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.
Highlights
Sigma model in Euclidean AdS3A convenient way to imagine Euclidean AdS3 is to consider it as a subspace of R3,1 defined by the equation
Most of the technique used in this paper was established in [10, 11] and the reader is encouraged to review them
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
Summary
A convenient way to imagine Euclidean AdS3 is to consider it as a subspace of R3,1 defined by the equation. In Euclidean AdS3 space this scalar equation is the cosh-gordon equation. Where Ah is the Hopf differential and Hh is the mean curvature of the surface described by the solution to the string equation of motion. Since we are concerned here with a minimal area surface (described by the string equations of motion) we will have a vanishing mean curvature and the second equation in (2.12) is equal to zero. The compatibility equation implies that Ah is a holomorphic function It leads to a generalized cosh-gordon equation αzz. Given a solution for the pair of equations (2.23), the Poincare coordinates are related to (2.23) by. Riemann theta function solution for the cosh-gordon equation was found in [10, 11], the system (2.23) was solved in terms of those theta functions.
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