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Computing the center of a fusion category

Fabian Mäurer, Ulrich Thiel

TL;DR

This work develops and implements an explicit algorithm to compute the Drinfeld center Z(C) of a pivotal fusion category, enabling construction of center objects as pairs (X, γ) and explicit derivation of F- and R-symbols. The core method relies on the induction functor I: C → Z(C) to generate center objects as subobjects of inductions I(X_i), with a refined approach using center-generating simples to manage computation. The authors implement a general software framework TensorCategories.jl on OSCAR and demonstrate the approach on the Ising category over a non-algebraically closed field, obtaining full center data and modular information, including non-splitting phenomena over splitting fields. They apply the method to compute centers for all multiplicity-free fusion categories up to rank 5 (and select rank-6 examples), producing comprehensive data sets that can feed further research in modular categories and anyon theories. The results illustrate both the feasibility and the richness of center computations across fields and splitting behavior, and they provide open-access computational tools and data through the AnyonWiki ecosystem.

Abstract

We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories.jl for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples $(X,γ)$ where $X$ is an object and $γ$ is a half-braiding. From these models we can compute the $F$-symbols and $R$-symbols. Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper).

Computing the center of a fusion category

TL;DR

This work develops and implements an explicit algorithm to compute the Drinfeld center Z(C) of a pivotal fusion category, enabling construction of center objects as pairs (X, γ) and explicit derivation of F- and R-symbols. The core method relies on the induction functor I: C → Z(C) to generate center objects as subobjects of inductions I(X_i), with a refined approach using center-generating simples to manage computation. The authors implement a general software framework TensorCategories.jl on OSCAR and demonstrate the approach on the Ising category over a non-algebraically closed field, obtaining full center data and modular information, including non-splitting phenomena over splitting fields. They apply the method to compute centers for all multiplicity-free fusion categories up to rank 5 (and select rank-6 examples), producing comprehensive data sets that can feed further research in modular categories and anyon theories. The results illustrate both the feasibility and the richness of center computations across fields and splitting behavior, and they provide open-access computational tools and data through the AnyonWiki ecosystem.

Abstract

We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories.jl for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples where is an object and is a half-braiding. From these models we can compute the -symbols and -symbols. Using the data from the AnyonWiki, we were able to compute the center together with its -symbols and -symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper).
Paper Structure (23 sections, 7 theorems, 54 equations, 2 figures, 5 algorithms)

This paper contains 23 sections, 7 theorems, 54 equations, 2 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Basic assumptions, notation, and facts
  3. Fusion categories
  4. Coends
  5. Dimensions
  6. Modular categories
  7. Structure of the center
  8. F-symbols and R-symbols
  9. Computing the center
  10. Computing half-braidings
  11. An algorithm based on the induction functor
  12. The induction functor and the basic idea
  13. Center-generating simple objects
  14. Computing Hom-spaces in $\mathcal{Z}(\mathcal{C})$
  15. Splitting of simple central objects
  16. ...and 8 more sections

Key Result

theorem 1

Let $\mathcal{C}$ be the Ising fusion category considered over $\mathbb{Q}(\sqrt 2)$. Then $\mathcal{Z}(\mathcal{C})$ has five simple objects with dimensions $1,1,2,2$ and $4\cdot \sqrt 2$. The multiplication table is given by and the $S$-matrix is given by In particular, $\mathcal{Z}(\mathcal{C})$ is a non-split modular category. It splits over $\mathbb Q(\xi_{16})$ where $\xi_{16}$ is a primit

Figures (2)

  • Figure 1: The actual center of $\Bbbk$-$\mathsf{Vec}_G$ for $G$ the symmetric group $S_3$ and $\Bbbk = \mathbb{Q}(\xi_3)$, where $\xi_3$ is a primitive third root of unity. The table lists the simple central objects $(X,\gamma_X)$. The first column lists the underlying object $X$ of $\Bbbk$-$\mathsf{Vec}_G$. The other columns are indexed by the simple objects $Y$ of $\Bbbk$-$\mathsf{Vec}_G$ and specify the half-braiding $\gamma_X(Y)$. The data in this table is readily provided by our software.
  • Figure 2: The ideal for half-braidings for the object $(23) \oplus (12) \oplus (13)$ in $\QQ$-$\mathsf{Vec}_{S_3}$. The second expression is a Gröbner basis with respect to the lexicographic ordering.

Theorems & Definitions (15)

  • theorem 1
  • theorem 2: Bruguières--Virelizier bruguieres2013center
  • remark 1
  • lemma 1: Müger subfactors
  • remark 2
  • lemma 2
  • proof
  • definition 1
  • lemma 3
  • proof
  • ...and 5 more