On Exactly Marginal Deformations of N=4 SYM and Type IIB Supergravity on AdS_5 * S^5

TL;DR

This paper investigates exactly marginal deformations of ${\cal N}=4$ SYM and their Type IIB AdS/CFT duals by perturbatively constructing deformed supergravity backgrounds around ${\rm AdS}_5\times S^5$. It identifies the field-theory marginality condition $(h\bar h)_{\bf 8}=0$ and translates it into a gravity obstruction that appears at third order in the deformation, with explicit second-order solutions verifying the AdS/CFT correspondence. The authors show that a logarithmic running of couplings at third order reproduces the leading beta-function behavior found in perturbative field theory, indicating a non-renormalization of the leading coefficient independent of the 't Hooft coupling in both weak and strong coupling limits. The moduli space near the ${\cal N}=4$ point is analyzed as ${\cal M}_c\simeq {\bf C}^2/{\cal T}$ with ${\cal T}=SL(2,{\bf Z}_3)$, connecting the gravity construction to the known Leigh-Strassler marginal deformations and suggesting a deeper non-renormalization structure in the AdS/CFT context. The results provide a concrete framework for studying ${\cal M}_c$ and pave the way for extensions to other AdS backgrounds such as ${\rm AdS}_4\times S^7$ and orbifolds.

Abstract

N=4 supersymmetric Yang-Mills theory with gauge group SU(n) (n>=3) is believed to have two exactly marginal deformations which break the supersymmetry to N=1. We discuss the construction of the string theory dual to these deformations, in the supergravity approximation, in a perturbation series around the AdS_5 * S^5 solution. We construct explicitly the deformed solution at second order in the deformation. We show that deformations which are marginal but not exactly marginal lead to a non-conformal solution with a logarithmically running coupling constant. Surprisingly, at third order in the deformation we find the same beta functions for the couplings in field theory and in supergravity, suggesting that the leading order beta functions (or anomalous dimensions) do not depend on the gauge coupling (the coefficient is not renormalized).