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Gauge fields - strings duality and the loop equation

A. Polyakov, V. Rychkov

TL;DR

This work investigates gauge fields–strings duality through loop equations and zigzag symmetry by analyzing Wilson loops in a 5D string background. Using the WKB/minimal-area approach in Lobachevsky space and the special case of wavy-line contours, the authors show that the string representation can reproduce the gauge theory loop equation precisely at the critical dimension $D=4$, providing a nontrivial consistency check between string dynamics and Yang–Mills Schwinger–Dyson equations. The results are currently limited to the WKB regime and to a restricted contour class, but they establish a framework for deriving OPE data and for extending the analysis beyond these limitations. The findings imply a potential universal structure for gauge/string duality in conformal settings and outline concrete paths to incorporate short-distance operator product expansions and non-conformal backgrounds in future work.

Abstract

We explore gauge fields - strings duality by means of the loop equations and the zigzag symmetry. The results are striking and incomplete. Striking - because we find that the string ansatz proposed in [A.M. Polyakov, hep-th/9711002] satisfies gauge theory Schwinger-Dyson equations precisely at the critical dimension D=4. Incomplete - since we get these results only in the WKB approximation and only for a special class of contours. The ways to go beyond these limitations and in particular the OPE for operators defined on the loop are also discussed.

Gauge fields - strings duality and the loop equation

TL;DR

This work investigates gauge fields–strings duality through loop equations and zigzag symmetry by analyzing Wilson loops in a 5D string background. Using the WKB/minimal-area approach in Lobachevsky space and the special case of wavy-line contours, the authors show that the string representation can reproduce the gauge theory loop equation precisely at the critical dimension , providing a nontrivial consistency check between string dynamics and Yang–Mills Schwinger–Dyson equations. The results are currently limited to the WKB regime and to a restricted contour class, but they establish a framework for deriving OPE data and for extending the analysis beyond these limitations. The findings imply a potential universal structure for gauge/string duality in conformal settings and outline concrete paths to incorporate short-distance operator product expansions and non-conformal backgrounds in future work.

Abstract

We explore gauge fields - strings duality by means of the loop equations and the zigzag symmetry. The results are striking and incomplete. Striking - because we find that the string ansatz proposed in [A.M. Polyakov, hep-th/9711002] satisfies gauge theory Schwinger-Dyson equations precisely at the critical dimension D=4. Incomplete - since we get these results only in the WKB approximation and only for a special class of contours. The ways to go beyond these limitations and in particular the OPE for operators defined on the loop are also discussed.
Paper Structure (8 sections, 87 equations)

This paper contains 8 sections, 87 equations.

Table of Contents

  1. Introduction
  2. The loop equation and the zigzag symmetry
  3. The string representation: D-branes and/or $\sigma$-model.
  4. Minimal area in the Lobachevsky space
  5. The theory of wavy lines
  6. The second derivative of the minimal area
  7. Interpretation and discussion of the result
  8. Conclusions and outlook