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Braided zesting and its applications

Colleen Delaney, César Galindo, Julia Plavnik, Eric C. Rowell, Qing Zhang

TL;DR

Zesting provides a systematic framework to construct new braided and ribbon fusion categories from an $A$-graded starting category by twisting the tensor product with invertible central objects. The authors develop a complete obstruction-and-parameterization theory for associative, braided, and twist zestings, relate these to group and Eilenberg–MacLane cohomology, and derive explicit modular-data formulas for the zested categories. They illustrate the approach with detailed applications, including cyclic zestings and modular categories from type $A$ quantum groups such as $SU(N)_k$, producing new modular data and analyzing Müger centers. The work connects zesting to gauging-like constructions, expands the catalog of known fusion categories, and provides concrete tools for constructing and classifying zestings with computable $S$- and $T$-matrices.

Abstract

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.

Braided zesting and its applications

TL;DR

Zesting provides a systematic framework to construct new braided and ribbon fusion categories from an -graded starting category by twisting the tensor product with invertible central objects. The authors develop a complete obstruction-and-parameterization theory for associative, braided, and twist zestings, relate these to group and Eilenberg–MacLane cohomology, and derive explicit modular-data formulas for the zested categories. They illustrate the approach with detailed applications, including cyclic zestings and modular categories from type quantum groups such as , producing new modular data and analyzing Müger centers. The work connects zesting to gauging-like constructions, expands the catalog of known fusion categories, and provides concrete tools for constructing and classifying zestings with computable - and -matrices.

Abstract

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.

Paper Structure

This paper contains 37 sections, 32 theorems, 185 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group cohomology
  4. Basic definitions on fusion and modular categories
  5. Pivotal and spherical fusion categories
  6. Premodular and modular tensor categories
  7. Graded fusion categories and tensor natural isomorphisms of the identity functor
  8. Conventions and Graphical calculus
  9. Associative zestings
  10. Relative centralizer of monoidal subcategories
  11. Associative zesting
  12. Rigidity of associative zesting
  13. Obstruction to associative zestings
  14. Braided zesting
  15. Equivalence of braided zestings
  16. ...and 22 more sections

Key Result

Proposition 2.7

Let ${\mathcal{C}}$ be a faithfully $G$-graded fusion category. The group homomorhism $\Phi:\widehat{G}\to \operatorname{Aut}_{{\otimes}}^G(\operatorname{Id}_{\mathcal{C}})$ is an isomorphism. In particular $\Phi:\widehat{U({\mathcal{C}}})\cong \operatorname{Aut}_{{\otimes}}(\operatorname{Id}_{\math

Figures (15)

  • Figure 1: Definition of $\chi$
  • Figure 2: Associative zesting constraint
  • Figure 3: Pentagon axiom that must be satisfied by the zested associators.
  • Figure 4: Equivalent formulation of the pentagon axiom from Figure 3.
  • Figure 5: 4-cocycle obstruction
  • ...and 10 more figures

Theorems & Definitions (82)

  • Example 2.1: Pointed fusion categories
  • Example 2.2
  • Remark 2.3
  • Example 2.4: Pointed braided fusion categories
  • Example 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Corollary 2.9
  • proof
  • ...and 72 more