Wilson loops and minimal area surfaces in hyperbolic space

Understand

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 $- \phi"(s) + \big[V_0+{1\over 2} (\lambda+{1 \over \lambda}) V_1 + {i\over 2} (\lambda-{1 \over \lambda}) V_2\big] \phi(s)=0$ 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.