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

TL;DR

This paper investigates the structure of two-loop supergravity amplitudes compactified on tori, revealing modular properties and connections with string theory, and finds agreement with string perturbation results for certain low-energy terms.

Contribution

It uncovers the modular and differential equation structure of two-loop supergravity amplitudes and their relation to string theory coefficients, extending understanding of supergravity-string connections.

Findings

Coefficients satisfy Poisson equations on moduli space.

Agreement with string perturbation theory for low-order terms.

Pattern of interrelated equations for higher-order coefficients.

Abstract

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. Although the detailed behaviour of the string amplitude is not generally expected to be reproduced by supergravity perturbation theory to all orders, for the terms considered here we find agreement with direct results from string perturbation theory. These results point to a fascinating pattern of interrelated Poisson equations for the IIB coefficients at higher orders in the momentum expansion which may have a significance beyond the particular methods by which they were motivated.