Wilson loops and minimal area surfaces in hyperbolic space

Abstract

The AdS/CFT correspondence relates Wilson loops in $N$=4 SYM theory to minimal area surfaces in AdS space. If the loop is a plane curve the minimal surface lives in hyperbolic space $H_3$ (or equivalently Euclidean AdS${}_3$ space). We argue that finding the area of such extremal surface can be easily done if we solve the following problem: given two real periodic functions $V_{0,1}(s)$, $V_{0,1}(s+2\pi )=V_{0,1}(s)$, a third periodic function $V_2(s)$ is to be found such that all solutions to the equation $-\varphi "(s)+[V_0+\frac{1}{2}(\lambda +\frac{1}{\lambda })V_1+\frac{i}{2}(\lambda -\frac{1}{\lambda })V_2]\varphi (s)=0$ are anti-periodic in $s\in [0,2\pi ]$ for any value of $\lambda$. This problem is equivalent to the statement that the monodromy matrix is trivial. It can be restated as that of finding a one complex parameter family of curves $X(\lambda ,s)$ where $X(\lambda =1,s)$ is the given shape of the Wilson loop and such that the Schwarzian derivative $\{X(\lambda ,s),s\}$ is meromorphic in $\lambda$ with only two simple poles. We present a formula for the area in terms of the functions $V_{0,1,2}$ and discuss solutions to these equivalent problems in terms of theta functions. Finally, we also consider the near circular Wilson loop clarifying its integrability properties and rederiving its area using the methods described in this paper.

LateX, 39 pages, 1 figure

References in corpus (8)

• Gluon scattering amplitudes at strong coupling
• An Operator Product Expansion for Polygonal null Wilson Loops
• Form factors at strong coupling via a Y-system
• Supersymmetric Wilson loops on S^3
• Generating AdS String Solutions
• The Spectrum of Excitations of Holographic Wilson Loops
• Moduli Dynamics of AdS_3 Strings
• Dressed Wilson Loops on S^2

Cited by in corpus (5)

• Quantum Corrections to Supergravity on AdS${}_2\times S^2$
• Dual conformal transformations of smooth holographic Wilson loops
• From algebraic curve to minimal surface and back
• Minimal area surfaces in AdS${}_3$ through integrability
• Holographic Calculations of Euclidean Wilson Loop Correlator in Euclidean anti-de Sitter Space