A QFT for non-semisimple TQFT

Abstract

$\def\Tank{\mathcal{T}^A_{n,k}}$$\def\Uqsln{U_q(\mathfrak{sl}_n)}$We construct a family of 3d quantum field theories $\Tank$ that conjecturally provide a physical realization—and derived generalization—of non-semisimple mathematical TQFT’s based on the modules for the quantum group $\Uqsln$ at an even root of unity $q = \operatorname{exp}(i \pi / k)$. The theories $\Tank$ 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 $\Tank$ 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 $\Tank$ 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 $\Uqsln$ modules.We analyze many other key features of $\Tank$ 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.