullback of this twisting bundle,  the pullback of this Dixme-Duriddy bundle.  And so if A is trivial, if A is just the complex numbers, then this S is equivalent to a spin-C structure on M.  In that way you recover the ordinary Baum-Douglas cycles.  But yeah, so more generally, so if A is some general Dixme-Duriddy bundle, the existence of S implies that you have this relation.  So the pullback of the Dixme-Duriddy invariant of A is this third integral C4-Widme class.  And the way you go from a d-cycle to a cycle in this Tristic K-Emolegy group is fairly simple.  So you take the fundamental class, you can compare with this vector bundle, and then the pair consisting of this map phi and the bimodule S  you define a push-forward map from the Tristic K-Emolegy of M to the Tristic K-Emolegy of your space X.  And so you can push forward to get a class, which shouldn't it be the search-geek-widney class of the normal bundle of M and X or something for this push-forward to work that way?  So we should use the normal bundle. Not sure.  I'm not sure. I'm not sure. So now I'll talk a bit about the free Hopkins-Tellerman theorem.  I should say I'm only talking about a very special case of the theorem, a very general theorem for arbitrary compactly groups.  I'm just going to be talking about the case where G is simply connected, simple and connected.  So we take the space X to be the group itself, and so there's an action of G by conjugation.  And so it's known in this case that third aquavirin homology is just isomorphic to the integers, and the lower homology is vanishing,  which means that the Dixmeid-Durradi invariant is just an integer.  So I'm going to use this notation A superscript L to just mean the Dixmeid-Durradi bundle with invariant L.  And so the free Hopkins-Tellerman theorem is an isomorphism between the Rlindering and some positive-level K.  And the twisted K homology, there's a version for K theory too, as stated by the Poincare duality, but here I've written it in terms of K homology.  And the Dixmeid-Durradi invariant is shifted a little bit, so it's shifted by this dual coccidant number.  And yeah, so more than this, so free Hopkins-Tellerman construct a very interesting map going from representations to twisted K theory.  So given a positive energy representation, they explain how to construct a cycle for twisted K theory.  So this is a very interesting part of their construction.  But going in the opposite direction is maybe a little bit less clear.  Given some cycle for twisted K homology, say D cycle, is there some nice description of the corresponding positive energy representation?  So I'm going to be talking about this.  So I'm going to bring things up into two main results.  So first I'm going to describe a map. I'm going to construct a map for you from twisted K homology to the verlandering.  And it'll be this particular avatar of the verlandering, so these formal alternating characters.  And then the second thing is a more explicit cycle-level description of what this map does to D cycles.  And it's going to turn out that this map is given by basically the index of a certain elliptic operator on a certain space.  And what I mean by index here, so the operator will be on some non-compact manifold.  So the kernel and the co-cernel will be infinite dimensional, but the multiplicities of the T-action, the tourist action on these Hilbert spaces will be finite.  So it will make sense to talk about the index of this operator as a formal character of T.  Okay, so I need to tell you briefly how to construct these bundles over a group using loop groups.  So PG is going to, I should say this particular construction is not due to me, I think it's probably well known.  So PG is going to denote the, for me it's going to denote the cosy periodic paths in G.  So these are maps from the real line to G such that this product is some constant independent of S, S is the parameter in R.  Is it still for any constant or is it fixed for that new PG?  For any constant, the constant can be any element of G, but it's independent of S.  Yeah, so different gammas will be sent to a different constant.  Okay, yeah.  Yeah, but and this gives you a map, so this gives you a map from PG to G.  So you can take this cosy periodic path gamma and map it to the corresponding constant that appears on the right side of equals.  And this makes, it's not too hard to see that this makes PG into a principal LG bundle over G.  So you've got a principal bundle over G.  And then you can build a Dixie 3D bundle by doing an associated bundle construction.  You take a positive energy representation.  So LG won't act on V, only some central extension of LG will act.  But when you take compound operators, the action will descend to LG and so you can build this associated bundle.  And yeah, so it's known that the Dixie 3D invariant of this is L.  That's your start with a level L positive energy representation.  Okay, right. So the next thing I want to this, it's not quite motivated yet, but I'll motivate it on the next slide.  So this, yeah, so I sort of, I think of this original Dixie 3D bundle as being very, very big.  The silvered spaces that you start with are very, very large.  So what I want to describe is much smaller model for the Dixie 3D bundle over the maximum torus.  It's just going to work over the maximum torus, but it's sort of much smaller and more manageable in some way.  So I'm going to describe that. This is going to be an important part of the map.  So, yeah, so remember the maximum torus and this lattice pi, they both sit inside LG and subgroups.  So of course you can pull back the central extension and you get some central extension of this group.  And then I'm going to take a certain representation of the central extension.  So the way that you build it, I think this is kind of the nicest way to build it,  you start with the regular representation of pi hat, the central extension of this lattice.  And then you restrict to the sub space where the central circle acts with a certain weight.  The weight should be minus L if you want to Dixie 3D bundle with, so if you want the numbers to work out, it should be weight minus L.  So this is a certain sub space of this slightly larger Hilbert space.  And yeah, so this carries a representation of this central extension.  And then you can build a Dixie 3D bundle in the same way, using the associated bundle.  So T here is the algebra of the maximum torus.  And yeah, so we're building a Dixie 3D bundle in essentially the same way, using the associated bundle.  And yeah, so it's not too hard to argue that when you take the Dixie 3D bundle over the whole group and restrict to the torus,  these become rate isomorphic when you restrict to the torus.  And this is essentially because you use the same central extension to build both of them.  