Generalised 6j symbols over the category of $G$-graded vector spaces

TL;DR

This paper develops and computes generalized 6j symbols for categories of G-graded vector spaces, providing explicit classifications and calculations relevant for 3-manifold invariants with defect data.

Contribution

It introduces a classification of module functors as matrices over G-graded vector spaces and computes the associated generalized 6j symbols, expanding the understanding of invariants in 3-manifold topology.

Findings

Classified module functors over G-graded vector spaces using matrices.

Calculated generalized 6j symbols for categories and bimodule categories over G-graded vector spaces.

Provided explicit examples and classifications for finite cyclic groups G.

Abstract

Any choice of a spherical fusion category defines an invariant of oriented closed 3-manifolds, which is computed by choosing a triangulation of the manifold and considering a state sum model that assigns a 6j symbol to every tetrahedron in this triangulation. This approach has been generalized to oriented closed 3-manifolds with defect data by Meusburger. In a recent paper, she constructed a family of invariants for such manifolds parametrised by the choice of certain spherical fusion categories, bimodule categories, finite bimodule functors and module natural transformations. Meusburger defined generalised 6j symbols for these objects, and introduces a state sum model that assigns a generalised 6j symbol to every tetrahedron in the triangulation of a manifold with defect data, where the type of 6j symbol used depends on what defect data occur within the tetrahedron. The present work provides non-trivial examples of suitable bimodule categories, bimodule functors and module natural transformation, all over categories of $G$-graded vector spaces. Our main result is the description of module functors in terms of matrices, which allows us to classify these functors when $G$ is a finite cyclic group. Furthermore, we calculate the generalised 6j symbols for categories of $G$-graded vector spaces, (bi-)module categories over such categories and (bi-)module functors.