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Wilson Loops and Minimal Surfaces

Nadav Drukker, David J. Gross, Hirosi Ooguri

TL;DR

The paper tests the AdS/CFT proposal by relating Wilson loops in N=4 SYM to minimal surfaces in AdS5×S^5 and by deriving the loop equation in both bosonic and supersymmetric settings. It shows that for a class of BPS loops, the area diverges but a Legendre transform yields a finite effective action that respects the loop equation; it analyzes boundary conditions, zig-zag symmetry, and the role of cusps and intersections. It then assesses how minimal-surface computations in AdS5×S^5 reproduce loop-equation predictions, finding agreement for smooth loops and nontrivial cusp contributions, with non-BPS loops requiring excited-string states and remaining challenging. Overall, the work provides substantial evidence for the Wilson loop – minimal surface correspondence while highlighting outstanding issues for general, non-BPS loops.

Abstract

The AdS/CFT correspondence suggests that the Wilson loop of the large N gauge theory with N=4 supersymmetry in 4 dimensions is described by a minimal surface in AdS_5 x S^5. We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultra-violet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in AdS_5 x S^5 gives a solution of the equation. We also discuss the zig-zag symmetry of the loop operator. In the N=4 gauge theory, we expect the zig-zag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface.

Wilson Loops and Minimal Surfaces

TL;DR

The paper tests the AdS/CFT proposal by relating Wilson loops in N=4 SYM to minimal surfaces in AdS5×S^5 and by deriving the loop equation in both bosonic and supersymmetric settings. It shows that for a class of BPS loops, the area diverges but a Legendre transform yields a finite effective action that respects the loop equation; it analyzes boundary conditions, zig-zag symmetry, and the role of cusps and intersections. It then assesses how minimal-surface computations in AdS5×S^5 reproduce loop-equation predictions, finding agreement for smooth loops and nontrivial cusp contributions, with non-BPS loops requiring excited-string states and remaining challenging. Overall, the work provides substantial evidence for the Wilson loop – minimal surface correspondence while highlighting outstanding issues for general, non-BPS loops.

Abstract

The AdS/CFT correspondence suggests that the Wilson loop of the large N gauge theory with N=4 supersymmetry in 4 dimensions is described by a minimal surface in AdS_5 x S^5. We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultra-violet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in AdS_5 x S^5 gives a solution of the equation. We also discuss the zig-zag symmetry of the loop operator. In the N=4 gauge theory, we expect the zig-zag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface.

Paper Structure

This paper contains 24 sections, 131 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Wilson Loops in N=4 Gauge Theory
  3. Definition
  4. Perturbation Theory
  5. Minimal surfaces in Anti de-Sitter Space
  6. Boundary Conditions and BPS Loop
  7. Calculating the Area
  8. Legendre Transformation
  9. Zig-Zag Symmetry
  10. Removing the Constraint
  11. The Loop Equation
  12. Bosonic Theories
  13. Supersymmetric Case
  14. Predictions
  15. Loop Equation in $AdS_5 \times S^5$
  16. ...and 9 more sections

Figures (7)

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  • ...and 2 more figures