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

Fabio Lischka

TL;DR

This work develops concrete, computable instances of generalized $6j$ symbols in the defect‑enriched 3‑manifold invariant framework by focusing on the tensor category $\mathrm{Vec}_G^{\omega}$. It provides an explicit matrix description of $\mathrm{Vec}_G^{\omega}$‑module functors and classifies simple functors for finite cyclic $G$, together with complete computations of generalized $6j$ symbols for $\mathrm{Vec}_G^{\omega}$, its bimodule categories, and module functors. The results yield a practical algebraic toolkit for defect Dijkgraaf–Witten theories, linking group cohomology, module theory and topological quantum field theory. The construction supports state‑sum models for 3‑manifolds with defect data and offers a pathway to explicit calculations in topological settings where defects play a central role.

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.

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

TL;DR

This work develops concrete, computable instances of generalized symbols in the defect‑enriched 3‑manifold invariant framework by focusing on the tensor category . It provides an explicit matrix description of ‑module functors and classifies simple functors for finite cyclic , together with complete computations of generalized symbols for , its bimodule categories, and module functors. The results yield a practical algebraic toolkit for defect Dijkgraaf–Witten theories, linking group cohomology, module theory and topological quantum field theory. The construction supports state‑sum models for 3‑manifolds with defect data and offers a pathway to explicit calculations in topological settings where defects play a central role.

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 -graded vector spaces. Our main result is the description of module functors in terms of matrices, which allows us to classify these functors when is a finite cyclic group. Furthermore, we calculate the generalised 6j symbols for categories of -graded vector spaces, (bi-)module categories over such categories and (bi-)module functors.
Paper Structure (30 sections, 43 theorems, 226 equations)

This paper contains 30 sections, 43 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group Cohomology
  4. Tensor categories
  5. The Deligne product of abelian categories
  6. Properties of semisimple categories
  7. Graphical calculus for spherical fusion categories
  8. The tensor category $\mathrm{Vec}_G^{\omega}$
  9. Monoidal structures and rigidity
  10. Pivotal structures on $\mathrm{Vec}_G^{\omega}$
  11. Module categories
  12. Definition and properties
  13. Module categories over $\mathrm{Vec}_G^{\omega}$
  14. $(\mathrm{Vec}_G^{\omega_G}, \mathrm{Vec}_H^{\omega_H})$-bimodule categories
  15. Module functors
  16. ...and 15 more sections

Key Result

Lemma 2.1

emNorm Every $n$-cochain $\eta$ with $\dif \eta$ normalized is cohomologous to a normalized $n$-cochain $\eta'$.

Theorems & Definitions (100)

  • Lemma 2.1
  • Lemma 2.2: Shapiro's Lemma
  • Remark 2.3
  • Lemma 2.4: Schur's Lemma
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • ...and 90 more