Phases of $\Nc= \infty$ QCD-like gauge theories on $S^3 \times S^1$ and nonperturbative orbifold-orientifold equivalences

TL;DR

The paper investigates phase diagrams of large-$N$ QCD-like theories with two-index fermions on $S^3\times S^1$, revealing rich structures where center and chiral symmetries may realize independently and where nonperturbative orbifold/orientifold equivalences relate distinct theories. It develops a matrix-model-like one-loop effective action for Wilson lines, analyzes both thermal and twisted partition functions, and shows that (i) perturbative deconfinement temperatures coincide across SYM and QCD-like relatives, and (ii) symmetry realizations depend crucially on boundary conditions along $S^1$. The work introduces a symmetry-based framework to map phases between orbifold BF and orientifold AS/S theories and ${\cal N}=1$ SYM, with large-$N$ equivalence holding in neutral sectors when the relevant $\mathbb{Z}_2$ symmetries remain unbroken. It also demonstrates how center-symmetry transitions on $S^3\times S^1$ potentially extrapolate smoothly to strong-coupling transitions on ${\mathbb R}^3\times S^1$, and discusses chiral vs center-symmetry disentanglement, chiral condensate behavior, and topological order-parameter classifications. Overall, the paper proposes a coherent large-$N$ perspective for predicting and understanding phase structures across related QCD-like theories with practical implications for lattice studies and holographic extensions.

Abstract

We study the phase diagrams of $\Nc= \infty$ vector-like, asymptotically free gauge theories as a function of volume, on $S^3\times S^1$. The theories of interest are the ones with fermions in two index representations [adjoint, (anti)symmetric, and bifundamental abbreviated as QCD(adj), QCD(AS/S) and QCD(BF)], and are interrelated via orbifold or orientifold projections. The phase diagrams reveal interesting phenomena such as disentangled realizations of chiral and center symmetry, confinement without chiral symmetry breaking, zero temperature chiral transitions, and in some cases, exotic phases which spontaneously break the discrete symmetries such as C, P, T as well as CPT. In a regime where the theories are perturbative, the deconfinement temperature in SYM, and QCD(AS/S/BF) coincide. The thermal phase diagrams of thermal orbifold QCD(BF), orientifold QCD(AS/S), and $\N=1$ SYM coincide, provided charge conjugation symmetry for QCD(AS/S) and $\Z_2$ interchange symmetry of the QCD(BF) are not broken in the phase continously connected to $\R^4$ limit. When the $S^1$ circle is endowed with periodic boundary conditions, the (nonthermal) phase diagrams of orbifold and orientifold QCD are still the same, however, both theories possess chirally symmetric phases which are absent in $\None$ SYM. The match and mismatch of the phase diagrams depending on the spin structure of fermions along the $S^1$ circle is naturally explained in terms of the necessary and sufficient symmetry realization conditions which determine the validity of the nonperturbative orbifold orientifold equivalence.