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Wilson loop correlators at strong coupling: from matrices to bubbling geometries

Jaume Gomis, Shunji Matsuura, Takuya Okuda, Diego Trancanelli

TL;DR

This work computes at strong coupling the large-$N$ correlation functions of supersymmetric Wilson loops in large representations with local ${\cal N}=4$ SYM operators, using a normal matrix model in field theory and bubbling geometries in AdS/CFT. The authors establish a Ward-identity–based relation between the Wilson loop correlator with the stress tensor and with a dimension-2 chiral primary, enabling a robust cross-check via S-duality: the semiclassical ’t Hooft-loop result maps exactly to the Wilson-loop result at strong coupling. They develop a detailed matrix-model framework (including complex and normal matrix models) to extract the moments that encode the Wilson-loop correlators, and show that the gauge-theory predictions precisely agree with the bubbling-supergravity computations through KK holography. The results provide a nontrivial quantitative test of electric-magnetic duality in ${\cal N}=4$ SYM and exemplify how bubbling geometries encode correlation data for Wilson loops in large representations. Overall, the paper strengthens the AdS/CFT dictionary for extended objects and illustrates exact gauge/gravity agreement across multiple complementary approaches.

Abstract

We compute at strong coupling the large N correlation functions of supersymmetric Wilson loops in large representations of the gauge group with local operators of N=4 super Yang-Mills. The gauge theory computation of these correlators is performed using matrix model techniques. We show that the strong coupling correlator of the Wilson loop with the stress tensor computed using the matrix model exactly matches the semiclassical computation of the correlator of the 't Hooft loop with the stress tensor, providing a non-trivial quantitative test of electric-magnetic duality of N=4 super Yang-Mills. We then perform these calculations using the dual bulk gravitational picture, where the Wilson loop is described by a "bubbling" geometry. By applying holographic methods to these backgrounds we calculate the Wilson loop correlation functions, finding perfect agreement with our gauge theory results.

Wilson loop correlators at strong coupling: from matrices to bubbling geometries

TL;DR

This work computes at strong coupling the large- correlation functions of supersymmetric Wilson loops in large representations with local SYM operators, using a normal matrix model in field theory and bubbling geometries in AdS/CFT. The authors establish a Ward-identity–based relation between the Wilson loop correlator with the stress tensor and with a dimension-2 chiral primary, enabling a robust cross-check via S-duality: the semiclassical ’t Hooft-loop result maps exactly to the Wilson-loop result at strong coupling. They develop a detailed matrix-model framework (including complex and normal matrix models) to extract the moments that encode the Wilson-loop correlators, and show that the gauge-theory predictions precisely agree with the bubbling-supergravity computations through KK holography. The results provide a nontrivial quantitative test of electric-magnetic duality in SYM and exemplify how bubbling geometries encode correlation data for Wilson loops in large representations. Overall, the paper strengthens the AdS/CFT dictionary for extended objects and illustrates exact gauge/gravity agreement across multiple complementary approaches.

Abstract

We compute at strong coupling the large N correlation functions of supersymmetric Wilson loops in large representations of the gauge group with local operators of N=4 super Yang-Mills. The gauge theory computation of these correlators is performed using matrix model techniques. We show that the strong coupling correlator of the Wilson loop with the stress tensor computed using the matrix model exactly matches the semiclassical computation of the correlator of the 't Hooft loop with the stress tensor, providing a non-trivial quantitative test of electric-magnetic duality of N=4 super Yang-Mills. We then perform these calculations using the dual bulk gravitational picture, where the Wilson loop is described by a "bubbling" geometry. By applying holographic methods to these backgrounds we calculate the Wilson loop correlation functions, finding perfect agreement with our gauge theory results.

Paper Structure

This paper contains 23 sections, 173 equations, 2 figures.

Table of Contents

  1. Introduction and outline
  2. Outline of the paper
  3. Symmetry analysis of the Wilson loop correlators
  4. Correlators with chiral primary operators
  5. Correlator with the stress tensor
  6. Correlator with the stress tensor from Ward identities
  7. Correlators from gauge theory
  8. Correlators from a matrix model
  9. Complex and normal matrix models
  10. Large N solution of the normal matrix model
  11. Correlators as moments in the normal matrix model
  12. Correlator of Wilson loop with the stress tensor from S-duality
  13. Correlators from supergravity
  14. Review of the bubbling solution
  15. Kaluza-Klein holography
  16. ...and 8 more sections

Figures (2)

  • Figure 1: We depict here, rotated and inverted, the Young tableau $R$ of the irreducible representation of $U(N)$ in which we take the Wilson loop operator. The tableau consists of $g$ blocks, the $I$-th one of them having $n_I$ rows of length $K_I$. All the edges of the diagram are taken to be long, meaning that $n_I$ and $K_I$ are both of order ${\cal O}(N)$ for all $I$. This guarantees that the dual bubbling geometry has small curvature everywhere.
  • Figure 2: (a) The branch cuts $[e_{2I}, e_{2I-1}]$ ($I=1,\ldots, g+1$) on the $\zeta$ plane ${\mathbb C}$ for the the hyperelliptic curve (\ref{['hyperel']}) in the $g=2$ case. (b) The eigenvalue density $\rho(\xi)$ of the hermitian matrix model (\ref{['Hermitian-R']}). (c) The corresponding eigenvalue distribution (droplets) of the normal matrix model (\ref{['Normal-RJ']}). The shape of a droplet is given by $\rho(\xi)$.