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 $D=4$, 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.