Constraining gravitational interactions in the M theory effective action

We consider purely gravitational interactions of the type D^{6n} R^4 in the effective action of M theory which are related to the type IIA interactions of the form e^{2n\phi_A} D^{6n} R^4 where \phi_A 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 (n+1) string amplitude of the type IIA D^{6n} R^4 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 (n+1) 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 n \leq 2, all possible source terms involve only BPS couplings. While the R^4 and D^6 R^4 M theory couplings agree with known results, the coefficient of the D^{12} R^4 interaction takes the form \zeta (2)^3 (\Omega_1+\Omega_2 \zeta (3)). We also analyze the D^{18} R^4 and D^{24} R^4 interactions, and show that their coefficients have at least the terms \zeta (2)^4 (\tilde\Omega_1 +\tilde\Omega_2\zeta (3) +\tilde\Omega_3\zeta (5)) and \zeta (2)^5 (\underline\Omega_1 +\underline\Omega_2\zeta (3) +\underline\Omega_3\zeta (5) + \underline\Omega_4\zeta (3)^2 + \underline\Omega_5\zeta (7) +\underline\Omega_6\zeta (3) \zeta (5) +\underline\Omega_7 \zeta (3)^3) respectively. The various undetermined constants have vanishing transcendentality.