Lax Pair for Strings in Lunin-Maldacena Background

TL;DR

This paper constructs a Lax pair for string theory in beta-deformed AdS_5 x S^5 backgrounds, extending integrability techniques to deformed supergravity solutions and exploring their dual gauge theories.

Contribution

It derives a local Lax pair for strings in the beta-deformed background using TsT transformations, and introduces multi-parameter deformations of AdS_5 x S^5.

Findings

Lax pair for bosonic strings in beta-deformed background

Multi-parameter deformations of AdS_5 x S^5

Correspondence between solutions and twisted boundary conditions

Abstract

Recently Lunin and Maldacena used an SL(3,R) transformation of the AdS_5 x S^5 background to generate a supergravity solution dual to a so-called beta-deformation of N = 4 super Yang-Mills theory. We use a T-duality-shift-T-duality (TsT) transformation to obtain the beta-deformed background for real beta, and show that solutions of string theory equations of motion in this background are in one-to-one correspondence with those in AdS_5 x S^5 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 beta-deformed background. We also perform a chain of three consecutive TsT transformations to generate a three-parameter deformation of AdS_5 x S^5. The three-parameter background is dual to a nonsupersymmetric marginal deformation of N=4 SYM. Finally, we combine the TsT transformations with SL(2,R) ones to obtain a 6+2 parameter deformation of AdS_5 x S^5.