Algebraic structures for Kapustin-Witten twisted gauge theories

Topological twisting is a technique for producing topological field theories from supersymmetric field theories -- one exciting application is Kapustin and Witten's 2006 discovery that the categories appearing in the geometric Langlands conjecture can be obtained as topological twists of N=4 supersymmetric gauge theories, and that these two categories are interchanged by S-duality.  There are, however, several incompatibilities between Kapustin and Witten's construction and the geometric representation theory literature. First, their techniques do not produce the right algebraic structures on the moduli spaces appearing in geometric Langlands, and secondly, their construction doesn't explain the singular support conditions Arinkin and Gaitsgory introduced in order to make the geometric Langlands correspondence possible.  In this talk I'll explain joint work with Philsang Yoo addressing both of these issues: how to understand topological twisting in (derived) algebraic geometry, and how to interpret singular support conditions as arising from the choice of a vacuum state.