Fractal phase structure of QCD under imaginary rotation

We show that in hot QCD rotating at an imaginary angular velocity ${\mathrm{\Omega }}_I$, the angular velocity selects not only the temperature of the bulk but whether charge conjugation $C$ is broken there. For QCD at temperature $T$ and imaginary quark chemical potential $\theta$, with ${\mathrm{\Omega }}_I\mathrm{/}2\pi =p\mathrm{/}q$ in lowest terms, the bulk far from the rotation axis is in the same thermal equilibrium state as nonrotating QCD at temperature $T\mathrm{/}q$ and imaginary quark chemical potential ${\theta }^{\mathrm{\prime }}=q\theta +\pi (p+q+1)$; this follows from the Euclidean boundary conditions and locality alone, and holds nonperturbatively. Consequently, even at zero quark chemical potential, the bulk is placed at the Roberge-Weiss (RW) point whenever $p$ and $q$ are both odd, and $C$ is spontaneously broken there for $T>q{T}_{\mathrm{R}\mathrm{W}}$, with $T_{\mathrm{R}\mathrm{W}}$ the RW endpoint temperature. The bulks at ${\mathrm{\Omega }}_I=2\pi \mathrm{/}3$ and $4\pi \mathrm{/}3$, for example, are both at temperature $T\mathrm{/}3$, yet only the former can break $C$. The set of imaginary angular velocities at which $C$ is broken has a fractal structure: raising $T$ adds rationals of ever larger denominator. At irrational ${\mathrm{\Omega }}_I\mathrm{/}2\pi$ the bulk corresponds to zero-temperature QCD at any $T$ and remains confined.