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Spectral curves, emergent geometry, and bubbling solutions for Wilson loops

Takuya Okuda, Diego Trancanelli

TL;DR

This paper analyzes supersymmetric circular Wilson loops in ${\mathcal N}=4$ SYM for large representations, deriving the Gaussian matrix-model spectral curve and showing it is a genus-$g$ hyperelliptic surface encoding the bubbling geometry characteristic of the dual gravity solution. The authors construct the resolvent via a meromorphic one-form on the spectral curve, relate holomorphic data ${\\mathcal A},{\\mathcal B}$ to matrix-model quantities, and match the curve to the bubbling geometry described by ${AdS_2\times S^2\times S^4}$ on a hyperelliptic base. They compute the Wilson loop expectation value from the matrix model in the large-$N$ limit and discuss the gravity side, proving the on-shell Lagrangian is a total derivative and evaluating contributions from the new cycles, while acknowledging the need for holographic renormalization to fully complete the gravity–gauge correspondence. The results provide a concrete example of emergent geometry from eigenvalue distributions and establish a precise gauge/gravity dictionary, with further work on boundary counterterms and correlators explored in companion work. Overall, the paper advances the understanding of how matrix-model spectral data encode bubbling holography for Wilson loops and lays groundwork for complete gravity computations of loop observables.

Abstract

We study the supersymmetric circular Wilson loops of N=4 super Yang-Mills in large representations of the gauge group. In particular, we obtain the spectral curves of the matrix model which captures the expectation value of the loops. These spectral curves are then proven to be precisely the hyperelliptic surfaces that characterize the bubbling solutions dual to the Wilson loops, thus yielding an example of a geometry emerging from an eigenvalue distribution. We finally discuss the Wilson loop expectation value from the matrix model and from supergravity.

Spectral curves, emergent geometry, and bubbling solutions for Wilson loops

TL;DR

This paper analyzes supersymmetric circular Wilson loops in SYM for large representations, deriving the Gaussian matrix-model spectral curve and showing it is a genus- hyperelliptic surface encoding the bubbling geometry characteristic of the dual gravity solution. The authors construct the resolvent via a meromorphic one-form on the spectral curve, relate holomorphic data to matrix-model quantities, and match the curve to the bubbling geometry described by on a hyperelliptic base. They compute the Wilson loop expectation value from the matrix model in the large- limit and discuss the gravity side, proving the on-shell Lagrangian is a total derivative and evaluating contributions from the new cycles, while acknowledging the need for holographic renormalization to fully complete the gravity–gauge correspondence. The results provide a concrete example of emergent geometry from eigenvalue distributions and establish a precise gauge/gravity dictionary, with further work on boundary counterterms and correlators explored in companion work. Overall, the paper advances the understanding of how matrix-model spectral data encode bubbling holography for Wilson loops and lays groundwork for complete gravity computations of loop observables.

Abstract

We study the supersymmetric circular Wilson loops of N=4 super Yang-Mills in large representations of the gauge group. In particular, we obtain the spectral curves of the matrix model which captures the expectation value of the loops. These spectral curves are then proven to be precisely the hyperelliptic surfaces that characterize the bubbling solutions dual to the Wilson loops, thus yielding an example of a geometry emerging from an eigenvalue distribution. We finally discuss the Wilson loop expectation value from the matrix model and from supergravity.

Paper Structure

This paper contains 15 sections, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectral curves and bubbling geometries
  3. A hyperelliptic surface as the spectral curve
  4. Parameters and constraints
  5. Comparison
  6. SU(N) gauge group
  7. Wilson loop expectation value
  8. Wilson loop expectation value from the matrix model
  9. Wilson loop expectation value from supergravity
  10. The on-shell Lagrangian is a total derivative
  11. Contributions from new cycles
  12. Conclusion
  13. Details on the bubbling geometry
  14. An explicit expression for the four-form
  15. Asymptotic behavior

Figures (2)

  • Figure 1: The Young tableau $R$ is shown rotated and inverted. It consists of $g$ blocks, the $I$-th one of them having $n_I$ rows of length $K_I$. We set $K_{g+1}\equiv 0$ and $n_{g+1}\equiv N-\sum_{I=1}^g n_I$.
  • Figure 2: The $A$- and $B$-cycles of the hyperelliptic surface $\Sigma$ of genus $g=2$.