A study of time reversal symmetry of abelian anyons

TL;DR

This paper investigates the time reversal symmetry properties of abelian anyons in 2+1 dimensions, highlighting the role of the group of symmetric anyons and quadratic refinements in understanding anomalies.

Contribution

It introduces a framework for analyzing time reversal symmetry of abelian anyons using quadratic refinements and the Arf invariant, providing concrete cases for odd and $( ext{Z}_2)^N$ groups.

Findings

The importance of the group of symmetric anyons modulo time-reversal composition.

The role of quadratic refinements in local Kramers degeneracy.

Explicit analysis for cases where the anyon group size is odd or $( ext{Z}_2)^N$.

Abstract

We perform a study of time reversal symmetry of abelian anyons $\mathcal{A}$ in 2+1 dimensions, in the spin structure independent cases. We will find the importance of the group $\mathcal{C}$ of time-reversal-symmetric anyons modulo anyons composed from an anyon and its time reversal. Possible choices of local Kramers degeneracy are given by quadratic refinements of the braiding phases of $\mathcal{C}$, and the anomaly is then given by the Arf invariant of the chosen quadratic refinement. We also give a concrete study of the cases when $|\mathcal{A}|$ is odd or $\mathcal{A}=(\mathbb{Z}_2)^N$.