And you can write down, yeah, so the rate of morphism will be given by a similar associated bundle.  So I like to think of this as sort of a small model for the bundle over the torus.  OK, so now briefly tell you what this map is. The first couple of steps are basically restricting to the maximum torus.  We restrict to some, so maybe one unfortunate aspect of this construction is we have to restrict to the neighborhood of the maximum torus.  It's not ideal, but anyway.  So the first step is we restrict to a tubular neighborhood, U, the maximum torus,  and then you apply tomics and morphism to go to the maximum torus.  Then we use this Morita isomorphism that I just talked about to switch to this smaller model for the DD bundle.  And then we're going to use some tools from analytic k-homology.  So it turns out, so one very nice feature of this DD bundle is that it has kind of an alternate description as a cross product algebra.  You're already confused. Pi was the lattice.  Yeah, Pi hat is the central expansion of the lattice that you get by.  So it's not commutative anymore?  No, it's not quite commutative anymore.  Almost commutative.  Right, yeah, so yeah, a nice feature of this small model for the DD bundle is when you look at this C star algebra,  it has this alternate description as a cross product.  And this is mixed available to you some tools from KK through.  And in particular, you end up getting an element in certain aqua variant k-homology groups, k-homology just of a vector space, just of the algebra of t.  We're now talking about a very simple space with respect to this group, so this central extension group that's playing a big role.  And once you're in this group, there's another tool from k-homology called the analytic assembly map.  This is kind of an integration over a T map, sort of a generalized index map going from this group into, so it goes into the k theory of the C star algebra of the group.  And you can sort of think of this as being roughly like a representation ring for this group.  Being something like a representation ring for this group.  Yeah, so as I said, we use some, I guess, much loved tools in analytic k-homology as part of the construction.  Right, and this is basically the description of the map.  So the last thing I should mention is that with a little bit of extra effort, you can keep track of what happens to the val group symmetry when you apply this map.  And actually all of the steps are equivalent under the normalizer of t, except for one.  So this tom isomorphism used way up here, not quite equivalent, but has an anti-symmetry property under the normalizer.  So if you keep track of the val symmetry, you can show that the range of this map is contained in the subgroup of this sort of generalized representation ring.  That's isomorphic to formal characters with this anti-symmetry property.  So going from here to here, we kind of are forgetting the pi action and just retaining the t action.  So that's the description of the map.  Right, and then, right, so for d-cycles, it gets somewhat more concrete.  So if your cycle is a d-cycle, so it's given by a four-tuple like this, so a manifold, mapping into your space, this marita bimodule.  So it gets a lot more concrete.  So first of all, what the corresponding analytics cycle looks like.  So the Hilbert space looks like L2 sections of a certain Hilbert bundle over your manifold.  And the Hilbert bundle has a Clifford module structure.  And this operator F is sort of a bounded version of a direct type operator, acting on smooth sections of this bundle.  So that's roughly what the cycle looks like.  And, okay, so the first nice thing happens when you apply this marita morphism to the small model of the Schmidt-Riedi bundle.  What happens, roughly speaking, is you replace these kind of big Hilbert spaces with these much smaller Hilbert spaces, which basically look like L2 of a lattice.  And then there's, yeah, so it tends to be something finite dimensional.  And then there's this nice correspondence between differential operators that act on the course of Hilbert spaces that look like L2 of a lattice,  and operators on covering spaces, on ordinary finite dimensional manifolds.  So in this step, when you pass to this smaller model, you end up getting something that looks like an ordinary direct type operator,  so on a finite dimensional bundle, but on a covering space.  Right, and then the last thing is you apply the analytic assembly map, and this ends up taking the index of the operator.  Yeah, in this kind of generalized sense that I talked about.  This is not quite immediate because the assembly map is defined in sort of a more abstract way,  but if you do a little bit of work, you can show that it boils down to this in this special case.  Okay, I'm not sure how much, probably, because I have a little bit of fun with that.  So yeah, so with the remaining time, I want to tell you what my initial motivation was for thinking about these things.  So initially I was motivated by an application to Hamiltonian loop group spaces.  Yeah, so let me tell you very briefly what a Hamiltonian loop group space is.  So the dual of the leogic of the loop group, we can think of it as a connection.  So the algebra of value one forms on the circle, and we can think of those as connections on the trivial G bundle over the circle.  And yeah, so the loop group acts on this space by gauge transformation.  That's the action we want to consider.  And then a Hamiltonian LG space is, so it's going to be an LG manifold, so smooth,  barric manifold with an action of LG with a symplectic form and a proper moment map from M into LG star.  And this moment map should be equivariated for the action that we talked about up here.  So that's a Hamiltonian LG space.  And so in this work with Yan Li Song and Akert, we were thinking about quantizing Hamiltonian LG spaces.  How to quantize Hamiltonian LG spaces.  And one maybe naive thing that you might try is to take some finite dimensional sub-manifold of the Hamiltonian LG space,  try to quantize it with sort of more usual means, you do it with derap operators or something,  and then get a positive energy representation by some kind of induction procedure.  That's maybe sort of naively how you might go about it.  So one space you might try is you might try to take inverse image of the Liegeber of the maximum torrents.  So because this subspace is finite dimensional, the subset that you end up with,  so if it's smooth, it will be some finite dimensional sub-manifold of M.  But the problem, well one problem with this is that it could be singular.  But we found a way around this, so we, some of this earlier work,  we showed that you can always find a small thickening of this singular subset that's still finite dimensional,  but it's smooth, and moreover we showed how to build the canonical spinacy structure.  