A manifestly Morita-invariant construction of Turaev-Viro invariants

TL;DR

This paper introduces a new Morita-invariant state sum construction for Turaev-Viro invariants using spherical module categories and pivotal bicategories, providing a more algebraically robust approach.

Contribution

It develops a Morita-invariant construction of Turaev-Viro invariants via pivotal bicategories, avoiding reliance on Reshetikhin-Turaev methods.

Findings

The construction recovers standard Turaev-Viro invariants.

Proves invariance under pivotal Morita equivalence.

Develops evaluation techniques for sphere graphs with module category labels.

Abstract

We present a state sum construction that assigns a scalar to a skeleton in a closed oriented three-dimensional manifold. The input datum is the pivotal bicategory $\mathbf{Mod}^{\mathrm{sph}}(\mathcal{A})$ of spherical module categories over a spherical fusion category $\mathcal{A}$. The interplay of algebraic structures in this pivotal bicategory with moves of skeleta ensures that our state sum is independent of the skeleton on the manifold. We show that the bicategorical invariant recovers the value of the standard Turaev-Viro invariant associated to $\mathcal{A}$, thereby proving the independence of the Turaev-Viro invariant under pivotal Morita equivalence without recurring to the Reshetikhin-Turaev construction. A key ingredient for the construction is the evaluation of graphs on the sphere with labels in $\mathbf{Mod}^{\mathrm{sph}}(\mathcal{A})$ that we develop in this article. A central tool are Nakayama-twisted traces on pivotal bimodule categories which we study beyond semisimplicity.