 Hi everybody. So first I would like to thank the organizers for allowing me to talk. So yeah, let's get right to it. So today I'm going to talk about a fairly old theorem that was proven in the 60s. So basically a TI and bot proved a theorem that allows you to compute equivariant indices of a manifold in terms of fixed points of a certain compactly group action. And my aim is to extend this to the infinite dimensional manifold of base loops of SU2. So what is the base loop group? Let G be a connected compactly group. So the base loops we generated by omega G are all smooth loops based on the identity. So it's going to start at the identity element of G and at the same point. So this is an infinite dimensional group. You can multiply point-wise. And there are two natural groups acting on this. So first we have the maximal torus inside G and this acts by point-wise conjugation. And we also have a circle group acting by rotation. However, we have to maintain the base point. So instead we have sort of a base rotation. So once you rotate you have to multiply it back to make sure the base is still at the identity. And these two actions will commute with each other. So you can see the paper of Tia Presley for more details. I'll give the references at the end. So in fact, what's interesting about omega G or the base loop groups is that it is in fact a symplastic manifold. And the T cross S1 action that we described earlier is actually Hamiltonian with respect to the symplastic structure. And the corresponding moment map is given by two components going into the algebra of T cross S1. So on the S1 component we have the energy function. So like some of you might be familiar with this, this basically measures the energy of a loop. So it's basically omega prime T squared. But because it's base you have to pull back to the origin. And the other one involves projecting the loops down to the the algebra of the maximal torus. So this component is called the momentum map. The moment map has an energy component and a momentum component. And in a Tia and Presley's paper, they cite this moment map and they prove a general generalization of the Gullman-Sternberg theorem, which says that moment maps of finite dimensional symplastic manifolds are convex polytopes. Because this is infinite dimensional, you basically get a paraboloid. You get a convex body that's not a finite polytop, but it just keeps going on to infinity. And I would like to take a brief detour here to talk about finite viral groups. So if you're familiar with finite dimensional groups, you may recall that the viral group associated to type A minus 1 is isomorphic to the symmetric group of N elements, Sn. And there is an affine version of this for infinite dimensional regroups. So we're going to denote the affine valve group by W bar. And it is isomorphic to the semi-direct product, W semi-direct product Q, where Q denotes the current lattice of S, U, N. Okay, so if you're in case you're not familiar with this, that's okay. What's important to remember is that there are certain generators for this. So for the finite valve group, you have N minus 1 generators, S1, S2 all the way to S, M minus 1. The only difference is that for the affine version, you have just one extra generator called S0. And as a small example, the simplest example is SU2. So that has a valve group just equal to S2, which is the group with two elements. It has code lattice isomorphic to Z. Oh, sorry, should be S1. So that's just the trigger group. So the affine valve group for the affine S02 is the semi-direct product, which is isomorphic to Z as a group. It has two generators, S1 and S0. And there's no relation between them. So except S0 squared equals 1 and S1 squared equals 1. So these two, you can think of them as reflections with no relations between them. So now getting to the meat of this whole thing, this was the theorem proven by Atiyah and Bart. So to phrase it in a simple way, assume that M is a manifold with the Hamiltonian G action and that the fixed point set M superscript G is finite. Then suppose you have any G-equivariant vector bundle V on M. You have the following equality. So on the left-hand side, you have the order characteristic. So this is the double complex, double cohomology associated to this vector bundle. So each cohomology group serves as a representation of the group G. Therefore, you can take the trace. And when you take alternating traces, you get a certain virtual representation because you have these minor signs. So that's what the left-hand side means. The right-hand side, he hears the beauty of it. Like a priori, this thing on the left-hand side is very hard to compute. But Atiyah and Bart showed that they were able to prove this in terms of data just around the fixed point. So just to explain this, because P is a fixed point and V is a G-equivariant vector bundle, that induces an action of G on B restricted to P. So you can take its trace. And also, there's an induced representation of G on the cotangent bundle of M at P. And this representation has certain eigenvalues. So what you do is you take the identity map minus this induced representation and you find its eigenvalues. You multiply them together. And Atiyah and Bart basically is saying that it's enough to compute this fairly simple thing on the right-hand side. And you have inequality. So that works for finite-dimensional manifolds. So our goal is to extend this to the base loops, which is