Equivariant Topological Quantum Field Theory and Symmetry Protected Topological Phases

TL;DR

This paper classifies low-dimensional bosonic symmetry protected topological phases using a generalized equivariant topological quantum field theory framework, extending existing cohomology classifications to unoriented cases.

Contribution

It generalizes Turaev's equivariant TQFT to unoriented cases and establishes classification results for invertible equivariant TQFTs in low dimensions.

Findings

Classified unoriented equivariant TQFTs by twisted group cohomology.

Classified oriented equivariant TQFTs by ordinary group cohomology.

Confirmed the group cohomology proposal for SPT phases.

Abstract

Short-range entangled topological phases of matter are closely connected to Topological Quantum Field Theory. We use this connection to classify bosonic Symmetry Protected Topological Phases in low dimensions, including the case when the symmetry involves time-reversal. To accomplish this, we generalize Turaev's description of equivariant TQFT to the unoriented case. We show that invertible unoriented equivariant TQFTs in one or less spatial dimensions are classified by twisted group cohomology, in agreement with the group cohomology proposal of Chen, Gu, Liu and Wen. We also show that invertible oriented equivariant TQFTs in spatial dimension two or less are classified by ordinary group cohomology.