Soft symmetries of topological orders

TL;DR

This paper explores exotic 'soft' symmetries in (2+1)D topological orders, revealing new types of topological defects and implications for classifying symmetry-enriched phases, with extensions to higher dimensions.

Contribution

It introduces the concept of soft symmetries in topological orders, linking them to gauged SPT states and expanding the understanding of symmetry actions beyond traditional permutations.

Findings

Identification of soft symmetries not permuting anyons

Construction of topological defects via gauged SPT states

Extension of phenomena to (3+1)D gauge theories with quaternion group

Abstract

(2+1)D topological orders possess emergent symmetries given by a group $\text{Aut}(\mathcal{C})$, which consists of the braided tensor autoequivalences of the modular tensor category $\mathcal{C}$ that describes the anyons. In this paper we discuss cases where $\text{Aut}(\mathcal{C})$ has elements that neither permute anyons nor are associated to any symmetry fractionalization but are still non-trivial, which we refer to as soft symmetries. We point out that one can construct topological defects corresponding to such exotic symmetry actions by decorating with a certain class of gauged SPT states that cannot be distinguished by their torus partition function. This gives a physical interpretation to work by Davydov on soft braided tensor autoequivalences. This has a number of important implications for the classification of gapped boundaries, non-invertible spontaneous symmetry breaking, and the general classification of symmetry-enriched topological phases of matter. We also demonstrate analogous phenomena in higher dimensions, such as (3+1)D gauge theory with gauge group given by the quaternion group $Q_8$.