A study of time reversal symmetry of abelian anyons

TL;DR

The paper develops a systematic framework for time-reversal symmetry on abelian anyons in 2+1D, introducing the obstruction and anomaly concepts and showing the anomaly can be captured by the Arf invariant of a quadratic refinement on the group \(\mathcal{C}\) of time-reversal-symmetric anyons. It provides a general theory tying symmetry actions to Moore-Seiberg data, and reduces the anomaly computation to a tractable finite sum over a quotiented group, with explicit expressions in terms of the topological spins and local Kramers degeneracies. Through two detailed case studies, it demonstrates that obstructions seem absent in all abelian cases studied (odd order and \(\mathcal{A}=(\mathbb{Z}_2)^N\)); the RP^4 anomaly can be controlled by choosing suitable \(\eta\). The work suggests that time-reversal on abelian anyon systems is typically unobstructed, and it clarifies how to determine and tune the anomaly, which has implications for symmetry-protected topological boundaries and related 1-form symmetry structures.

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 $\mathcal{A}=(\mathbb{Z}_2)^N$.