Gauging Categorical Symmetries in 3d Topological Orders and Bulk Reconstruction

TL;DR

This paper develops a categorical framework for gauging and ungauging nonabelian anyons in 3d topological orders, enabling bulk reconstruction and the analysis of modular invariants through condensation and symmetry considerations.

Contribution

It introduces a categorical condensation procedure for gauging nonabelian anyons and provides a method for bulk reconstruction in 3d topological orders, expanding understanding of symmetry and modular invariants.

Findings

A new procedure for gauging nonabelian anyons using categorical condensation.

A framework for bulk reconstruction via ungauging and Drinfeld center construction.

Explicit examples of constructing parent theory S-matrices from child theories.

Abstract

We use the language of categorical condensation to give a procedure for gauging nonabelian anyons, which are the manifestations of categorical symmetries in three spacetime dimensions. We also describe how the condensation procedure can be used in other contexts such as for topological cosets and constructing modular invariants. By studying a generalization of which anyons are condensable, we arrive at representations of congruence subgroups of the modular group. We finally present an analysis for ungauging anyons, which is related to the problem of constructing a Drinfeld center for a fusion category; this procedure we refer to as bulk reconstruction. We introduce a set of consistency relations regarding lines in the parent theory and wall category. Through use of these relations along with the $S$-matrix elements of the child theory, we construct $S$-matrix elements of a parent theory in a number of examples.