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, T^2, is analyzed in detail. The Schwinger parameter integrations are re-expressed as integration over the moduli space of a second torus, \hat T^2, which enables the leading low-momentum contribution to be evaluated in terms of maps of \hat T^2 into 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 D^4 R^4 in the effective M theory action (where R^4 denotes a contraction of four Weyl tensors), thereby extending earlier results for the 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, D^6 R^4, is problematic, which may indicate that it mixes with higher-loop effects in eleven-dimensional supergravity.