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In gauge theory, it is commonly stated that time-reversal symmetry only exists at =0 or for a 2-periodic 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 angle, i. e. , =/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 ||=1 on the conformal manifold.
Choi et al. (Tue,) studied this question.