Near BPS Wilson Loop in beta-deformed Theories

Abstract

We propose a definition of the Wilson loop operator in the N=1 beta-deformed supersymmetric Yang-Mills theory. Although the operator is not BPS, it has a finite expectation value at least up to order (g^2 N)^2. This does not happen generally for a generic non-BPS Wilson loop whose expectation value is UV divergent. For this reason we call this a near-BPS Wilson loop and conjecture that its exact expectation value is finite. We derive the general form of the boundary condition satisfied by the dual string worldsheet and find that it is deformed. Finiteness of the expectation value of the Wilson loop, together with some rather remarkable properties of the Lunin-Maldacena metric and the B-field, fixes the boundary condition to be one which is characterized by the vielbein of the deformed supergravity metric. The Wilson loop operators provide natural candidates as dual descriptions to some of the existing D-brane configurations in the Lunin-Maldacena background. We also construct the string dual configuration for a near-1/4 BPS circular Wilson loop operator. The string lies on a deformed three-sphere instead of a two-sphere as in the undeformed case. The expectation value of the Wilson loop operator is computed using the AdS/CFT correspondence and is found to be independent of the deformation. We conjecture that the exact expectation value of the Wilson loop is given by the same matrix model as in the undeformed case.