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

Understand

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.