Description
VOA's often appear on holomorphic boundary conditions of 3d topological QFT's. A classic example involves WZW models on the boundary of Chern-Simons theories. In such a setup, algebraic structures in the boundary VOA (e.g. categories of modules and conformal blocks) can be used to reconstruct structures of the bulk 3d TQFT. I will discuss a modern generalization of this boundary-bulk correspondence, involving logarithmic VOA's and topological twists of 3d supersymmetric gauge theories. Two subtle and beautiful new features that arise are the presence of dg/derived structure, and deformations over certain moduli spaces of connections. I will also discuss a key example, introduced in recent work with T. Creutzig, N. Garner, and N. Geer, relating Feigin-Tipunin algebras, a supersymmetric generalization of Chern-Simons theory, and the derived category of representations of a quantum group at a root of unity.