So our thinking was to look at the Dirac operator on this space.  You can twist by a prequantum line bubble.  And then essentially you look at the index of this operator, except not quite.  So because we passed through a thickening of this space, you should,  instead of looking at just the index, you should take an index pairing with a k-theory class  that plays the role of a pointer-a dual to this x.  Yeah, but there's some nice index pairing to study,  and you prove that it gives you an, yeah, so when you take this index pairing,  it gives you an element of this group.  So again the kernel, core kernel, are infinite dimensional,  but the multiplicities of the torus are finite,  and in fact you get a character with this anti-symmetry property under the affine value.  So you could define the quantization of this Hamiltonian loop group space  as the unique element of the Rindring corresponding to this formal character.  And one attractive feature of this approach is that you can do the wind deformation with it.  This was really original motivation for doing this.  And there's another approach that's a few years older to mostly to Eckert  and also some collaborators.  So you could, so this is another approach based more on,  so it's based more on twisted k-homology.  So you could, so the base loop group, this is sort of a, yeah,  so I mentioned this is a subgroup of algae,  so it turns out to act freely on n.  So it acts freely on algae star and by equivalence it also acts freely on n.  And you can take the quotient of m by this subgroup  and you get a finite dimensional manifold,  finite dimensional compact manifold called the quasi-Hematonian g-space.  And this actually gives, there's actually a one to one correspondence between these things.  If you put appropriate data on m, there's a one to one correspondence between these things.  So you could try to define the quantization of the loop group space  in terms of some appropriate quantization of m, whatever that should be.  And Eckert and collaborators found a nice way of quantizing quasi-Hematonian spaces.  So they noticed that these quasi-Hematonian spaces,  they don't always have spincy structures,  but they do always give rise to certain canonical d-segals,  or certain canonical, yeah, so there's always this canonical rate of bimodular, this d-segal,  at level the dual-coxeter number.  So they called this a twisted spincy structure.  And they're in their earlier papers.  And yeah, so in a fairly natural way, if you have a pre-quantum line bundle in your loop group space,  this also has sort of analog downstairs on the quasi-Hematonian space.  So it takes the form of a marita morphism between a trivial line bundle and,  sorry, trivial d-d bundle and the pullback of, yeah,  so the corresponding d-d bundle on g.  You start with a level k pre-quantum line bundle, you get this sort of marita morphism.  And then, so Eckert gave a definition of the quantization of m as,  well, basically the class of this d-segal, yeah, so you take the tensor product of this,  so this is the analog of the spincy structure,  and you take the tensor product with the analog of the pre-quantum line bundle,  and this gives you a d-segal in this group.  So you could, right, yeah, so you could then define the quantization of the loop group space  as the corresponding element in the Verlind ring that Friedhawk can tell him if he gives you.  I'm not sure I said that in the most clear way.  Feel free to ask me about it.  Okay, so now, right, so now I can, yeah, so the initial question that I wanted to understand  was how these two pictures are related to each other.  Initially, it seemed quite different to me,  but I thought there should be some simple relationship between the two.  So now I want to explain how that works.  So it's almost an immediate consequence of what I've told you so far.  So let X be this push forward of this d-segal.  Yeah, so this is the quantization of the q-Hamiltonian space, the quasi-Hamiltonian space.  And then, so according to the first theorem I told you,  when I apply this map I, that's this complicated map that I told you about when you apply this map,  you get some element, some formal character of t.  And yeah, so this first theorem I told you about says that this element equals the image of X  under the free Hopkins' element isomorphism.  And then the second theorem I mentioned, so this was the explicit description in the case that you have a d-segal.  So the second theorem tells you that I of X is given by the index of some first order elliptic operator on some space.  And it turns out that this operator is exactly the operator that you get when you look at this index pair.  So it's exactly the same operator.  And then the last step, oh yeah, I didn't read it, but the last step is just when you apply the analytic assembly map,  this is what connects the index of this operator to I of X, to the image of this under the free Hopkins' element isomorphism.  Okay, yeah, and that's sort of the end of the proof that these two approaches give you the same thing.  Take off.  Thank you.  Thank you.  That's a good point.  No, it's not.  No, unfortunately, sorry, sorry.  Yes, yeah, yeah, yeah, yeah, yeah, unfortunately I can't give an alternate proof.  Yeah, at least not yet.  I would love to be able to do this.  So the free token cinema isomorphism is about GV covariance.  Yeah, yeah, I was sort of implicitly using that, but I didn't say that.  So everything in the description should be GAC overranged.  So the manifold should have a G action.  This vector model should be GAC overranged.  The bimodule should have a G action.  Yeah, and this map should be GAC overranged.  Is it clear that like classes in GV covariance and K theory can always be realized versus the notion of insight?  So I don't think it's known for sure yet.  So the latest thing I heard was that, so maybe if Thomas Schick is here, we can answer.  I don't know, but the latest thing I heard is that, so there are four authors including Thomas Schick have announced the proof that you can do this at least in the non-equivariant case.  Yeah, I don't know about the equivalent case.  So as far as I know, it's not, for my construction, I don't need it.  So for my construction, I really just use it as a nice geometric package for producing cycle.  It exists.  Certainly for the groups I was talking about, it exists.  I'm not sure about it in China.  Is there any more questions?  Let's give this to you again.  Thank you.
