Four-graviton scattering in eleven-dimensional supergravity is considered at one loop compactified on one, two and three-dimensional tori. The dependence on the toroidal geometry determines the known perturbative and non-perturbative terms in the corresponding processes in type II superstring theories in ten, nine and eight dimensions. The ultra-violet divergence must be regularized so that it has a precisely determined finite value that is consistent both with T-duality in nine dimensions and with eleven-dimensional supersymmetry.Comment: harvmac (b) 12 pages. typos corrected and a reference adde

Identities based on monodromy for integrations in string theory are used to derive relations between different color-ordered tree-level amplitudes in both bosonic and supersymmetric string theory. These relations imply that the color-ordered tree-level n-point gauge theory amplitudes can be expanded in a minimal basis of (n-3)! amplitudes. This result holds for any choice of polarizations of the external states and in any number of dimensions.

Dualities of M-theory are used to determine the exact dependence on the coupling constant of the D 6 R 4 interaction of the IIA and IIB superstring effective action. Upon lifting to eleven dimensions this determines the coefficient of the D 6 R 4 interaction in eleven-dimensional M -theory. These results are obtained by considering the four-graviton two-loop scattering amplitude in eleven-dimensional supergravity compactified on a circle and on a two-torus -extending earlier results concerning lower-derivative interactions. The torus compactification leads to an interesting SL(2, Z)-invariant function of the complex structure of the torus (the IIB string coupling) that satisfies a Laplace equation with a source term on the fundamental domain of moduli space. The structure of this equation is in accord with general supersymmetry considerations and immediately determines treelevel and one-loop contributions to D 6 R 4 in perturbative IIB string theory that agree with explicit string calculations, and two-loop and three-loop contributions that have yet to be obtained in string theory. The complete solution of the Laplace equation contains infinite series' of single D-instanton and double D-instanton contributions, in addition to the perturbative terms. General considerations of the higher loop diagrams of elevendimensional supergravity suggest extensions of these results to interactions of higher order in the low energy expansion.