Vanishing of the $H^3$ obstruction for time-reversal symmetry in (2+1)D abelian bosonic TQFTs

Abstract

In $(2+1)$-dimensional topological quantum field theories (TQFTs), the action of a global symmetry group on the anyon system is one of the central topics of research. Owing to the subtle categorical nature of anyons, a global symmetry acting on them is generally realized in a projective manner. Symmetry fractionalization encodes this projective realization. The obstruction to defining symmetry fractionalization is captured by a cohomology class, known as the $H^3$ obstruction, whose nontriviality signals a failure to define symmetry fractionalization consistently. In this short note, we prove that the $H^3$ obstruction for time-reversal symmetry always vanishes in abelian bosonic TQFTs.