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.

United States

Sannah Ziama

Holographic calculations of Euclidean Wilson loop correlator in Euclidean anti-de Sitter space

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS 3 ). If the Wilson loop is given by a boundary curve $ \overrightarrow{X} $ ( s ) we define, using the integrable properties of the system, a family of curves $ \overrightarrow{X} $ ( λ, s ) depending on a complex parameter λ known as the spectral parameter. This family has remarkable properties. As a function of λ , $ \overrightarrow{X} $ ( λ, s ) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, $ \overrightarrow{X} $ ( λ, s ) has an essential singularity at the origin λ = 0. The coefficients of the expansion of $ \overrightarrow{X} $ ( λ, s ) around λ = 0, when appropriately integrated along the curve give the area of the corresponding minimal area surface.