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

As shown by Hashimoto and Itzhaki in hep-th/9911057, the perturbative degre es 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 resor t to a Morita equivalent (or T-dual) description appropriate for that energ y. In this note, we study NCYM on a non-commutative torus with an irrationa l deformation parameter theta. In that case, an infinite tower of dual desc riptions is generically needed in order to describe the UV regime. We const ruct a hierarchy of dual descriptions in terms of the continued fraction ap proximations of theta. We encounter different descriptions depending on the level of the irrationality of theta and the amount of non-locality tolerat ed. The behavior turns out to be isomorphic to that found for the phase str ucture of the four-dimensional Villain Z(N) lattice gauge theories, which w e revisit as a warm-up. At large 't Hooft coupling, using the AdS/CFT corre spondance, we find that there are domains of the radial coordinate U where no T-dual description makes the derivative expansion converge. The radial d irection obtains multifractal characteristics near the boundary of AdS.