Noninvertible Time-Reversal Symmetry

Abstract

In gauge theory, it is commonly stated that time-reversal symmetry only exists at θ=0 or π for a 2π-periodic θ angle. In this Letter, we point out that in both the free Maxwell theory and massive QED, there is a noninvertible time-reversal symmetry at every rational θ angle, i.e., θ=πp/N. The noninvertible time-reversal symmetry is implemented by a conserved, antilinear operator without an inverse. It is a composition of the naive time-reversal transformation and a fractional quantum Hall state. We also find similar noninvertible time-reversal symmetries in non-Abelian gauge theories, including the N=4 SU(2) super Yang-Mills theory along the locus |τ|=1 on the conformal manifold.