In gauge theory, it is commonly stated that time-reversal symmetry only exists at $\theta =0$ or $\pi$ for a $2\pi$-periodic $\theta$ 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 $\theta$ angle, i.e., $\theta =\pi p/N$. 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 $|\tau |=1$ on the conformal manifold.

• Received 13 December 2022
• Accepted 24 February 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.131602

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Published by the American Physical Society