Lax Pair for Strings in Lunin-Maldacena Background

TL;DR

The paper develops a TsT-based approach to the Lunin–Maldacena $\beta$-deformed background with real deformation $\gamma$, deriving a local Lax pair for the bosonic string and revealing a precise map to $AdS_5\times S^5$ with twisted $U(1)$ boundary conditions. It provides explicit constructions of a $su(2)_\gamma$ subsector Lax pair and extends the deformation to a three-parameter family (and a $6+2$ parameter family) via successive TsT and SL(2,R) transformations, corresponding to nonsupersymmetric marginal deformations of ${\cal N}=4$ SYM. The results establish integrability in the deformed setting and lay groundwork for string Bethe equations and their gauge-theory counterparts, while highlighting rich connections to toric deformations and potential stability analyses. Overall, the work broadens the landscape of integrable AdS/CFT-compatible backgrounds and offers concrete tools for exploring their holographic duals.

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.