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Four-graviton couplings in the low energy effective action of type II stringvacua compactified on tori are strongly constrained by supersymmetry andU-duality. While the $R^4$ and $D^4 R^4$ couplings are known exactly in termsof Langlands-Eisenstein series of the U-duality group, the $D^6 R^4$ couplingsare not nearly as well understood. Exploiting the coincidence of the U-dualitygroup in $D=6$ with the T-duality group in $D=5$, we propose an exact formulafor the $D^6 R^4$ couplings in type II string theory compactified on $T^4$, interms of a genus-two modular integral plus a suitable Eisenstein series. Thesame modular integral computes the two-loop correction to $D^6 R^4$ in 5dimensions, but here provides the non-perturbative completion of the knownperturbative terms in $D=6$. This proposal hinges on a systematic re-analysisof the weak coupling and large radius of the $D^6 R^4$ in all dimensions $D\geq3$, which fills in some gaps and resolves some inconsistencies in earlierstudies.

$D^6 R^4$ amplitudes in various dimensions