Categorical 4-manifold invariants from trisection diagrams

TL;DR

This paper introduces a new 4-manifold invariant based on trisection diagrams and algebraic structures like bimodule categories and fusion categories, unifying and extending previous invariants.

Contribution

It defines a novel 4-manifold invariant using bimodule categories and fusion categories, with a diagrammatic calculus, generalizing existing invariants such as Hopf algebraic and Bärenz-Barrett invariants.

Findings

Includes Hopf algebraic 4-manifold invariants as a special case

Recovers Bärenz and Barrett's invariants via a pivotal functor

Provides a simple diagrammatic description of the invariant

Abstract

We use Gay and Kirby's description of 4-manifolds in terms of trisections and trisection diagrams to define a new 4-manifold invariant. The algebraic data are an indecomposable finite semisimple bimodule category over a pair of spherical fusion categories, equipped with a bimodule trace, and a pivotal functor from another spherical fusion category into the spherical fusion category of its bimodule endofunctors and natural transformations between them. The 4-manifold invariant has a simple description in terms of a diagrammatic calculus for this data, in which the three spherical fusion categories correspond to the three colours of the trisection diagram. It includes the Hopf algebraic 4-manifold invariants by Chaidez, Cotler and Cui, which arise when the bimodule category is the category of finite-dimensional complex vector spaces. We also recover the 4-manifold invariants of B\"arenz and Barrett defined by a pivotal functor from a spherical fusion category into a modular fusion category.