201804161403-Loizides_lrv_0_47.png 201804161403-Loizides_lrv_48_95.png 201804161403-Loizides_lrv_96_575.png 201804161403-Loizides_lrv_576_659.png 201804161403-Loizides_lrv_660_707.png 201804161403-Loizides_lrv_708_767.png 201804161403-Loizides_lrv_768_959.png 201804161403-Loizides_lrv_960_1043.png 201804161403-Loizides_lrv_1044_1091.png 201804161403-Loizides_lrv_1092_3035.png 201804161403-Loizides_lrv_3036_3263.png 201804161403-Loizides_lrv_3264_3347.png 201804161403-Loizides_lrv_3348_3959.png 201804161403-Loizides_lrv_3960_6587.png 201804161403-Loizides_lrv_6588_6719.png 201804161403-Loizides_lrv_6720_7451.png 201804161403-Loizides_lrv_7452_8219.png 201804161403-Loizides_lrv_8220_8567.png 201804161403-Loizides_lrv_8568_9407.png 201804161403-Loizides_lrv_9408_11627.png 201804161403-Loizides_lrv_11628_12023.png 201804161403-Loizides_lrv_12024_17591.png 201804161403-Loizides_lrv_17592_22175.png 201804161403-Loizides_lrv_22176_27215.png 201804161403-Loizides_lrv_27216_27491.png 201804161403-Loizides_lrv_27492_27599.png 201804161403-Loizides_lrv_27600_27875.png 201804161403-Loizides_lrv_27876_28367.png 201804161403-Loizides_lrv_28368_33671.png 201804161403-Loizides_lrv_33672_34595.png 201804161403-Loizides_lrv_34596_35687.png 201804161403-Loizides_lrv_35688_39323.png 201804161403-Loizides_lrv_39324_43931.png 201804161403-Loizides_lrv_43932_44651.png 201804161403-Loizides_lrv_44652_49391.png 201804161403-Loizides_lrv_49392_50495.png 201804161403-Loizides_lrv_50496_52979.png 201804161403-Loizides_lrv_52980_54527.png 201804161403-Loizides_lrv_54528_57443.png 201804161403-Loizides_lrv_57444_60479.png 201804161403-Loizides_lrv_60480_60611.png 201804161403-Loizides_lrv_60612_61139.png 201804161403-Loizides_lrv_61140_61751.png 201804161403-Loizides_lrv_61752_61823.png 201804161403-Loizides_lrv_61824_61919.png 201804161403-Loizides_lrv_61920_62483.png 201804161403-Loizides_lrv_62484_63995.png 201804161403-Loizides_lrv_63996_64036.png
Thank you very much to the organizers for giving me the opportunity to speak at this interesting workshop. So my talk is mainly going to be based on an article that's not quite available on Archive yet, but I put a copy on my webpage in case you want to look at it. And it's very much inspired and motivated by earlier joint work with Akron Meinrank and Yanisang. Oh great, yeah. Okay, yeah, so my talk is mainly going to be based on the... Yeah, so the first thing listed here, but as I said, it's very much motivated by joint work with Yanisang and Akron Meinrank. Right. Ah, okay. Right, so my talk is related to the Freight Hopkins Tellment Theorem, so I'm going to start with a very brief introduction to that theorem. And part of the Freight Hopkins Tellment Theorem is a map going from representation theory of the loop group, LG, of some compact lead group, to twisted K theory, or twisted phenomenology of G. And so one thing I'm going to do in the talk is describe a map going in the opposite direction, going from twisted geomology to representation theory. And I'm going to be especially interested in what that map does to what are called d-cycles. This is the terminology of Baumkering-Wang. So d-cycles are sort of a nice geometric package that gives you cycles in twisted geomology. And then time permitting, I'll talk about sort of the original motivation that I had for thinking about this stuff, which has to do with Hamiltonian-LG space. Okay, so let me give you kind of a light mean overview of the loop group. So throughout my talk, G is going to be a compact, simply connected, simple loop group. And I'm going to fix the maximum torsity inside G. LG is going to be the loop group, so the maps from the circle into the lead group. And there are some subgroups of the loop group that are going to come up in my talk. So base loop groups, so these are loops which begin and end at the density element. And then sort of complementary to that, there's a copy of G sitting inside the loop group as constant loops. And another subgroup that will come up is the integral lattice. So you look at the kernel of the exponential map for this maximal torus. This is a lattice which we call it pi. And this naturally sits inside the loop group as the exponential loops formula that we did. And the loop group has a famous U1 central extension. And so I put the formula for the Lie algebra cosec of the central extension. And it involves this, V is the so-called basic inner product on the Lie algebra. It's an invariant inner product on the Lie algebra with a particularly nice normalization. And yeah, so one other thing I should mention about this central extension. It has a symmetry. So as one acts on the loop group by reprametrization, by rigid rotation of loops. And this symmetry lifts to the central extension. I am confused by this high definition. So that's the natural map from T, I guess it's going to, from the Lie algebra of T to T itself. Right. So how's the kernel of that inside omega G? So I'm identifying it with a subset of omega G. So if you have some elements in the kernel, you take, I guess this should be XT. Look at the loops that you get by exponentially. The one parameter subgroup generated by that element. So maybe instead of inclusion, I should put like an error, an inclusion error. Okay, and so let me tell you a little bit about the representations. So the group has an interesting family of projective representations called positive energy representations. And these are representations of the central extension with some conditions. So the representation should extend to an action of this semi-direct product group by this rotation circle. And the weights of the action of this rotation circle should be bounded below my zero. That's the positive energy condition. And then we say that one of these representations is at level K if the central circle. So the other circle acts with weight K. K is some integer. So these are positive energy representations. And they have a nice theory, very much parallel to the representation theory of G, the finite dimensionality group. So in particular, irreducible positive energy representations are parameterized by their highest weights. And another interesting thing that happens is that if you fix a level, so if you look at the fixed level K, it turns out there are only finitely many of these irreducible positive energy representations. And they're parameterized by the so-called level K weights. So my notation for this is pi K star. So pi star is the weight lattice, the dual of the integral lattice. And pi K star, you take the intersection of pi star with the K-fold inflation of the fundamental alpha. So I should also mention here and throughout my talk, I'm going to be using the basic inner product to identify G with its dual. So the alcove sits naturally inside the algebra of the maximum torus, but I'm thinking of it as a subset of the dual space. Right. And then so the verlimeter ring is sort of the analog of the representation ring for the loop group. So the level K verlimeter ring. So as a group, you can think of it as the phyabilian group