An Exact Prediction of N = 4 SUSYM Theory for String Theory

We show that the expectation value of a circular BPS-Wilson loop in N = 4 SUSYM can be calculated exactly, to all orders in a 1/N expansion and to all orders in g2N . Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in α′ and to all orders in gs. We then compare this result with string theory. We find that the gauge theory calculation, for large g2N and to all orders in the 1/N2 expansion does agree with the leading string theory calculation, to all orders in gs and to lowest order in α′. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory.