REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: The Turaev-Viro theory is a 2+1 dimensional topological quantum field theory (TQFT), meaning it associates a finite dimensional Hilbert space Z(S) to each oriented surface S and linear maps Z(S) --> Z(S') to 3-dimensional cobordisms relating S and S'. Roughly speaking, Turaev-Viro takes as input the category of representations of a quantum group U_q(g) with its braiding forgotten, and the vector space Z(S) is some flavor of skein module associated to g.

In this talk I will discuss forthcoming work (joint with Dave Rose and Paul Wedrich) in which we define an appropriate "skein dg category" to each oriented surface, providing a categorical analogue of the 2-dimensional part of Turaev-Viro for SL(2). Our construction is formulated in the language of Khovanov's arc rings, and indeed our main idea is the extension of these rings from the disk to arbitrary oriented surfaces, as I will demonstrate.