  We study quantum chromodynamics including the two-index symmetric or
anti-symmetric quark (QCD(Sym/ASym)) on small $\mathbb{R}^2\times T^2$ with a
suitable magnetic flux. We first discuss the 't Hooft anomaly of these theories
and claim that discrete chiral symmetry should be spontaneously broken
completely to satisfy the anomaly matching condition. The $T^2$
compactification with the magnetic flux preserves the 't Hooft anomaly, and the
$2$d effective theory is constrained by the same anomaly of $4$d QCD(Sym/ASym).
We demonstrate the spontaneous breakdown of chiral symmetry using the dilute
gas of center vortices, which confirms the prediction of the 't Hooft anomaly.
We also find that each vacuum maintains the charge conjugation symmetry, and
this gives affirmative support for the nonperturbative large-$N$ orientifold
equivalence between QCD(Sym/ASym) and $\mathcal{N}=1$ supersymmetric $SU(N)$
Yang-Mills theory.

High Energy Physics - Theory