(In)validity of large N orientifold equivalence

TL;DR

This work tests the proposed large-N orientifold equivalence between ${\cal N}=1$ SYM and QCD(AS/S) by analyzing symmetry realizations on ${\mathbb R}^3\times S^1$ and in high-temperature regimes. The authors show that at small circle radius, charge conjugation symmetry is spontaneously broken in QCD(AS/S) but not in SYM, causing the orientifold equivalence to fail in that regime; at sufficiently high temperature, the symmetry is restored and the leading large-N thermodynamics agree, allowing equivalence for thermal observables. The analysis reframes the claim as a dyadic, neutral-sector equivalence arising from projections of a common parent theory, emphasizing that the key condition for validity is unbroken symmetry in the ground state. The paper also discusses extensions to multiple flavors and various compactifications, drawing analogies to orbifold cases and arguing that symmetry realization, not merely the presence of twisted sectors, governs the applicability of large-N mappings. Overall, the results establish that orientifold equivalence is contingent on specific symmetry dynamics and is not a universal feature of large-N gauge theories.

Abstract

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 non-perturbative 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 $R^3 \times S^1$, 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. 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 non-perturbative 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 $R^4$ (or $ R^3 \times S^1$ for sufficiently large radius) is not currently known.