G-crossed Modularity of Symmetry-Enriched Topological Phases

TL;DR

This paper explores the mathematical structure of symmetry-enriched topological phases in 2+1 dimensions using G-crossed extensions of unitary modular tensor categories, linking topological properties with symmetry actions and defects.

Contribution

It establishes the relation between G-crossed UMTCs and topological state spaces on complex surfaces, defining operators for mapping class transformations, and proves that faithful G-crossed extensions are G-crossed modular.

Findings

Defined operators representing mapping class transformations

Connected G-crossed UMTCs with topological state spaces on surfaces

Proved that faithful G-crossed extensions are G-crossed modular

Abstract

The universal properties of (2 + 1)D topological phases of matter enriched by a symmetry group G are described by G-crossed extensions of unitary modular tensor categories (UMTCs). While the fusion and braiding properties of quasiparticles associated with the topological order are described by a UMTC, the G-crossed extensions further capture the properties of the symmetry action, fractionalization, and defects arising from the interplay of the symmetry with the topological order. We describe the relation between the G-crossed UMTC and the topological state spaces on general surfaces that may include symmetry defect branch lines and boundaries that carry topological charge. We define operators in terms of the G-crossed UMTC data that represent the mapping class transformations for such states on a torus with one boundary, and show that these operators provide projective representations of the mapping class groups. This allows us to represent the mapping class group on general surfaces and ensures a consistent description of the corresponding symmetry-enriched topological phases on general surfaces. Our analysis also enables us to prove that a faithful G-crossed extension of a UMTC is necessarily G-crossed modular.