Planar limit of orientifold field theories and emergent center symmetry

We consider orientifold field theories [i.e., $\mathrm{SU}(N)$ Yang-Mills theories with fermions in the two-index symmetric or antisymmetric representations] on $R_3\times S_1$ where the compact dimension can be either temporal or spatial. These theories are planar equivalent to supersymmetric Yang-Mills theory. The latter has $Z_N$ center symmetry. The famous Polyakov criterion establishing confinement-deconfinement phase transition as that from $Z_N$ symmetric to $Z_N$ broken phase applies. At the Lagrangian level the orientifold theories have at most a $Z_2$ center. We discuss how the full $Z_N$ center symmetry dynamically emerges in the orientifold theories in the limit $N\to \infty$. In the confining phase the manifestation of this enhancement is the existence of stable $k$ strings in the large-$N$ limit of the orientifold theories. These strings are identical to those of supersymmetric Yang-Mills theories. We argue that critical temperatures (and other features) of the confinement-deconfinement phase transition are the same in the orientifold daughters and their supersymmetric parent up to $1/N$ corrections. We also discuss the Abelian and non-Abelian confining regimes of four-dimensional QCD-like theories.

Figure 1

$Z_N$ symmetric vacuum fields $v_k$. For definitions see Eq. (3).