$D^6 R^4$ amplitudes in various dimensions

TL;DR

This paper proposes an exact formula for the $D^6 R^4$ couplings in type II string theory compactified on $T^4$, using genus-two modular integrals and Eisenstein series, advancing understanding of non-perturbative effects in string amplitudes.

Contribution

It introduces a novel exact formula for $D^6 R^4$ couplings in various dimensions, connecting genus-two integrals with U-duality and T-duality symmetries.

Findings

Provides an exact formula for $D^6 R^4$ couplings in $D=6$

Connects two-loop corrections in $D=5$ with non-perturbative completions in $D=6$

Resolves inconsistencies in previous analyses of these couplings

Abstract

Four-graviton couplings in the low energy effective action of type II string vacua compactified on tori are strongly constrained by supersymmetry and U-duality. While the $R^4$ and $D^4 R^4$ couplings are known exactly in terms of Langlands-Eisenstein series of the U-duality group, the $D^6 R^4$ couplings are not nearly as well understood. Exploiting the coincidence of the U-duality group in $D=6$ with the T-duality group in $D=5$, we propose an exact formula for the $D^6 R^4$ couplings in type II string theory compactified on $T^4$, in terms of a genus-two modular integral plus a suitable Eisenstein series. The same modular integral computes the two-loop correction to $D^6 R^4$ in 5 dimensions, but here provides the non-perturbative completion of the known perturbative terms in $D=6$. This proposal hinges on a systematic re-analysis of the weak coupling and large radius of the $D^6 R^4$ in all dimensions $D\geq 3$, which fills in some gaps and resolves some inconsistencies in earlier studies.