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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^{2nϕ_A} D^{6n} R^4 where ϕ_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 ζ(2)^3 (Ω_1+Ω_2 ζ(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 ζ(2)^4 (\tildeΩ_1 +\tildeΩ_2ζ(3) +\tildeΩ_3ζ(5)) and ζ(2)^5 (\underlineΩ_1 +\underlineΩ_2ζ(3) +\underlineΩ_3ζ(5) + \underlineΩ_4ζ(3)^2 + \underlineΩ_5ζ(7) +\underlineΩ_6ζ(3) ζ(5) +\underlineΩ_7 ζ(3)^3) respectively. The various undetermined constants have vanishing transcendentality.