  The QCD phase diagram in the space of temperature and imaginary baryon
chemical potential has been an interesting subject in numerical lattice QCD
simulations because of the absence of the sign problem and its deep structure
related to confinement/deconfinement. We study constraints on the phase diagram
by using an 't Hooft anomaly. The relevant anomaly is an anomaly in the space
of imaginary chemical potential. We compute it in the UV, and discuss how it is
matched by the pion effective field theory at low temperatures. Then we study
implications of the anomaly to the phase diagram. There must be a line of phase
transition studied in the past by Roberge and Weiss such that the expectation
value of the Polyakov loop is not smooth when we cross the line. Moreover, if
the greatest common divisor of the color and flavor numbers is greater than
one, the phase transition across the Roberge-Weiss line must be either a first
order phase transition, or a second order phase transition described by a
nontrivial interacting three-dimensional CFT.