infinite-dimensional. And to start with, we will write this in a slightly different way. So instead of having the denominator be the product of some eigenvalues, we're going to write R sub w. So this stands for a rational function, which depends on w. And I write R w because if you remember the T cross S1 action, it was proven that the fixed points of this action are indexed by the fellow group. So if we think back to the paraboloid I mentioned earlier, the fixed points are basically the vertices of that paraboloid. So basically, for every fixed point, which is every element in the fellow group, we get a certain rational function. So in the finite-dimensional case, this rational function would just be given by the product of all eigenvalues of identity minus this induced representation. So yeah, as you can see here, this denominator. And in this case, the character of the tangent cone around that point is exactly equal to this product here. And this all works for smooth manifolds. But as we'll see, our situation isn't so simple. So the first difficulty is that this is infinite-dimensional, so the theorem doesn't directly apply. However, we can take a filtration. So we have x0 and x1, but so on and so on, where their union is equal to omega g. And each one of these is the finite-dimensional space. And these spaces are called Schubert varieties. So this might be familiar to you if you know something about Schubert varieties. This is basically an infinite-dimensional vision. So which brings us to the second difficulty. The second difficulty is that Schubert varieties are generally singular. And in the case of omega su2, only the simplest one, the two-dimensional one, is actually smooth. Everything else is singular. So again, this Atiyah Baugh theorem doesn't directly apply. But we are saved by the fact that each Schubert variety has a well-known desingularization. And topologically, they can be thought of as twisted products of spheres. And so they're built on top of each other by twisting a certain projectivized line bundle. So the four-dimensional versions of this are basically Herzberg services. The two-dimensional version is just a sphere. And then you keep adding on, like, more twisted line bundles. And these are known as thought Samuelson manifolds. So because these are smooth, this is where we will apply the Atiyah Baugh formula. Okay, so another quick detour. I'm sorry about this typo. I'm going to talk about pre-quantum line bundles. So a pre-quantum line bundle over a simplistic manifold is a permission line bundle with connection such that the curvature of the connection is cohomologous to omega. So usually, like, there's a normalization here. I think it's over times i over 2 pi. But for us, we're going to ignore that because it doesn't matter much for the purposes of our calculation. And suppose we have a movement map psi, then we can define this quantum operator given by the fundamental measure field and this movement map. So the point is that if M is scalar, then we can use the scalar polarization to get a representation of G on the space of holomorphic sections. And that's what we'll do for omega G. So for omega G, we can do this with the positive energy representations of omega G with highest weight lambda. So these are irreducible representations which correspond to certain line bundles. We call these L of lambda. So we have this proposition by Shrawan Kumar who says that line bundles on omega G are parameterized by integers. So therefore, the prevaric group is isomorphic to the integers. So they're basically multiples of the zero with fundamental weight of this alpha and the algebra. So once we're given these line bundles, and we have this filtration by Schubert varieties, we can of course restrict these bundles to each Schubert variety XW. And since we have this desingularization, maybe I can draw some. So we have a desingularization from the Boss-Danlison manifolds to the Schubert varieties. We can pull back any line bundle on the Schubert varieties to get another line bundle, a prevaric line bundle. And so in that way, we can consider the holomorphic sections of this line bundle here and use the Atiyah-Bach theorem on this line bundle. And so our results will mostly focus on the case where G equals SU2. And for this group, omega of SU2, you can consider the reduced alpha and alpha group. And this is isomorphic to Z. And we have an ascending chain. So this is given by, I remember, the two generators S0 and S1, S0, S1, S0, S0, S10. And you just keep adding alternating generators to the front. So we have this infinite ascending chain. And this chain corresponds to a filtration of Schubert varieties. So you will have X sub S0, X sub S1, S0, and so on. And each one of them will have a complex dimension, the length of this word. We'll denote the length n elements of this chain by Wn. And in this way, remember what I said earlier about fixed points, in the case of omega SU2, they're indexed by the integers. Oh, sorry. So this is the Schubert variety filtration. And for any reduced word, by reduced, I mean there are no sub words that are equal to this. And we can define this sort of weird looking quantity as W given by e to the, I guess, prefixes of this word applied to the simple root a ij. So don't worry if it looks complicated. This is just sort of bookkeeping notation to write a nice theorem at the end. And we're going to denote SWn by Sn for simplicity. And I remember the rational functions