Volume independence in large Nc QCD-like gauge theories

TL;DR

This work unifies and extends the concept of large-$N_c$ volume independence by casting it as a generalized orbifold equivalence, encompassing spacetime and theory-space reductions. It shows that the validity of these equivalences hinges on unbroken discrete symmetries, with adjoint matter (under periodic boundary conditions) preserving center symmetry and enabling zero-volume reduction, while fundamental and tensor representations require careful handling via quiver constructions or special conditions. By connecting volume reduction, orientifold dualities, and quiver theory space, the paper provides a framework whereby small-volume or reduced models can reproduce leading large-$N_c$ dynamics of widely different theories in infinite volume, with concrete mappings for expectation values and connected correlators in neutral sectors. The results have practical implications for studying nonperturbative large-$N_c$ QCD and related gauge theories, potentially enabling more efficient analytic and numerical investigations by exploiting reductions in either spacetime volume or theory space.

Abstract

Volume independence in large $\Nc$ gauge theories may be viewed as a generalized orbifold equivalence. The reduction to zero volume (or Eguchi-Kawai reduction) is a special case of this equivalence. So is temperature independence in confining phases. In pure Yang-Mills theory, the failure of volume independence for sufficiently small volumes (at weak coupling) due to spontaneous breaking of center symmetry, together with its validity above a critical size, nicely illustrate the symmetry realization conditions which are both necessary and sufficient for large $\Nc$ orbifold equivalence. The existence of a minimal size below which volume independence fails also applies to Yang-Mills theory with antisymmetric representation fermions [QCD(AS)]. However, in Yang-Mills theory with adjoint representation fermions [QCD(Adj)], endowed with periodic boundary conditions, volume independence remains valid down to arbitrarily small size. In sufficiently large volumes, QCD(Adj) and QCD(AS) have a large $\Nc$ ``orientifold'' equivalence, provided charge conjugation symmetry is unbroken in the latter theory. Therefore, via a combined orbifold-orientifold mapping, a well-defined large $\Nc$ equivalence exists between QCD(AS) in large, or infinite, volume and QCD(Adj) in arbitrarily small volume. Since asymptotically free gauge theories, such as QCD(Adj), are much easier to study (analytically or numerically) in small volume, this equivalence should allow greater understanding of large $\Nc$ QCD in infinite volume.