Explicit Factorization of a Categorical Center

TL;DR

This paper provides an explicit construction of the inverse to a known equivalence in braided fusion categories, using quantum field theory insights, and examines cases where the equivalence does not hold.

Contribution

It introduces a concrete method to invert the natural map from the square of a braided fusion category to its Drinfeld center, clarifying the construction and limitations.

Findings

Explicit inverse construction for the natural map

Insights from quantum field theory inform the construction

Analysis of degenerate cases where the equivalence fails

Abstract

Given a braided fusion category $C$, it is well known that the natural map $C \boxtimes C^{bop} \to Z(C)$ from the square of $C$ to the (Drinfeld) categorical center $Z(C)$ is an equivalence if and only if $C$ is modular. However, it is not clear how to construct the inverse and the natural isomorphisms. In this work, we provide an explicit construction using insights from a specific quantum field theory, and explore how the equivalence fails for the degenerate cases.