Modular properties of two-loop maximal supergravity and connections with string theory

Understand

The low-momentum expansion of the two-loop four-graviton scattering amplitude in eleven-dimensional supergravity compactified on a circle and a two-torus is considered up to terms of order S^6R^4 (where S is a Mandelstam invariant and R is the linearized Weyl curvature).

• In the case of the toroidal compactification the coefficient of each term in the low energy expansion is generically a sum of a number of SL(2,Z)-invariant functions of the complex structure of the torus.
• Each such function satisfies a separate Poisson equation on moduli space with particular source terms that are bilinear in coefficients of lower order terms, consistent with qualitative arguments based on supersymmetry.
• Comparison is made with the low-energy expansion of type II string theories in ten and nine dimensions.