from earlier. So these are these rational functions. For each Wn, there is a rational function. And we're going to call that Rn for simplicity. And for each Wn, there's a Schubert variety where we'll call that Xn for simplicity. Okay, so here we have our theorem. So the rational functions in the localization formula for omega g are given by, so in the denominator, we have this product over all positive roots of the Fnv algebra SL2. And on top, we have an alternating, sorry, we have a signed quality involving this Sn. And we have this extra term with e to the w applied to alpha 1. Oh, sorry. This should really be Wm. So the word with length n applied to this simple root here. And so how do we prove this? So remember, the Schubert varieties themselves are singular. So we must prove the formula for Bach-Samuelson varieties or manifolds using the Atiyah-Bach formula. And it's important to note that it suffices to do this on the Bach-Samuelson manifolds because the space of holomorphic sections of this, the pullback of this line bundle is isomorphic to restricting the line bundle to Schubert varieties. So even though the Schubert varieties are singular, we can still carry out the computation on their desingularizations. And we get the exact same thing as representations of T cross s1. And to prove, so this is what we get for each individual Schubert variety. So again, this is derived by doing it on the Bach-Samuelson manifolds first. Because strictly speaking, the Atiyah-Bach theorem does not apply to Schubert varieties. And we get this, I guess, more messy looking formula. And what's important to note here is that on the top, in addition to this quantity Sm and this extra term here, we have a sum over what we call restricted partitions. So we define this restricted partition P sub A B C to be the number of integer partitions of C into at most A parts, all of which are at most B. So this is a very restrictive version instead of the normal integer partitions. But yes, this formula will involve these restricted partitions. So the idea is that if you have one of the line bundles I mentioned earlier, and you want to compute the order characteristic, so you would use these as the rational functions. And in the simple case of Schubert manifolds, the denominator would just be a product of eigenvalues. And the numerator would just be one. But in this case, because the Schubert varieties are singular, you get something a bit more complicated. And one remark I will make is that so the order characteristics of these Schubert varieties are isomorphic to what are called D my zero modules. So these are final dimensional sub modules of the of the entire irreducible representation. So in the past, in the literature, there was a formula to compute the characters of these D my zero modules. But they were none of them were explicit or effective. So basically, you had to define these divided difference operators. So those are sort of like a discrete version of differential operators. And you apply them iteratively to the highest weight. And that does give you the the character of the module. But in our approach, we have these rational functions, you can multiply by explicitly. So in this way, we can get a very effective formula for the same thing. Instead of having to iteratively apply these divide difference operators. And just as in the finite dimensional case, you might recall that there's a cost on multiplicity formula for calculating multiplicities of any given representation. So we have something similar here. First, let's define capital sub beta mu to be the number of solutions to the equation. And sub alpha times alpha equals mu, where alpha is any positive that not equal to beta. And all these coefficients have to be non negative. So basically, this is a, I guess, fancy version of an integer partition, except we're excluding certain certain integers from being used. And of course, this is not just one dimensional, this is more like in in ZN, how many vectors, how many ways are there of adding vectors to get a certain value. And by reading off the rational functions derived in the previous sections. So if we recall, we have these rational functions. We can treat these as geometric series. And just read off any multiplicities in front of each weight. Once we do that, we obtain the following cost on multiplicity formula for omega s u two. So our proposition is that for K positive and lambda equals K times the zero with fundamental weight, then the multiplicity of the weight alpha in the irreducible representation, our lambda is given by this following sum here. So this is nothing fancy. We just did this by reading off this one right here. And a priori, this is an infinite sum. But in fact, for each alpha, this will always be finite because if you look inside this bracket here, so only finally, many of these are actually positive weights. So therefore, only finally, many of these will actually have non zero partitions. And and so no matter what happens, the right hand side will always be a finite sum. That's why this multiplicity actually makes sense. So I should say that so in principle, the same methods can be used to compute for the case of g equals s u n, not just s u two, in which case, so let me go back here. So we expect that the formula will be almost exactly the same, except that in this bracket, we would have more terms at the end. So one for each alpha and so on. So this is our conjecture for g equals s u n.