# State Sum Invariants of Three Manifolds from Spherical Multi-fusion Categories

## TL;DR

This paper introduces a new family of quantum invariants for closed 3-manifolds derived from spherical multi-fusion categories, extending existing TQFTs and encompassing higher gauge theories with a richer label structure.

## Contribution

It defines spherical multi-fusion categories and constructs associated 3-manifold invariants, generalizing the Turaev-Viro-Barrett-Westbury TQFTs and including higher gauge theories.

## Key findings

- Defines spherical multi-fusion categories with weakened sphericity.
- Constructs state sum invariants that include labels on 0- and 1-simplices.
- Shows the invariants generalize and include recent higher gauge theory TQFTs.

## Abstract

We define a family of quantum invariants of closed oriented $3$-manifolds using spherical multi-fusion categories. The state sum nature of this invariant leads directly to $(2+1)$-dimensional topological quantum field theories ($\text{TQFT}$s), which generalize the Turaev-Viro-Barrett-Westbury ($\text{TVBW}$) $\text{TQFT}$s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the $\text{TVBW}$ approach in that here the labels live not only on $1$-simplices but also on $0$-simplices. It is shown that a multi-fusion category in general cannot be a spherical fusion category in the usual sense. Thus we introduce the concept of a spherical multi-fusion category by imposing a weakened version of sphericity. Besides containing the $\text{TVBW}$ theory, our construction also includes the recent higher gauge theory $(2+1)$-$\text{TQFT}$s given by Kapustin and Thorngren, which was not known to have a categorical origin before.