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Two universal results for Wilson loops at strong coupling

Sean A. Hartnoll

TL;DR

The paper establishes two universal, non-supersymmetric results for Wilson loops in strongly coupled gauge theories with IIB duals of the form $M \times S^5$. First, a D5-brane construction expresses a loop in the rank $k$ antisymmetric representation in terms of the fundamental loop via a simple renormalisation, with the key relation $\left. S_{D5} \right|_{renor.} = \frac{2N}{3\pi} \sin^3 \theta_0 \left. S_{F1} \right|_{renor.}$ and a defining equation for $\theta_0$. Second, every fundamental-string embedding gives rise to an infinite family of D3-brane embeddings $\\Sigma \\times \\Upsilon \\subset M \\times S^5$ for any minimal surface $\\Upsilon \\subset S^5$, with renormalised actions $\\left. S_{D3,k} \\right|_{renor.} = \frac{4N}{\\sqrt{\\lambda}} \sqrt{\\sin^4\\alpha_0 + \\kappa^2} \\left. S_{F1} \\right|_{renor.}$ and a general construction using minimal immersions of $S^2$ into $S^4$ tied to totally isotropic holomorphic curves in $\\mathbb{CP}^4$. Together, these results reveal a universal, potentially integrable structure underlying strong-coupling Wilson loops and suggest deep links between holographic Wilson loops, matrix models, and algebraic geometry of minimal immersions. The work also points to rich future directions, including field-theory interpretations and stability analyses of the D3-brane solutions.

Abstract

We present results for Wilson loops in strongly coupled gauge theories. The loops may be taken around an arbitrarily shaped contour and in any field theory with a dual IIB geometry of the form M x S^5. No assumptions about supersymmetry are made. The first result uses D5 branes to show how the loop in any antisymmetric representation is computed in terms of the loop in the fundamental representation. The second result uses D3 branes to observe that each loop defines a rich sequence of operators associated with minimal surfaces in S^5. The action of these configurations are all computable. Both results have features suggesting a connection with integrability.

Two universal results for Wilson loops at strong coupling

TL;DR

The paper establishes two universal, non-supersymmetric results for Wilson loops in strongly coupled gauge theories with IIB duals of the form . First, a D5-brane construction expresses a loop in the rank antisymmetric representation in terms of the fundamental loop via a simple renormalisation, with the key relation and a defining equation for . Second, every fundamental-string embedding gives rise to an infinite family of D3-brane embeddings for any minimal surface , with renormalised actions and a general construction using minimal immersions of into tied to totally isotropic holomorphic curves in . Together, these results reveal a universal, potentially integrable structure underlying strong-coupling Wilson loops and suggest deep links between holographic Wilson loops, matrix models, and algebraic geometry of minimal immersions. The work also points to rich future directions, including field-theory interpretations and stability analyses of the D3-brane solutions.

Abstract

We present results for Wilson loops in strongly coupled gauge theories. The loops may be taken around an arbitrarily shaped contour and in any field theory with a dual IIB geometry of the form M x S^5. No assumptions about supersymmetry are made. The first result uses D5 branes to show how the loop in any antisymmetric representation is computed in terms of the loop in the fundamental representation. The second result uses D3 branes to observe that each loop defines a rich sequence of operators associated with minimal surfaces in S^5. The action of these configurations are all computable. Both results have features suggesting a connection with integrability.

Paper Structure

This paper contains 5 sections, 3 theorems, 33 equations, 1 table.

Table of Contents

  1. Introduction
  2. A universal result for antisymmetric loops
  3. Some D3 brane embeddings
  4. D3 brane immersions and minimal surfaces in $S^5$
  5. Concluding remarks

Key Result

Theorem 1

(Calabi Calabi, Barbosa Barbosa) The minimal immersion lies in an extremal $S^4 \subset S^5$. Either it is a totally geodesic $S^2$ with area $4\pi$ or it has area $4\pi m$, with $m \in {{\Bbb Z}}$ and $m \geq 3$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3