generated by these level K weights, generated by these irreducible positive energy representations at level K. And it also has some other descriptions. So it has a nice description as a quotient of the ordinary representation ring of G by the so-called verlimeter idea. And this shows you its ring structure. And it has another description that's going to be important for my talk. And yeah, so I apologize, the notation is a bit unearthly. So it has another description as certain formal characters for the torus, for the maximum torus. So formal infinite sums of irreducible characters of the torus with a certain anti-symmetry property under the action of the affine valgur. So to understand this, it's maybe helpful to think by analogy with the case of a compact loop group. So if you have irreducible representation of a compact loop group and you restrict it to the torus, you get a formula for it, the val character formula. And it's given by some numerator which is alternating under a certain action of the val group, divided by some universal denominator that doesn't depend on the representation. There's a similar story for the loop group. They also have characters given by something that looks like the val character formula. And again, it takes a form of a numerator which is alternating under a certain action of the affine val groups. This is sort of the loop group, analog of the val group, divided by some universal denominator that doesn't depend on the representation. So that's this isomorphism here. Okay, then I have a picture to kind of maybe give you a bit of a feel for what this affine valgur action looks like. By the way, this K plus H check has to do with, so this is describing what this action is. The H check that appears there is the dual coaxial number. And this is quite important in the free-throwing. So maybe I'll spend a minute to talk about it. By the way, this image I have to think accurate and David Lee Bland for letting use this image. So this is a picture for SU3 with K plus 3. So this level K is this 3. And the level K weights are the black dots inside this red triangle. So there's finally many from here. And so the green hyperplanes, this affine valgur action is generated by reflections and all of these green hyperplanes. So an element of this group of formal characters, it has a multiplicity function, which is a function from the weight lattice to the integers. And it should be alternating under reflections in all of these hyperplanes. So notice that the fundamental domain for this action, this reflection action, it's a bit bigger. So this is where the dual coaxial number comes in. So the fundamental domain for this action is a bit larger than the set of level K weights. So maybe you can see what this group is. Okay, so that's what I wanted to say about loop groups. The other side of the free talk in the theorem is twisted k-homology or k-theory. So for me, the twist of k-homology is going to be what's called the Dixie-Mute-Rede bundle. So it's a bundle of c-star algebras over your space x that you're interested in, with fibers equal to the compact operators on some Hilbert space. So locally trivial bundle, locally it just looks like an open set times compact operators, but globally there can be some interesting twisting. And so these bundles are classified up to isomorphism by something called the Dixie-Mute-Rede invariant, which is a certain k-homology class that has a component in degree three, which is maybe the most important, and then there can also be a component in degree one for some extra grading information. And so in the equivalent situation where we have a group acting, these are equivalent k-homology classes. So the compact operator is on a z-quadered Hilbert space here, right? Yes, yes. And... I'm going to say something else at the moment. Right, so I should mention, so different bundles will give you different groups. So we're talking about sort of very sparsible groups than you could get. So for me, I'm going to use the analytic definition of the twisted k-homology group. It's known to be equivalent to definitions of topology. So for me, the twisted k-theory of space x is going to be the k-homology, the analytic k-homology of the algebra of continuous sections of this bundle, vanishing at infinity if you're in a non-compact space. So again, I should emphasize, you get different groups maybe from different Dixme de Rady bundles, different C-style bundles. And just to remind you what the cycles in this group look like. So they look like triples h-row f, where h is some z-quadered Hilbert space. The row is a representation of your algebra on this space. And f is, you can think of as sort of an abstract, zeroth order, liptic operator. And what that really means is that if you look at these three operators, they should be compact. So this first condition maybe you can think of as sort of locality condition. And this says that f is close to being self-adjoint, and this is sort of a compact, resolvent type condition. So that's sort of an abstract definition. The fundamental example is coming from the Clifford-Algebra bundle. So if you take any even dimensional Riemannian metaphor, you can look at the Clifford-Algebra bundle. This is a finite dimensional Dixme-Druity bundle. This Dixme-Druity invariant is the third integral Stiefel-Wittney class. So this is the abstraction to the existence of a spin-z structure on M. And there's a class in this group that you can think of as being like the fundamental class of the manifold M. And it's basically the Durand-Durac operator acting on differential formulas. To make it exactly match this definition, you need to do something like apply a functional calculus to get a bounded operator. It's basically the Durand-Durac operator. You're assuming that was right. You're assuming that was right. Right. Yeah. I should have put it back. Right. So don't you need to assume this is spin-z structure in order to get a fundamental class of k-theta? So this is going to live in twisted k-theta. I guess that's part of the idea. So this won't live in the ordinary k-theta of M, but it will live in the a-theta of k-theta of M. So this is maybe one useful aspect of twisted k-theta. Okay. So now I'm going to describe a nice geometric package for generating cycles in this group. So these are called d-cycles or Baum-Carry-Wine-Covins d-cycles. And these are a twisted analog of Baum-Douglas cycles in geometric technology, which we've seen that before. So a d-cycle is a four-tuple consisting of a compact-oriented even dimensional Riemannian manifold. I'm just describing an even case. There's a similar version for the odd case. And then you have a vector bundle on M, a continuous map from your manifold into your space X. And the last piece of data is maybe the most interesting. So you have a marita