Advances in Theoretical and Mathematical Physics

Volume 28 (2024)

Number 1

A QFT for non-semisimple TQFT

Pages: 161 – 405

DOI: https://dx.doi.org/10.4310/ATMP.2024.v28.n1.a4

Authors

Thomas Creutzig (Department of Mathematics, FAU Erlangen, Germany)

Tudor Dimofte (School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom; and Department of Mathematics and Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, Calif., U.S.A.)

Niklas Garner (Department of Mathematics and Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, Calif., U.S.A.; and Department of Physics, University of Washington, Seattle, Wash., U.S.A.)

Nathan Geer (Department of Mathematics and Statistics, Utah State University, Logan, Utah, U.S.A.)

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.

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Published 19 August 2024