(In)validity of largeNorientifold equivalence

It has been argued that the bosonic sectors of supersymmetric $SU(N)$ Yang-Mills theory, and of QCD with a single fermion in the antisymmetric (or symmetric) tensor representation, are equivalent in the $N\to \infty$ limit. If true, this correspondence can provide useful insight into properties of real QCD (with fundamental representation fermions), such as predictions [with $O(1/N)$ corrections] for the nonperturbative vacuum energy, the chiral condensate, and a variety of other observables. Several papers asserting to have proven this large $N$ “orientifold equivalence” have appeared. By considering theories compactified on ${\mathbb{R}}^3\times S^1$, we show explicitly that this large $N$ equivalence fails for sufficiently small radius, where our analysis is reliable, due to spontaneous symmetry breaking of charge-conjugation symmetry in QCD with an antisymmetric (or symmetric) tensor representation fermion. This theory is also chirally symmetric for small radius, unlike super-Yang-Mills theory. The situation is completely analogous to large-$N$ equivalences based on orbifold projections: simple symmetry realization conditions are both necessary and sufficient for the validity of the large $N$ equivalence. Whether these symmetry realization conditions are satisfied depends on the specific nonperturbative dynamics of the theory under consideration. Unbroken charge-conjugation symmetry is necessary for validity of the large $N$ orientifold equivalence. Whether or not this condition is satisfied on ${\mathbb{R}}^4$ (or ${\mathbb{R}}^3\times S^1$ for sufficiently large radius) is not currently known.