Large-N volume independence in conformal and confining gauge theories

TL;DR

Large-$N$ volume independence asserts that certain gauge theories on ${\mathbb R}^{d-k} \times (S^1)^k$ retain their IR, scale-invariant, and long-distance properties independent of the compactification radii provided center and translation symmetries remain unbroken. The authors develop a physical picture based on holonomy eigenvalues and the resulting KK spectrum, highlighting distinct regimes where the KK spacing is $2\pi/(NL)$ or vanishing as $N\to\infty$, and showing how this underpins volume independence for both confining and conformal theories. They apply the framework to ${\cal N}=4$ SYM and massive QCD(adj), revealing a rich phase structure and nontrivial limits where infinite-volume behavior can be inferred from small-volume data, especially in the large-$N$ limit. The results have practical implications for lattice studies of the conformal window and for reductions to matrix models, offering a revised, $N$-dependent perspective on when finite-volume effects are suppressed.

Abstract

Consequences of large $N$ volume independence are examined in conformal and confining gauge theories. In the large $N$ limit, gauge theories compactified on $\R^{d-k} \times (S^1)^k$ are independent of the $S^1$ radii, provided the theory has unbroken center symmetry. In particular, this implies that a large $N$ gauge theory which, on $\R^d$, flows to an IR fixed point, retains the infinite correlation length and other scale invariant properties of the decompactified theory even when compactified on $\R^{d-k} \times (S^1)^k$. In other words, finite volume effects are $1/N$ suppressed. In lattice formulations of vector-like theories, this implies that numerical studies to determine the boundary between confined and conformal phases may be performed on one-site lattice models. In $N=4$ supersymmetric Yang-Mills theory, the center symmetry realization is a matter of choice: the theory on $\R^{4-k}\times (S^1)^k$ has a moduli space which contains points with all possible realizations of center symmetry. Large $N$ QCD with massive adjoint fermions and one or two compactified dimensions has a rich phase structure with an infinite number of phase transitions coalescing in the zero radius limit.