**Title:** Surface braid groups, quantum groups, and quantum D-modules

**Speaker:** David Jordan

**Speaker Info:**

**Brief Description:**

**Special Note**:

**Abstract:**

Given a surface S, it's nth braid group B_n(S) is pi_1(Conf_n(S)), the fundamental group of the space of size-n subsets of S; for S=R^2, this gives the usual braid group. A celebrated application of quantum groups is the construction of representations of the usual braid group on nth tensor powers of of a module V.In this talk I will ask, and answer, the following natural question: what extra data is needed from the theory of quantum groups, in order to extend the usual braid group action to an action of the surface braid groups B_n(S), for higher genus surfaces S?

The answer for a torus, S=T^2 turns out to be the structure of a quantum D-module on G, i.e. a module over the algebra D_q(G) of quantum differential operators on G, a certain q-deformation of the algebra D(G). The algebra D_q(G), in turn, inherits many interesting symmetries from its affiliation with T^2: for instance, it has Fourier transform automorphism, a quantum moment map, and (for G=GL_n) a Hamiltonian reduction to the spherical DAHA of type A_{n-1}.

Current projects in this direction include relations with KZB connections in genus one (with Adrien Brochier), an interpretation of D_q(G) in terms of TFT's and factorization homology (with David Ben-Zvi and Adrien Brochier), and a q-deformation of the Springer Sheaf (with David Ben-Zvi and Tom Nevins).

Copyright © 1997-2024 Department of Mathematics, Northwestern University.