Wilson loops and minimal surfaces

The AdS-CFT correspondence suggests that the Wilson loop of the large N gauge theory with $\mathcal{N}=4\mathrm{}$ supersymmetry in four dimensions is described by a minimal surface in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5.$ We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultraviolet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ gives a solution of the equation. We also discuss the zigzag symmetry of the loop operator. In the $\mathcal{N}=4\mathrm{}$ gauge theory, we expect the zigzag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface.