A QFT for non-semisimple TQFT

Abstract

We construct a family of 3d quantum field theories ${\mathcal T}_{n,k}^A$ that conjecturally provide a physical realization --- and derived generalization --- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(i\pi/k)$. The theories ${\mathcal T}_{n,k}^A$ are defined as topological twists of certain 3d ${\mathcal N}=4$ Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern-Simons theories, coupled to a twisted ${\mathcal N}=4$ matter sector (the source of non-semisimplicity). We show that ${\mathcal T}_{n,k}^A$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $\mathfrak{sl}_n$-type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in ${\mathcal T}_{n,k}^A$ to the derived category of modules for a boundary Feigin-Tipunin algebra, and --- using a logarithmic Kazhdan-Lusztig-like correspondence that has been established for $n=2$ and expected for general $n$ --- to the derived category of $U_q(\mathfrak{sl}_n)$ modules. We analyze many other key features of ${\mathcal T}_{n,k}^A$ and match them from quantum-group and VOA perspectives, including deformations by flat $PGL(n,\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.