Two loops in eleven dimensions

The two-loop Feynman diagram contribution to the four-graviton amplitude of eleven-dimensional supergravity compactified on a two-torus $\mathcal{T}{}^2$ is analyzed in detail. The Schwinger parameter integrations are reexpressed as an integration over the moduli space of a second torus $\mathcal{T̂}{}^2,$ which enables the leading low-momentum contribution to be evaluated in terms of maps of $\mathcal{T̂}{}^2$ into $\mathcal{T}{}^2.$ The ultraviolet divergences associated with boundaries of moduli space are regularized in a manner that is consistent with the expected duality symmetries of string theory. This leads to an exact expression for terms of order ${\partial }^4{\mathcal{R}}^4$ in the effective M-theory action (where ${\mathcal{R}}^4$ denotes a contraction of four Weyl tensors), thereby extending earlier results for the ${\mathcal{R}}^4$ term that were based on the one-loop eleven-dimensional amplitude. Precise agreement is found with terms in type-IIA and -IIB superstring theory that arise from the low-energy expansion of the tree-level and one-loop string amplitudes and predictions are made for the coefficients of certain two-loop string theory terms as well as for an infinite set of D-instanton contributions. The contribution at the next order in the derivative expansion ${\partial }^6{\mathcal{R}}^4$ is problematic, which may indicate that it mixes with higher-loop effects in eleven-dimensional supergravity.