Volume independence for Yang-Mills fields on the twisted torus

TL;DR

This paper reviews the concept of volume independence in SU(N) Yang-Mills theories on twisted tori, highlighting perturbative and non-perturbative results, and discussing stability issues and numerical findings in lower dimensions.

Contribution

It provides a comprehensive review of volume independence in gauge theories on twisted tori, including perturbative, non-perturbative, and numerical insights, with emphasis on stability and non-commutative relations.

Findings

Volume independence depends on the combination l N^{2/d} for even d.

Perturbative results are valid to all orders under certain twist conditions.

Numerical results in 2+1 dimensions support non-perturbative aspects.

Abstract

We review some recent results related to the notion of volume independence in SU(N) Yang-Mills theories. The topic is discussed in the context of gauge theories living on a d-dimensional torus with twisted boundary conditions. After a brief introduction reviewing the formalism for introducing gauge fields on a torus, we discuss how volume independence arises in perturbation theory. We show how, for appropriately chosen twist tensors, perturbative results to all orders in the 't Hooft coupling depend on a specific combination of the rank of the gauge group (N) and the periods of the torus (l) given by l N^{2/d}, for d even.We discuss the well-known relation to non-commutative field theories and address certain threats to volume independence associated to the occurrence of tachyonic instabilities at one-loop order. We end by presenting some numerical results in 2+1 dimensions that extend these ideas to the non-perturbative domain.