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The String Dual of a Confining Four-Dimensional Gauge Theory

Joseph Polchinski, Matthew J. Strassler

TL;DR

This work provides the first concrete holographic description of a four-dimensional confining gauge theory via the ${ m N}=1^*$ mass deformation of ${ m N}=4$ Super Yang–Mills. By activating Myers dielectric effects, the dual geometry avoids naked singularities, replacing them with extended 5-brane sources and enabling calculable descriptions of flux tubes, baryons, domain walls, and various condensates. The analysis connects perturbations around the ${ m AdS}_5 imes{ m S}^5$ background to field-theory operators, and demonstrates a rich network of dual descriptions across D5-NS5 5-branes, $(c,d)$ bound states, and S-/T-dual frames. It also outlines extensions to QCD-like theories and nonsupersymmetric cases, offering a blueprint for using brane dynamics to study nonperturbative phenomena in strongly coupled gauge theories. The results illuminate how holography can encode confinement and nonperturbative physics in a controlled setting, with implications for brane physics, moduli spaces, and potential links to phenomenological QCD-like models.

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.

The String Dual of a Confining Four-Dimensional Gauge Theory

TL;DR

This work provides the first concrete holographic description of a four-dimensional confining gauge theory via the mass deformation of Super Yang–Mills. By activating Myers dielectric effects, the dual geometry avoids naked singularities, replacing them with extended 5-brane sources and enabling calculable descriptions of flux tubes, baryons, domain walls, and various condensates. The analysis connects perturbations around the background to field-theory operators, and demonstrates a rich network of dual descriptions across D5-NS5 5-branes, bound states, and S-/T-dual frames. It also outlines extensions to QCD-like theories and nonsupersymmetric cases, offering a blueprint for using brane dynamics to study nonperturbative phenomena in strongly coupled gauge theories. The results illuminate how holography can encode confinement and nonperturbative physics in a controlled setting, with implications for brane physics, moduli spaces, and potential links to phenomenological QCD-like models.

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.

Paper Structure

This paper contains 35 sections, 163 equations, 3 figures.

Table of Contents

  1. Introduction
  2. ${\cal N}=1^*$ Ground States
  3. Field Theory Background
  4. Brane Representations
  5. Perturbations on $A{ d}S_5 \times S^5$
  6. $AdS$/CFT and its Deformations
  7. Field Equations and Background
  8. Fermion Masses and Tensor Spherical Harmonics
  9. Linearized Solutions
  10. Five-brane Probes
  11. The D5 Probe Action
  12. The $(c,d)$ Probe Action
  13. The Probe Potential and Minima
  14. The Full Problem
  15. The Warped Geometry
  16. ...and 20 more sections

Figures (3)

  • Figure 1: D4/D2 system projected on the 45-plane. The D4-brane, which is also extended in 0123, wraps multiple times ($\tan\theta$). The D2-brane, which is oriented in the 05 directions, can partly dissolve, leaving a piece connecting two leaves of the D4-brane.
  • Figure 2: a) A small baryon vertex, at $r> r_0$: the $S^5$ baryon vertex is outside the $S^2$ of the vacuum brane (the vacuum brane is also extended in 123, while the baryon is not). b) The baryon $S^5$ contracted to $r < r_0$: a 3-brane (shaded) has been created. c) The $S^5$ has contracted to nothing, leaving a 3-brane baryon filling the $S^2$.
  • Figure 3: Triple 5-brane junction, corresponding to a domain wall. Depicted is $z(x^1)$; the full geometry is obtained by translating in $x^{2,3}$ and rotating in the transverse $SO(3)$ symmetry. At $x^1 < 0$ the system is in the vacuum corresponding to 5-brane A; at $x^1 > 0$ it is in the vacuum corresponding to 5-brane B, with different radius and orientation. The domain wall lies in the shaded plane $x^1 = 0$ and is a 5-brane of type C. The bending of branes A and B is described by the BPS equation (\ref{['dwbps']}).