bimodule linking a Clifford algebra bundle under your Riemannian manifold with the pullback of this twisting bundle, the pullback of this Dixme-Duriddy bundle. And so if A is trivial, if A is just the complex numbers, then this S is equivalent to a spin-C structure on M. In that way you recover the ordinary Baum-Douglas cycles. But yeah, so more generally, so if A is some general Dixme-Duriddy bundle, the existence of S implies that you have this relation. So the pullback of the Dixme-Duriddy invariant of A is this third integral C4-Widme class. And the way you go from a d-cycle to a cycle in this Tristic K-Emolegy group is fairly simple. So you take the fundamental class, you can compare with this vector bundle, and then the pair consisting of this map phi and the bimodule S you define a push-forward map from the Tristic K-Emolegy of M to the Tristic K-Emolegy of your space X. And so you can push forward to get a class, which shouldn't it be the search-geek-widney class of the normal bundle of M and X or something for this push-forward to work that way? So we should use the normal bundle. Not sure. I'm not sure. I'm not sure. So now I'll talk a bit about the free Hopkins-Tellerman theorem. I should say I'm only talking about a very special case of the theorem, a very general theorem for arbitrary compactly groups. I'm just going to be talking about the case where G is simply connected, simple and connected. So we take the space X to be the group itself, and so there's an action of G by conjugation. And so it's known in this case that third aquavirin homology is just isomorphic to the integers, and the lower homology is vanishing, which means that the Dixmeid-Durradi invariant is just an integer. So I'm going to use this notation A superscript L to just mean the Dixmeid-Durradi bundle with invariant L. And so the free Hopkins-Tellerman theorem is an isomorphism between the Rlindering and some positive-level K. And the twisted K homology, there's a version for K theory too, as stated by the Poincare duality, but here I've written it in terms of K homology. And the Dixmeid-Durradi invariant is shifted a little bit, so it's shifted by this dual coccidant number. And yeah, so more than this, so free Hopkins-Tellerman construct a very interesting map going from representations to twisted K theory. So given a positive energy representation, they explain how to construct a cycle for twisted K theory. So this is a very interesting part of their construction. But going in the opposite direction is maybe a little bit less clear. Given some cycle for twisted K homology, say D cycle, is there some nice description of the corresponding positive energy representation? So I'm going to be talking about this. So I'm going to bring things up into two main results. So first I'm going to describe a map. I'm going to construct a map for you from twisted K homology to the verlandering. And it'll be this particular avatar of the verlandering, so these formal alternating characters. And then the second thing is a more explicit cycle-level description of what this map does to D cycles. And it's going to turn out that this map is given by basically the index of a certain elliptic operator on a certain space. And what I mean by index here, so the operator will be on some non-compact manifold. So the kernel and the co-cernel will be infinite dimensional, but the multiplicities of the T-action, the tourist action on these Hilbert spaces will be finite. So it will make sense to talk about the index of this operator as a formal character of T. Okay, so I need to tell you briefly how to construct these bundles over a group using loop groups. So PG is going to, I should say this particular construction is not due to me, I think it's probably well known. So PG is going to denote the, for me it's going to denote the cosy periodic paths in G. So these are maps from the real line to G such that this product is some constant independent of S, S is the parameter in R. Is it still for any constant or is it fixed for that new PG? For any constant, the constant can be any element of G, but it's independent of S. Yeah, so different gammas will be sent to a different constant. Okay, yeah. Yeah, but and this gives you a map, so this gives you a map from PG to G. So you can take this cosy periodic path gamma and map it to the corresponding constant that appears on the right side of equals. And this makes, it's not too hard to see that this makes PG into a principal LG bundle over G. So you've got a principal bundle over G. And then you can build a Dixie 3D bundle by doing an associated bundle construction. You take a positive energy representation. So LG won't act on V, only some central extension of LG will act. But when you take compound operators, the action will descend to LG and so you can build this associated bundle. And yeah, so it's known that the Dixie 3D invariant of this is L. That's your start with a level L positive energy representation. Okay, right. So the next thing I want to this, it's not quite motivated yet, but I'll motivate it on the next slide. So this, yeah, so I sort of, I think of this original Dixie 3D bundle as being very, very big. The silvered spaces that you start with are very, very large. So what I want to describe is much smaller model for the Dixie 3D bundle over the maximum torus. It's just going to work over the maximum torus, but it's sort of much smaller and more manageable in some way. So I'm going to describe that. This is going to be an important part of the map. So, yeah, so remember the maximum torus and this lattice pi, they both sit inside LG and subgroups. So of course you can pull back the central extension and you get some central extension of this group. And then I'm going to take a certain representation of the central extension. So the way that you build it, I think this is kind of the nicest way to build it, you start with the regular representation of pi hat, the central extension of this lattice. And then you restrict to the sub space where the central circle acts with a certain weight. The weight should be minus L if you want to Dixie 3D bundle with, so if you want the numbers to work out, it should be weight minus L. So this is a certain sub space of this slightly larger Hilbert space. And yeah, so this carries a representation of this central extension. And then you can build a Dixie 3D bundle in the same way, using the associated bundle. So T here is the algebra of the maximum torus. And yeah, so we're building a Dixie 3D bundle in essentially the same way, using the associated bundle. And yeah, so it's not too hard to argue that when you take the Dixie 3D bundle over the whole group and restrict to the torus, these become rate isomorphic when you restrict to the torus. And this is essentially because you use the same central