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

Understand

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.