Large-N volume independence in conformal and confining gauge theories

Abstract

Consequences of large N volume independence are examined in conformal and confining gauge theories. In the large N limit, gauge theories compactified on \( {\mathbb{R}^{d - k}} \times {\left( {{S^1}} \right)^k} \) 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 \( {\mathbb{R}^d} \), flowstoan IR fixed point, retains the infinite correlation length and other scale invariant properties of the decompactified theory even when compactified on \( {\mathbb{R}^{d - k}} \times {\left( {{S^1}} \right)^k} \). 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 \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory, the center symmetry realization is a matter of choice: the theory on \( {\mathbb{R}^{4 - k}} \times {\left( {{S^1}} \right)^k} \) 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.