3-dimensional TQFTs from derived categories of quantum group representations

Abstract

For any finite modular tensor category A, we show that the associated derived $\infty$-category D(A) supports a topological quantum field theory in dimension 3. This TQFT takes the form of a symmetric monoidal functor from an $\infty$-category of surfaces with markings by objects in D(A), and appropriately decorated bordisms, to the $\infty$-category of dg vector spaces. We show that the state spaces in this theory are naturally identified with linearized mapping spaces for D(A). Though our TQFT requires markings from the derived $\infty$-category, we show that all markings can be removed after taking a homotopy truncation. The resulting unmarked TQFT produces projective mapping class group actions on cohomology in dimension 2, and in particular a projective SL_2(Z)-action on Hochschild cohomology. We expect these mapping class group actions to recover those of Lentner et al. arxiv:2003.06527 and Schweigert-Woike arxiv:2004.14343. In dimension 3 we obtain power-series valued knot invariants, and power-series valued invariants for modular tensor categories. We also obtain power-series invariants for closed 3-manifold, though these can already be calculated at the abelian level. Our derived field theories are proposed as mathematical formalizations for topological A-twists of certain N=4 supersymmetric QFTs, in dimension 3. Following physical principles, we discuss the possibility of deforming our TQFTs along local systems via an analogous (conjectural) deformation of quantum group representations along the Langlands dual group.