Notes on Euclidean Wilson loops and Riemann Theta functions

TL;DR

This paper constructs explicit analytic solutions for Euclidean Wilson loops in AdS3 using Riemann theta functions, demonstrating their properties and deformations, and establishing a connection with Riemann surface geometry.

Contribution

It introduces a new class of analytic solutions for Euclidean Wilson loops in AdS3 via Riemann theta functions, including their deformations and geometric interpretations.

Findings

Solutions expressed in terms of Riemann theta functions

Existence of a one-parameter family of area-preserving deformations

Mapping between Wilson loops and curves on Riemann surfaces

Abstract

The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS5 space. In this paper we consider the case of Euclidean flat Wilson loops which are related to minimal area surfaces in Euclidean AdS3 space. Using known mathematical results for such minimal area surfaces we describe an infinite parameter family of analytic solutions for closed Wilson loops. The solutions are given in terms of Riemann theta functions and the validity of the equations of motion is proven based on the trisecant identity. The world-sheet has the topology of a disk and the renormalized area is written as a finite, one-dimensional contour integral over the world-sheet boundary. An example is discussed in detail with plots of the corresponding surfaces. Further, for each Wilson loops we explicitly construct a one parameter family of deformations that preserve the area. The parameter is the so called spectral parameter. Finally, for genus three we find a map between these Wilson loops and closed curves inside the Riemann surface.