Some finite terms from ladder diagrams in three and four loop maximal supergravity

Abstract

We consider the finite part of the leading local interactions in the low energy expansion of the four graviton amplitude from the ladder skeleton diagrams in maximal supergravity on T 2 , at three and four loops. At three loops, we express the D 8 R 4 and D 10 R 4 amplitudes as integrals over the moduli space of an underlying auxiliary geometry. These amplitudes are evaluated exactly for special values of the the moduli of the auxiliary geometry, where the integrand simplifies. We also perform a similar analysis for the D 8 R 4 amplitude at four loops that arise from the ladder skeleton diagrams for a special value of a parameter in the moduli space of the auxiliary geometry. While the dependence of the amplitudes on the volume of the T 2 is very simple, the dependence on the complex structure of the T 2 is quite intricate. In some of the cases, the amplitude consists of terms each of which factorizes into a product of two SL(2, Z) invariant modular forms. While one of the factors is a non-holomorphic Eisenstein series, the other factor splits into a sum of modular forms each of which satisfies a Poisson equation on moduli space with source terms that are bilinear in the Eisenstein series. This leads to several possible perturbative contributions unto genus 5 in type II string theory on S 1 . Unlike the one and two loop supergravity analysis, these amplitudes also receive non-perturbative contributions from bound states of three D-(anti)instantons in the IIB theory. 1 email address: anirbanbasu@hri.res.in 2 For theories with self-dual field strengths, what we mean are covariant equations of motion.

Poisson equation for the Mercedes diagram in string theory at genus one

“…Also, it would be interesting to generalize the analysis to amplitudes at higher genus in string theory. Explicit expressions for the fourgraviton amplitude at three and four loops in maximal supergravity [22,[39][40][41][42] might provide useful hints about the structure of the corresponding string amplitudes.…”

Perturbative type II amplitudes for BPS interactions

“…One could also use these techniques to analyze the structure of the non-BPS amplitudes, like the D 8 4  interaction to start with. While higher genus expressions for such amplitudes are not known, multi-loop supergravity analysis might provide useful hints about their perturbative structure, along the lines of [52][53][54].…”

Non-BPS interactions from the type II one loop four graviton amplitude

“…It would be interesting to extend the analysis to lower dimensions in detail, and also at higher string loops. This could give insight into the structure of non-BPS operators in maximally supersymmetric string theories, which are not well understood apart from some constraints based on supersymmetry and the explicit multi-loop structure of maximal supergravity [13,[40][41][42][43][44].…”