Non-semisimple topological field theory and $\widehat{Z}$-invariants from $\mathfrak{osp}(1 \vert 2)$

TL;DR

This paper constructs non-semisimple 3D topological field theories from the unrolled quantum group of $rak{osp}(1|2)$, relating them to $ ext{Z}$-invariants and providing new computational formulas for their invariants.

Contribution

It introduces a novel non-semisimple topological field theory based on $rak{osp}(1|2)$ and establishes a connection with $ ext{Z}$-invariants, including a Verlinde formula for state space dimensions.

Findings

Constructed 3D non-semisimple TQFTs from $rak{osp}(1|2)$

Derived a Verlinde formula for topological state spaces

Linked $ ext{Z}$-invariants of $rak{osp}(1|2)$ to 3-manifold invariants

Abstract

We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$. More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 \pi \sqrt{-1}}{r}}$, where $r$ is a positive integer greater than $2$, and the construction applies when $r$ is not congruent to $4$ modulo $8$. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When $r$ is congruent to $\pm 1$ or $\pm 2$ modulo $8$, we relate the resulting $3$-manifold invariants with physicists' $\widehat{Z}$-invariants associated to $\mathfrak{osp}(1 \vert 2)$. Finally, we establish a relation between $\widehat{Z}$-invariants associated to $\mathfrak{sl}(2)$ and $\mathfrak{osp}(1 \vert 2)$ which was conjectured in the physics literature.