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Calibrated Surfaces and Supersymmetric Wilson Loops

A. Dymarsky, S. Gubser, Z. Guralnik, J. Maldacena

TL;DR

This work identifies calibrated, pseudo-holomorphic minimal surfaces in $AdS_5\times S^5$ that end on supersymmetric Wilson loops and shows their finite-area contribution vanishes, in line with non-renormalization theorems. By constructing a calibration two-form $J$ with an associated almost complex structure, the authors recast the Wilson-loop problem into finding pseudo-holomorphic curves, enabling exact area relations $\mathrm{Area}(\Sigma)=\int J = (1/\epsilon) \oint |\dot x|$ and hence $A_{\rm reg}=0$ for these loops. They analyze existence, counting, and supersymmetry of solutions, and provide explicit U(1) isometry-based ansätze leading to annulus and single-boundary surfaces, including coincident-boundary configurations that exhibit zero modes. The results connect BRST/topological twisting perspectives with geometric calibration in the gravity dual, reinforcing the non-renormalization paradigm for a broad class of BPS Wilson loops. Altogether, the paper advances the geometric and topological understanding of AdS/CFT for supersymmetric Wilson loops and offers concrete constructions of their dual calibrated surfaces.

Abstract

We study the dual gravity description of supersymmetric Wilson loops whose expectation value is unity. They are described by calibrated surfaces that end on the boundary of anti de-Sitter space and are pseudo-holomorphic with respect to an almost complex structure on an eight-dimensional slice of AdS_5 x S^5. The regularized area of these surfaces vanishes, in agreement with field theory non-renormalization theorems for the corresponding operators.

Calibrated Surfaces and Supersymmetric Wilson Loops

TL;DR

This work identifies calibrated, pseudo-holomorphic minimal surfaces in that end on supersymmetric Wilson loops and shows their finite-area contribution vanishes, in line with non-renormalization theorems. By constructing a calibration two-form with an associated almost complex structure, the authors recast the Wilson-loop problem into finding pseudo-holomorphic curves, enabling exact area relations and hence for these loops. They analyze existence, counting, and supersymmetry of solutions, and provide explicit U(1) isometry-based ansätze leading to annulus and single-boundary surfaces, including coincident-boundary configurations that exhibit zero modes. The results connect BRST/topological twisting perspectives with geometric calibration in the gravity dual, reinforcing the non-renormalization paradigm for a broad class of BPS Wilson loops. Altogether, the paper advances the geometric and topological understanding of AdS/CFT for supersymmetric Wilson loops and offers concrete constructions of their dual calibrated surfaces.

Abstract

We study the dual gravity description of supersymmetric Wilson loops whose expectation value is unity. They are described by calibrated surfaces that end on the boundary of anti de-Sitter space and are pseudo-holomorphic with respect to an almost complex structure on an eight-dimensional slice of AdS_5 x S^5. The regularized area of these surfaces vanishes, in agreement with field theory non-renormalization theorems for the corresponding operators.

Paper Structure

This paper contains 14 sections, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. BPS Wilson loops
  3. Minimal surfaces in $AdS_5 \times S^5$
  4. Pseudo-holomorphic curves in $AdS_5 \times S^5$
  5. More general cases
  6. Existence and counting of solutions
  7. Supersymmetry preserved by the surface
  8. Pseudo-holomorphic surfaces with a $U(1)$ isometry
  9. The first ansatz
  10. The second ansatz
  11. Coincident boundaries
  12. Acknowledgements
  13. The Wilson loop operator in the topologically twisted theory
  14. Wilson loops in the topological string

Figures (2)

  • Figure 1: $V_{\rm eff}(\eta)$ versus $\eta$ for different values of $p_\beta$. The dark curve has $p_\beta = 1$. The ones above it have $p_\beta > 1$, and the ones below it have $0 < p_\beta < 1$.
  • Figure 2: In (a) we see two coincident straight Wilson lines. We have separated them to ease visualization, but they are on top of each other. In (b) we plot two of the directions transverse to the D-brane and we see a string ending on the D-brane (represented by a cross) and a string leaving the D-brane. In (c) the two strings from (b) combine and move in a direction transverse to the D-brane. In (c) we see two coincident and oppositely oriented circular Wilson loops. Again we have separated them just to ease visualization.