Minimal area surfaces, Wilson loops and Riemann theta functions

The AdS/CFT correspondence has produced remarkable connections between different areas of physics and mathematics. In this talk I will start by reviewing the relation between the Wilson loop observable in gauge theories and the problem of finding minimal area surfaces in hyperbolic (or AdS) space. Until recently only very few explicit solutions were known to such problem but I will show how, using Riemann theta functions an infinite parameter family of new solutions can be found. Moreover a one parameter family of deformations is identified such that the area is preserved. The solution to this problem uses methods from the theory of solitons and non-linear differential equations and sheds a new interesting light on the connection between the Wilson loop observable in gauge theories and integrable systems.