extension to build both of them. And you can write down, yeah, so the rate of morphism will be given by a similar associated bundle. So I like to think of this as sort of a small model for the bundle over the torus. OK, so now briefly tell you what this map is. The first couple of steps are basically restricting to the maximum torus. We restrict to some, so maybe one unfortunate aspect of this construction is we have to restrict to the neighborhood of the maximum torus. It's not ideal, but anyway. So the first step is we restrict to a tubular neighborhood, U, the maximum torus, and then you apply tomics and morphism to go to the maximum torus. Then we use this Morita isomorphism that I just talked about to switch to this smaller model for the DD bundle. And then we're going to use some tools from analytic k-homology. So it turns out, so one very nice feature of this DD bundle is that it has kind of an alternate description as a cross product algebra. You're already confused. Pi was the lattice. Yeah, Pi hat is the central expansion of the lattice that you get by. So it's not commutative anymore? No, it's not quite commutative anymore. Almost commutative. Right, yeah, so yeah, a nice feature of this small model for the DD bundle is when you look at this C star algebra, it has this alternate description as a cross product. And this is mixed available to you some tools from KK through. And in particular, you end up getting an element in certain aqua variant k-homology groups, k-homology just of a vector space, just of the algebra of t. We're now talking about a very simple space with respect to this group, so this central extension group that's playing a big role. And once you're in this group, there's another tool from k-homology called the analytic assembly map. This is kind of an integration over a T map, sort of a generalized index map going from this group into, so it goes into the k theory of the C star algebra of the group. And you can sort of think of this as being roughly like a representation ring for this group. Being something like a representation ring for this group. Yeah, so as I said, we use some, I guess, much loved tools in analytic k-homology as part of the construction. Right, and this is basically the description of the map. So the last thing I should mention is that with a little bit of extra effort, you can keep track of what happens to the val group symmetry when you apply this map. And actually all of the steps are equivalent under the normalizer of t, except for one. So this tom isomorphism used way up here, not quite equivalent, but has an anti-symmetry property under the normalizer. So if you keep track of the val symmetry, you can show that the range of this map is contained in the subgroup of this sort of generalized representation ring. That's isomorphic to formal characters with this anti-symmetry property. So going from here to here, we kind of are forgetting the pi action and just retaining the t action. So that's the description of the map. Right, and then, right, so for d-cycles, it gets somewhat more concrete. So if your cycle is a d-cycle, so it's given by a four-tuple like this, so a manifold, mapping into your space, this marita bimodule. So it gets a lot more concrete. So first of all, what the corresponding analytics cycle looks like. So the Hilbert space looks like L2 sections of a certain Hilbert bundle over your manifold. And the Hilbert bundle has a Clifford module structure. And this operator F is sort of a bounded version of a direct type operator, acting on smooth sections of this bundle. So that's roughly what the cycle looks like. And, okay, so the first nice thing happens when you apply this marita morphism to the small model of the Schmidt-Riedi bundle. What happens, roughly speaking, is you replace these kind of big Hilbert spaces with these much smaller Hilbert spaces, which basically look like L2 of a lattice. And then there's, yeah, so it tends to be something finite dimensional. And then there's this nice correspondence between differential operators that act on the course of Hilbert spaces that look like L2 of a lattice, and operators on covering spaces, on ordinary finite dimensional manifolds. So in this step, when you pass to this smaller model, you end up getting something that looks like an ordinary direct type operator, so on a finite dimensional bundle, but on a covering space. Right, and then the last thing is you apply the analytic assembly map, and this ends up taking the index of the operator. Yeah, in this kind of generalized sense that I talked about. This is not quite immediate because the assembly map is defined in sort of a more abstract way, but if you do a little bit of work, you can show that it boils down to this in this special case. Okay, I'm not sure how much, probably, because I have a little bit of fun with that. So yeah, so with the remaining time, I want to tell you what my initial motivation was for thinking about these things. So initially I was motivated by an application to Hamiltonian loop group spaces. Yeah, so let me tell you very briefly what a Hamiltonian loop group space is. So the dual of the leogic of the loop group, we can think of it as a connection. So the algebra of value one forms on the circle, and we can think of those as connections on the trivial G bundle over the circle. And yeah, so the loop group acts on this space by gauge transformation. That's the action we want to consider. And then a Hamiltonian LG space is, so it's going to be an LG manifold, so smooth, barric manifold with an action of LG with a symplectic form and a proper moment map from M into LG star. And this moment map should be equivariated for the action that we talked about up here. So that's a Hamiltonian LG space. And so in this work with Yan Li Song and Akert, we were thinking about quantizing Hamiltonian LG spaces. How to quantize Hamiltonian LG spaces. And one maybe naive thing that you might try is to take some finite dimensional sub-manifold of the Hamiltonian LG space, try to quantize it with sort of more usual means, you do it with derap operators or something, and then get a positive energy representation by some kind of induction procedure. That's maybe sort of naively how you might go about it. So one space you might try is you might try to take inverse image of the Liegeber of the maximum torrents. So because this subspace is finite dimensional, the subset that you end up with, so if it's smooth, it will be some finite dimensional sub-manifold of M. But the problem, well one problem with this is that it could be singular. But we found a way around this, so we, some of this earlier work, we showed that you can always find a small thickening of this singular subset that's still finite dimensional, but it's smooth, and moreover we showed how to build the canonical