We propose that the expectation value of a circular BPS-Wilson loop in [script N] = 4 supersymmetric Yang–Mills 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 alpha[prime] 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 alpha[prime]. 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
