MIT Infinite Dimensional Algebra Seminar (Fall 2013)

This is a joint work with M. Finkelberg. Quasimaps' space Z_d (also known as Drinfeld's zastava space) is a remarkable compactification of the space of based degree d maps from the projective line to the flag variety of type A. The space Z_d has a natural Poisson structure, which goes back to Atiyah and Hitchin. We describe the Quasimaps' space as some quiver variety, and define the Atiyah-Hitchin Poisson structure in quiver terms. This gives a natural way to quantize this Poisson structure. The quantization of the coordinate ring of the Quasimaps' space turns to be some natural subquotient of the Yangian of type A. I will also discuss some generalization of this result to the BCD types.

Motivated, on the one hand, by R. Bezrukavnikov's conjectures in the representation theory of the rational Cherednik algebra in characteristic p, and, on the other hand, by the geometric Langlands duality for the trivial local system, we prove the following theorem. The result of the k-th step of the Drinfeld-Laumon construction applied to a local system corresponding to the irreducible representation of the symmetric group S_n indexed by a partition \mu=(n_1,n_2,...,n_m) (such that the n_i satisfy a set of inequalities) can be described explicitly as follows. It is a perverse sheaf on the stack SCoh_{k+1} (of coherent sheaves of rank k+1 with a section) which is supported on the closure of the preimage of the Harder-Narasimhan stratum in Coh_{k+1} with subquotients of rank one and of degrees which are shifts of the n_i; the monodromy of the corresponding local system is the irreducible representation of the symmetric group which is indexed by the partition obtained by deleting the k largest parts of \mu.

It is a well-known fact that the characters of integrable highest weight modules over affine Lie superalgebras can be expressed in terms of classical theta functions, consequently they are modular invariant. In the Lie superalgebra case the characters can be expressed in terms of Appell's elliptic functions of the third kind, which are not modular invariant. However it has been shown more recently by Zwegers that Appell's functions can be modified by adding a mysterious real analytic correction to restore modular invariance. In a recent paper by Wakimoto and myself we show how to construct modular invariant families of characters of affine Lie superalgebras, using Zwegers' corrections. Applying the quantum Hamiltonian reductions we find interesting modules over the N=2 superconformal algebra (important in mirror symmetry) with central charge c=3(1-(2m+2)/M) (the case m=0 is the well-known minimal series, but m>0 is new).

All relevant material on Lie superalgebras and theta functions will be explained.

The proper way to express symmetries in differential geometry is Lie groupoids rather than Lie groups. What replaces the Lie algebra of a group is the Lie algebroid of a groupoid. This construction lies at the foundations of calculus of variations. We shall present the classical theorems of Emmy Noether in this perspective , inspired by problems in mathematical physics (Einstein vs. Hilbert).

The degeneracy loci of a Poisson structure are the subspaces on which the dimensions of its symplectic leaves drop. I will describe joint work with Marco Gualtieri that is concerned with the local and global properties of these degeneracy loci, and gives new evidence for a conjecture of Bondal regarding Poisson structures on Fano varieties. I will then explain how knowledge of Poisson structures on projective three-space can be combined with deformation quantization to make progress on an open problem in noncommutative algebra: the classification of four-dimensional Artin--Schelter regular algebras.