spinacy structure. So our thinking was to look at the Dirac operator on this space. You can twist by a prequantum line bubble. And then essentially you look at the index of this operator, except not quite. So because we passed through a thickening of this space, you should, instead of looking at just the index, you should take an index pairing with a k-theory class that plays the role of a pointer-a dual to this x. Yeah, but there's some nice index pairing to study, and you prove that it gives you an, yeah, so when you take this index pairing, it gives you an element of this group. So again the kernel, core kernel, are infinite dimensional, but the multiplicities of the torus are finite, and in fact you get a character with this anti-symmetry property under the affine value. So you could define the quantization of this Hamiltonian loop group space as the unique element of the Rindring corresponding to this formal character. And one attractive feature of this approach is that you can do the wind deformation with it. This was really original motivation for doing this. And there's another approach that's a few years older to mostly to Eckert and also some collaborators. So you could, so this is another approach based more on, so it's based more on twisted k-homology. So you could, so the base loop group, this is sort of a, yeah, so I mentioned this is a subgroup of algae, so it turns out to act freely on n. So it acts freely on algae star and by equivalence it also acts freely on n. And you can take the quotient of m by this subgroup and you get a finite dimensional manifold, finite dimensional compact manifold called the quasi-Hematonian g-space. And this actually gives, there's actually a one to one correspondence between these things. If you put appropriate data on m, there's a one to one correspondence between these things. So you could try to define the quantization of the loop group space in terms of some appropriate quantization of m, whatever that should be. And Eckert and collaborators found a nice way of quantizing quasi-Hematonian spaces. So they noticed that these quasi-Hematonian spaces, they don't always have spincy structures, but they do always give rise to certain canonical d-segals, or certain canonical, yeah, so there's always this canonical rate of bimodular, this d-segal, at level the dual-coxeter number. So they called this a twisted spincy structure. And they're in their earlier papers. And yeah, so in a fairly natural way, if you have a pre-quantum line bundle in your loop group space, this also has sort of analog downstairs on the quasi-Hematonian space. So it takes the form of a marita morphism between a trivial line bundle and, sorry, trivial d-d bundle and the pullback of, yeah, so the corresponding d-d bundle on g. You start with a level k pre-quantum line bundle, you get this sort of marita morphism. And then, so Eckert gave a definition of the quantization of m as, well, basically the class of this d-segal, yeah, so you take the tensor product of this, so this is the analog of the spincy structure, and you take the tensor product with the analog of the pre-quantum line bundle, and this gives you a d-segal in this group. So you could, right, yeah, so you could then define the quantization of the loop group space as the corresponding element in the Verlind ring that Friedhawk can tell him if he gives you. I'm not sure I said that in the most clear way. Feel free to ask me about it. Okay, so now, right, so now I can, yeah, so the initial question that I wanted to understand was how these two pictures are related to each other. Initially, it seemed quite different to me, but I thought there should be some simple relationship between the two. So now I want to explain how that works. So it's almost an immediate consequence of what I've told you so far. So let X be this push forward of this d-segal. Yeah, so this is the quantization of the q-Hamiltonian space, the quasi-Hamiltonian space. And then, so according to the first theorem I told you, when I apply this map I, that's this complicated map that I told you about when you apply this map, you get some element, some formal character of t. And yeah, so this first theorem I told you about says that this element equals the image of X under the free Hopkins' element isomorphism. And then the second theorem I mentioned, so this was the explicit description in the case that you have a d-segal. So the second theorem tells you that I of X is given by the index of some first order elliptic operator on some space. And it turns out that this operator is exactly the operator that you get when you look at this index pair. So it's exactly the same operator. And then the last step, oh yeah, I didn't read it, but the last step is just when you apply the analytic assembly map, this is what connects the index of this operator to I of X, to the image of this under the free Hopkins' element isomorphism. Okay, yeah, and that's sort of the end of the proof that these two approaches give you the same thing. Take off. Thank you. Thank you. That's a good point. No, it's not. No, unfortunately, sorry, sorry. Yes, yeah, yeah, yeah, yeah, yeah, unfortunately I can't give an alternate proof. Yeah, at least not yet. I would love to be able to do this. So the free token cinema isomorphism is about GV covariance. Yeah, yeah, I was sort of implicitly using that, but I didn't say that. So everything in the description should be GAC overranged. So the manifold should have a G action. This vector model should be GAC overranged. The bimodule should have a G action. Yeah, and this map should be GAC overranged. Is it clear that like classes in GV covariance and K theory can always be realized versus the notion of insight? So I don't think it's known for sure yet. So the latest thing I heard was that, so maybe if Thomas Schick is here, we can answer. I don't know, but the latest thing I heard is that, so there are four authors including Thomas Schick have announced the proof that you can do this at least in the non-equivariant case. Yeah, I don't know about the equivalent case. So as far as I know, it's not, for my construction, I don't need it. So for my construction, I really just use it as a nice geometric package for producing cycle. It exists. Certainly for the groups I was talking about, it exists. I'm not sure about it in China. Is there any more questions? Let's give this to you again. Thank you.
joint work with Yanisang and Akron Meinrank. going to do in the talk is describe a map going the natural map from T, I guess it's going to, representation ring for the loop group. So the irreducible representation of a compact loop case where G is simply connected, simple and Hamiltonian loop group space is. So the dual of more on, so it's based more on twisted these two pictures are related to each other.