N = 4 Yang-Mills and N = 8 supergravity as limits of string theories

The formulation of supersymmetric string theories in ten dimensions is generalized to incorporate compactified dimensions. Expressions for the one-loop four-particle S-matrix elements of N=4 Yang-Mills and N=8 supergravity in four dimensions are obtained by studying the string-theory loop amplitudes in the limit that the radii of the compactified dimensions and the Regge slope parameter simultaneously approach zero. If certain patterns that emerge should persist in the higher orders of perturbation theory, then N=4 Yang-Mills in four dimensions would be ultraviolet finite to all orders, whereas N=8 supergravity in four dimensions would have ultraviolet divergences starting at three loops