On the short-distance structure of irrational non-commutative gauge theories

TL;DR

This paper investigates the complex short-distance structure of irrational non-commutative gauge theories, revealing a hierarchy of dual descriptions and multifractal features in the AdS/CFT framework.

Contribution

It introduces a hierarchy of dual descriptions for irrational non-commutative Yang-Mills theories based on continued fraction approximations of the deformation parameter.

Findings

Infinite tower of dual descriptions needed for UV regime

Multifractal characteristics of the radial coordinate in AdS

Behavior analogous to Villain Z_N lattice gauge theories

Abstract

As shown by Hashimoto and Itzhaki in hep-th/9911057, the perturbative degrees of freedom of a non-commutative Yang-Mills theory (NCYM) on a torus are quasi-local only in a finite energy range. Outside that range one may resort to a Morita equivalent (or T-dual) description appropriate for that energy. In this note, we study NCYM on a non-commutative torus with an irrational deformation parameter $\theta$. In that case, an infinite tower of dual descriptions is generically needed in order to describe the UV regime. We construct a hierarchy of dual descriptions in terms of the continued fraction approximations of $\theta$. We encounter different descriptions depending on the level of the irrationality of $\theta$ and the amount of non-locality tolerated. The behavior turns out to be isomorphic to that found for the phase structure of the four-dimensional Villain $Z_N$ lattice gauge theories, which we revisit as a warm-up. At large 't Hooft coupling, using the AdS/CFT correspondance, we find that there are domains of the radial coordinate $U$ where no T-dual description makes the derivative expansion converge. The radial direction obtains multifractal characteristics near the boundary of AdS.