Non-invertible Time-reversal Symmetry

Understand

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 paper, we point out that in both the free Maxwell theory and massive QED, there is a non-invertible time-reversal symmetry at every rational $\theta$-angle, i.e., $\theta= \pi p/N$.
• The non-invertible time-reversal symmetry is implemented by a conserved, anti-linear operator without an inverse.
• It is a composition of the naive time-reversal transformation and a fractional quantum Hall state.