Gauge-string duality for superconformal deformations of N=4 Super Yang-Mills theory

TL;DR

The paper demonstrates that the remarkable gauge–string correspondence for ${\cal N}=4$ SYM persists under a real beta deformation, by establishing an explicit map between semiclassical string states in the TsT-generated Lunin–Maldacena background and 1-loop anomalous dimensions of 2-scalar operators in the beta-deformed ${\cal N}=4$ SYM. It shows that integrable structures survive: the deformed worldsheet theory, the Landau–Lifshitz action for coherent spin states, and the XXZ-type spin chain all align via a Bethe Ansatz framework in the thermodynamic limit and for 1/J corrections. The authors derive string Bethe equations from the Lax representation and verify that they match the thermodynamic Bethe equations of the gamma-deformed spin chain at one- and two-loop orders in the ${\rm su}(2)_\gamma$ sector, with 1/J corrections agreeing between the sigma-model and spin-chain pictures. These results suggest a broader applicability of integrable structures under marginal deformations of ${\cal N}=4$ SYM and hint at a unified description of semiclassical strings in a large class of deformed backgrounds.

Abstract

We analyze in detail the relation between an exactly marginal deformation of N=4 SYM - the Leigh-Strassler or ``beta-deformation'' - and its string theory dual (recently constructed in hep-th/0502086) by comparing energies of semiclassical strings to anomalous dimensions of gauge-theory operators in the two-scalar sector. We stress the existence of integrable structures on the two sides of the duality. In particular, we argue that the integrability of strings in AdS_5 x S^5 implies the integrability of the deformed world sheet theory with real deformation parameter. We compare the fast string limit of the worldsheet action in the sector with two angular momenta with the continuum limit of the coherent state action of an anisotropic XXZ spin chain describing the one-loop anomalous dimensions of the corresponding operators and find a remarkable agreement for all values of the deformation parameter. We discuss some of the properties of the Bethe Ansatz for this spin chain, solve the Bethe equations for small number of excitations and comment on higher loop properties of the dilatation operator. With the goal of going beyond the leading order in the 't Hooft expansion we derive the analog of the Bethe equations on the string-theory side, and show that they coincide with the thermodynamic limit of the Bethe equations for the spin chain. We also compute the 1/J corrections to the anomalous dimensions of operators with large R-charge (corresponding to strings with angular momentum J) and match them to the 1-loop corrections to the fast string energies. Our results suggest that the impressive agreement between the gauge theory and semiclassical strings in AdS_5 x S^5 is part of a larger picture underlying the gauge/gravity duality.