A manifestly Morita-invariant construction of Turaev–Viro invariants

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 is Nakayama-twisted traces on pivotal bimodule categories, which we study beyond semisimplicity.