Fractal phase structure of QCD under imaginary rotation

We show that in hot QCD rotating at an imaginary angular velocity ΩI, 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 $θ$, with ΩI/2π=p/q in lowest terms, the bulk far from the rotation axis is in the same thermal equilibrium state as nonrotating QCD at temperature $T/q$ and imaginary quark chemical potential $θ'=qθ+π(p+q+1)$; 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>qTRW, with TRW the RW endpoint temperature. The bulks at ΩI=2π/3 and $4π/3$, for example, are both at temperature $T/3$, 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 ΩI/2π the bulk corresponds to zero-temperature QCD at any T and remains confined.