Notes on Euclidean Wilson loops and Riemann theta functions

Abstract

The AdS/CFT correspondence relates Wilson loops in $\mathcal{N}=4$ super Yang-Mills theory to minimal area surfaces in ${\mathrm{AdS}}_{5}$ space. In this paper we consider the case of Euclidean flat Wilson loops, which are related to minimal area surfaces in Euclidean ${\mathrm{AdS}}_{3}$ 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.