Noninvertible Time-Reversal Symmetry
Yichul Choi, Ho Tat Lam, Shu-Heng Shao
Abstract
In gauge theory, it is commonly stated that time-reversal symmetry only exists at $\ensuremath{\theta}=0$ or $\ensuremath{\pi}$ for a $2\ensuremath{\pi}$-periodic $\ensuremath{\theta}$ 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 $\ensuremath{\theta}$ angle, i.e., $\ensuremath{\theta}=\ensuremath{\pi}p/N$. 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 $\mathcal{N}=4$ SU(2) super Yang-Mills theory along the locus $|\ensuremath{\tau}|=1$ on the conformal manifold.