[Paper Review] The String Dual of a Confining Four-Dimensional Gauge Theory

Joseph Polchinski, Matthew J. Strassler|ArXiv.org|Mar 15, 2000

Mathematics and ApplicationsMathematics412 citations

TL;DR

This paper establishes the first supergravity dual of a four-dimensional confining gauge theory by showing that mass-deformed N=4 Yang-Mills theory (N=1*) has a nonsingular string theory dual via Myers' dielectric effect, where D3-branes polarize into D5-branes wrapping S² in AdS₅×S⁵. The confining vacuum features an NS5-brane with a fundamental string bound to it, providing a quantitative description of flux tubes, baryons, and other nonperturbative phenomena.

ABSTRACT

We study N=1 gauge theories obtained by adding finite mass terms to N=4 Yang-Mills theory. The Maldacena dual is nonsingular: in each of the many vacua, there is an extended brane source, arising from Myers' dielectric effect. The source consists of one or more (p,q) 5-branes. In particular, the confining vacuum contains an NS5-brane; the confining flux tube is a fundamental string bound to the 5-brane. The system admits a simple quantitative description as a perturbation of a state on the N=4 Coulomb branch. Various nonperturbative phenomena, including flux tubes, baryon vertices, domain walls, condensates and instantons, have new, quantitatively precise, dual descriptions. We also briefly consider two QCD-like theories. Our method extends to the nonsupersymmetric case. As expected, the N=4 matter cannot be decoupled within the supergravity regime.

Motivation & Objective

• To resolve the apparent naked singularity in the Maldacena dual of a confining N=4 Yang-Mills theory with mass deformations.
• To show that the singularity is actually a smooth extended brane source due to Myers' dielectric effect in the presence of Ramond-Ramond flux.
• To provide a quantitative, supergravity-based description of nonperturbative phenomena—flux tubes, baryon vertices, domain walls, condensates, and instantons—in confining N=1* gauge theories.
• To extend the method to nonsupersymmetric QCD-like theories and demonstrate the stability of the construction under small supersymmetry-breaking masses.
• To establish a bridge between field theory and string theory by showing that holomorphic and BPS quantities can be matched across different regimes using the dual description.

Proposed method

• Use the Maldacena duality to map N=1* gauge theory (mass-deformed N=4) to IIB string theory on AdS₅×S⁵ with a perturbation from finite mass terms.
• Apply Myers' dielectric effect: in the presence of RR 3-form flux (dual to mass terms), D3-branes polarize into D5-branes wrapping S², forming a smooth extended source instead of a singularity.
• Construct supergravity solutions where each vacuum of the N=1* theory corresponds to multiple D5-branes with D3-brane charge n_i, each at a distinct AdS radius r_i ∝ n_i.
• Identify the confining flux tube as a fundamental string bound to the NS5-brane, with a tension calculable from the geometry.
• Use the D5-brane worldvolume theory to describe baryon vertices (as D3-brane boundaries on the NS5-brane sphere), domain walls (as 5-branes interpolating between vacua), and instantons (as strings wrapping the S²).
• Leverage the SL(2,Z) duality of the N=4 theory to explore the full vacuum structure and relate different vacua via dual descriptions.

Experimental results

Research questions

• RQ1 Can the Maldacena dual of a confining N=4 Yang-Mills theory with mass deformations be free of naked singularities?
• RQ2 How do nonperturbative field theory phenomena—flux tubes, baryons, condensates, instantons—emerge in the supergravity dual?
• RQ3 What is the role of Myers' dielectric effect in resolving the apparent singularity in the dual geometry?
• RQ4 How do the multiple vacua of the N=1* theory map to distinct supergravity solutions with D5-branes at different AdS radii?
• RQ5 Can this construction be extended to nonsupersymmetric QCD-like theories, and what is the stability of the dual description under small supersymmetry breaking?

Key findings

• The apparent naked singularity in the supergravity dual of the N=1* theory is resolved by the formation of a smooth, extended D5-brane source via Myers' dielectric effect.
• The confining vacuum contains an NS5-brane, and the flux tube is a fundamental string bound to this NS5-brane, with a tension that can be computed from the geometry.
• Each vacuum of the N=1* theory corresponds to a distinct supergravity solution with multiple D5-branes, each with D3-brane charge n_i and located at an AdS radius r_i ∝ n_i.
• Baryon vertices are described as D3-branes ending on the NS5-brane's S², with the boundary of the D3-brane on the sphere acting as the baryon vertex.
• Domain walls are realized as 5-branes interpolating between different vacua, with the transition mediated by the D5-brane structure.
• Instantons in the large-N limit are resummed into strings wrapping the NS5-brane's two-sphere, providing a dual description of the instanton expansion of the superpotential.