Holographic Calculations of Euclidean Wilson Loop Correlator in Euclidean anti-de Sitter Space

TL;DR

This paper uses holographic methods to compute Euclidean Wilson loop correlators in anti-de Sitter space, employing advanced Riemann surface techniques to analyze their behavior at various separations.

Contribution

It introduces a novel approach using genus three hyperelliptic Riemann surfaces and theta functions to calculate Wilson loop correlators holographically.

Findings

Correlators depend on the branch points of the Riemann surface.

Large separation limit approximates loops as local operators.

Small loops can be treated as light supergravity modes.

Abstract

The correlation functions of two or more Euclidean Wilson loops of various shapes in Euclidean anti-de Sitter space are computed by considering the minimal area surfaces connecting the loops. The surfaces are parametrized by Riemann theta functions associated with genus three hyperelliptic Riemann surfaces. In the case of two loops, the distance $L$ by which they are separated can be adjusted by continuously varying a specific branch point of the auxiliary Riemann surface. When $L$ is much larger than the characteristic size of the loops, then the loops are approximately regarded as local operators and their correlator as the correlator of two local operators. Similarly, when a loop is very small compared to the size of another loop, the small loop is considered as a local operator corresponding to a light supergravity mode.