D6ℛ4 amplitudes in various dimensions

Four-graviton couplings in the low energy effective action of type II string vacua compactified on tori are strongly constrained by supersymmetry and U-duality. While the ℛ4 and D4ℛ4 couplings are known exactly in terms of Langlands-Eisenstein series of the U-duality group, the D6ℛ4 couplings are not nearly as well understood. Exploiting the coincidence of the U-duality group in D = 6 with the T-duality group in D = 5, we propose an exact formula for the D6ℛ4 couplings in type II string theory compactified on T4, in terms of a genus-two modular integral plus a suitable Eisenstein series. The same modular integral computes the two-loop correction to D6ℛ4 in 5 dimensions, but here provides the non-perturbative completion of the known perturbative terms in D = 6. This proposal hinges on a systematic re-analysis of the weak coupling and large radius of the D6ℛ4 in all dimensions D ≥ 3, which fills in some gaps and resolves some inconsistencies in earlier studies.