Lax pair for strings in Lunin-Maldacena background

Abstract

Recently Lunin and Maldacena used an SL(3,) transformation of the AdS5 × S5 background to generate a supergravity solution dual to a so-called β-deformation of = 4 super Yang-Mills theory. We use a T-duality-shift-T-duality (TsT) transformation to obtain the β-deformed background for real β ≡ γ, and show that solutions of string theory equations of motion in this background are in one-to-one correspondence with those in AdS5 × S5 with twisted boundary conditions imposed on the U(1) isometry fields. We then apply the TsT transformation to derive a local and periodic Lax pair for the bosonic part of string theory in the γ-deformed background. We also perform a chain of three consecutive TsT transformations to generate a three-parameter deformation of AdS5 × S5. The three-parameter background is dual to a nonsupersymmetric marginal deformation of = 4 SYM. Finally, we combine the TsT transformations with SL(2,) ones to obtain a 6+2 parameter deformation of AdS5 × S5.