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

TL;DR

The paper investigates the small-momentum expansion of the two-loop four-graviton amplitude in eleven-dimensional supergravity compactified on S^1 and T^2, revealing that the coefficients of higher-derivative terms are modular functions satisfying Poisson equations on moduli space. By computing analytic and nonanalytic contributions up to S^6R^4 and mapping to type II string theories in ten and nine dimensions, the authors demonstrate substantial agreement with known string perturbation data and uncover a pattern where modular-coefficient functions obey inhomogeneous Laplace equations with lower-order bilinear sources. This establishes a robust, symmetry-driven framework linking M-theory loop amplitudes to perturbative string theory across dimensions and suggests a general mechanism by which maximal supersymmetry constrains the structure of higher-derivative couplings. The work lays groundwork for understanding nonperturbative effects and higher-dimensional extensions, with implications for organizing the low-energy effective action in duality-complete theories.

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.

Figures (4)

• Figure 1: The two-loop four-graviton amplitude in eleven dimensions. (a) The $S$-channel planar diagram reduces to $S^2\, {\cal R}^4$ multiplying a scalar field theory double-box diagram. (b) The $S$-channel nonplanar diagram reduces to $S^2\, {\cal R}^4$ multiplying a nonplanar scalar field theory two-loop diagram. (c) The triangle diagram with a one-loop counterterm at one vertex that subtracts a sub-divergence. (d) A new two-loop primitive divergence.
• Figure 2: (a) A planar diagram is represented by the skeleton with a pair of external states connected to each of two internal lines. (b) The nonplanar configuration in which one pair of external states is attached to a single line and the other states are each attached to separate lines. Integrating the positions of the four states over the whole network generates the sum of all Feynman diagrams.
• Figure 3: (a) The region of integration of $\tau=\tau_1+i\tau_2$ is equivalent to a fundamental domain of $\Gamma_0(2)$. Ultraviolet divergences arise on the boundary of this region. (b) The integrand can be mapped into a threefold cover of the fundamental domain, ${\cal F}$, of $SL(2,{\mathbb Z})$. The ultraviolet divergences arise from the $\tau_1=0$ axis.
• Figure 4: The double subdivergence of the three-loop diagrams that contributes at order $E_{7/2}\, S^4\,{\cal R}^4$ in ten-dimensional type IIB.