Constraining gravitational interactions in the M theory effective action

Abstract

We consider purely gravitational interactions of the type in the effective action of M theory in 11 dimensional flat spacetime. The duality between M theory on S1 and type IIA string theory relates them to the type IIA interactions of the form where is the type IIA dilaton. The coefficients of the M theory interactions are determined by the strongly coupled type IIA theory. Given the nature of the dilaton dependence, it is plausible that for low values of n, the coefficient has a similar structure as the genus string amplitude of the type IIA interaction, namely the transcendental nature. Assuming this, and focussing on the even–even spin structure part of the type IIA string amplitude, this coefficient is given by the type IIB genus amplitude, which we constrain using supersymmetry, S-duality and maximal supergravity. The source terms of the Poisson equations satisfied by the S-duality invariant IIB couplings play a central role in the analysis. This procedure yields partial contributions to several multi-loop type IIB string amplitudes, from which we extract the transcendental nature of the corresponding M theory couplings. For , all possible source terms involve only BPS couplings. While the and M theory couplings agree with known results, the coefficient of the interaction takes the form . We also analyze the and interactions, and show that their coefficients have at least the terms and respectively. The various undetermined constants have vanishing transcendentality.