Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies

TL;DR

This paper develops a systematic framework using path integrals on non-orientable manifolds to classify and analyze reflection and time-reversal symmetry-enriched topological phases in 2+1 dimensions, revealing their anomalies and relation to 3+1D SPT phases.

Contribution

It introduces a general formula for the path integral on non-orientable manifolds and connects ground state degeneracy with symmetry fractionalization, advancing the understanding of anomalies in SET phases.

Findings

Derived a formula for path integrals on non-orientable manifolds.

Established a criterion for detecting anomalies in (2+1)D SET phases.

Connected (2+1)D SET phases to (3+1)D SPT phases via boundary analysis.

Abstract

We study symmetry-enriched topological (SET) phases in 2+1 space-time dimensions with spatial reflection and/or time-reversal symmetries. We provide a systematic construction of a wide class of reflection and time-reversal SET phases in terms of a topological path integral defined on general space-time manifolds. An important distinguishing feature of different topological phases with reflection and/or time-reversal symmetry is the value of the path integral on non-orientable space-time manifolds. We derive a simple general formula for the path integral on the manifold $\Sigma^2 \times S^1$, where $\Sigma^2$ is a two-dimensional non-orientable surface and $S^1$ is a circle. This also gives an expression for the ground state degeneracy of the SET on the surface $\Sigma^2$ that depends on the reflection symmetry fractionalization class, generalizing the Verlinde formula for ground state degeneracy on orientable surfaces. Consistency of the action of the mapping class group on non-orientable manifolds leads us to a constraint that can detect when a time-reversal or reflection SET phase is anomalous in (2+1)D and, thus, can only exist at the surface of a (3+1)D symmetry protected topological (SPT) state. Given a (2+1)D reflection and/or time-reversal SET phase, we further derive a general formula that determines which (3+1)D reflection and/or time-reversal SPT phase hosts the (2+1)D SET phase as its surface termination. A number of explicit examples are studied in detail.