Duality and higher derivative terms in M theory

Dualities of M-theory are used to determine the exact dependence on thecoupling constant of the D^6R^4 interaction of the IIA and IIB superstringeffective action. Upon lifting to eleven dimensions this determines thecoefficient of the D^6R^4 interaction in eleven-dimensional M-theory. Theseresults are obtained by considering the four-graviton two-loop scatteringamplitude in eleven-dimensional supergravity compactified on a circle and on atwo-torus -- extending earlier results concerning lower-derivativeinteractions. The torus compactification leads to an interestingSL(2,Z)-invariant function of the complex structure of the torus (the IIBstring coupling) that satisfies a Laplace equation with a source term on thefundamental domain of moduli space. The structure of this equation is in accordwith general supersymmetry considerations and immediately determines tree-leveland one-loop contributions to D^6R^4 in perturbative IIB string theory thatagree with explicit string calculations, and two-loop and three-loopcontributions that have yet to be obtained in string theory. The completesolution of the Laplace equation contains infinite series' of singleD-instanton and double D-instanton contributions, in addition to theperturbative terms. General considerations of the higher loop diagrams ofeleven-dimensional supergravity suggest extensions of these results tointeractions of higher order in the low energy expansion.Comment: harvmac. 41 pages. 3 figures. v2 typos corrected and reference list updated. v3. Significant new subsection deriving the non-zero coefficient of the IIB string theory three-loop contributio