[Paper Review] Marginal perturbations of N=4 Yang-Mills as deformations of AdS$_5 imes S^5$

Ansar Fayyazuddin, S. Mukhopadhyay|ArXiv.org|Apr 5, 2002

Black Holes and Theoretical Physics7 references8 citations

TL;DR

This paper constructs a one-parameter family of supersymmetric, constant-dilaton solutions in type IIB supergravity with 5-form and 3-form flux, preserving U(1)×Z₃ isometry, that realize a continuous deformation of the AdS₅×S⁵ background. The solution is dual to a marginal deformation of N=4 Yang-Mills theory preserving N=1 supersymmetry, with the N=4 theory recovered at a single point (d=0), providing a supergravity realization of a continuous line of conformal field theories beyond the original N=4 theory.

ABSTRACT

We study constant dilaton supersymmetric solutions of type IIB Supergravity with 5-form and 3-form flux with isometry group U(1) $ imes$$Z_3$. Some of these solutions correspond to marginal perturbations of N=4 Yang-Mills. We find one line of solutions in particular of AdS$_5$ fibred over an $S^5$. This line is described by a single complex parameter. AdS$_5 imes$S$^5$ is obtained when this partameter is tuned to zero.

Motivation & Objective

• To construct supersymmetric, constant-dilaton solutions in type IIB supergravity that are dual to marginal deformations of N=4 Yang-Mills theory.
• To identify a continuous family of supergravity backgrounds that generalize the AdS₅×S⁵ geometry while preserving supersymmetry.
• To explore the supergravity realization of the Leigh-Strassler marginal deformations, which reduce R-symmetry and preserve N=1 supersymmetry.
• To determine whether the full three-dimensional space of marginal couplings in Leigh-Strassler theory can be captured by supergravity solutions.
• To understand the role of discrete symmetries, particularly Z₃, in the supergravity duals and why they are not manifest in the constructed solutions.

Proposed method

• Solving the Killing spinor equations under the assumption of U(1)×Z₃ isometry to constrain the metric, gauge fields, and fluxes in type IIB supergravity.
• Imposing source equations and Bianchi identities on the supersymmetric field configurations to close the system of equations.
• Using a constrained Kähler metric and a holomorphic 3-form to describe the internal geometry, with the function Ω relating to the flux structure.
• Constructing a solution with a single complex parameter d, where d=0 corresponds to the standard AdS₅×S⁵ background.
• Verifying isometries including SO(4,2) (AdS isometry), U(1), and Z₃ under cyclic permutations of the chiral fields.
• Analyzing the Lie derivative of the metric and gauge field under special conformal transformations to confirm invariance.

Experimental results

Research questions

• RQ1 Can a continuous family of supersymmetric, constant-dilaton solutions in type IIB supergravity be constructed that generalize AdS₅×S⁵ and are dual to marginal deformations of N=4 Yang-Mills theory?
• RQ2 How does the single complex parameter d in the supergravity solution relate to the three marginal couplings λ₁, λ₂, λ₃ in the Leigh-Strassler superpotential?
• RQ3 Why is the second Z₃ symmetry (phase rotation) not preserved in the constructed supergravity solutions, despite being present in the field theory?
• RQ4 Can the full three-dimensional space of marginal couplings in Leigh-Strassler theory be realized in supergravity through more general ansätze?
• RQ5 What is the physical interpretation of the remaining marginal direction not captured by the current solution?

Key findings

• A one-parameter family of supersymmetric, constant-dilaton solutions in type IIB supergravity is constructed, with the AdS₅×S⁵ background recovered when the complex parameter d is set to zero.
• The solution preserves U(1)×Z₃ isometry, with Z₃ acting as cyclic permutation of the chiral fields, and exhibits SO(4,2) invariance corresponding to AdS isometries.
• The solution is dual to a continuous deformation of N=4 Yang-Mills theory that preserves N=1 supersymmetry and features two marginal couplings parameterized by τ and d.
• The supergravity solution is explicitly shown to be invariant under Poincaré, scale, and special conformal transformations via Lie derivative calculations on the metric and gauge field.
• The constructed solution does not preserve the second Z₃ symmetry (phase rotation) of the Leigh-Strassler theory, suggesting a possible discrete anomaly or symmetry realization issue in the dual field theory.
• The solution provides a supergravity realization of a continuous line of conformal field theories beyond N=4, though the full three-dimensional space of marginal deformations remains to be captured.