$\theta$-dependence and center symmetry in Yang-Mills theories

We investigate the relation between the realization of center symmetry and the dependence on the topological parameter $\theta$ in $SU(N)$ Yang-Mills theories, exploiting trace deformations as a tool to regulate center symmetry breaking in a theory with a small compactified direction. We consider, in particular, $SU(4)$ gauge theory, which admits two possible independent deformations, and study, as a first step, its phase diagram in the deformation plane for two values of the inverse compactified radius going up to $L^{-1} \sim 500$ MeV, comparing the predictions of the effective 1-loop potential of the Polyakov loop with lattice results. The $\theta$-dependence of the various phases is then addressed, up to the fourth order in $\theta$, by numerical simulations: results are found to coincide, within statistical errors, with those of the standard confined phase iff center symmetry is completely restored and independently of the particular way this happens, i.e. either by local suppression of the Polyakov loop traces or by long range disorder.