A note on non-perturbative R^4 couplings

TL;DR

This paper proves that the exact non-perturbative R^4 couplings in type II string theory are uniquely determined by supersymmetry and U-duality, ruling out contributions from cusp forms and confirming their expression as an Eisenstein series.

Contribution

It provides a rigorous proof that supersymmetry constrains the R^4 couplings to be an Eisenstein series, excluding cusp form contributions, using the D=8 N=2 superfield formalism.

Findings

R^4 couplings are eigenmodes of the Laplacian on the scalar manifold.

Exact R^4 threshold is identified with an order-3/2 Eisenstein series.

Cusp form contributions are ruled out by supersymmetry and U-duality.

Abstract

Exact non-perturbative results have been conjectured for R^4 couplings in type II maximally supersymmetric string theory. Strong evidence has already been obtained, but contributions of cusp forms, invisible in perturbation theory, have remained an open possibility. In this note, we use the D=8 N=2 superfield formalism of Berkovits to prove that supersymmetry requires the exact R^4 threshold to be an eigenmode of the Laplacian on the scalar manifold with a definite eigenvalue. Supersymmetry and U-duality invariance then identify the exact result with the order-3/2 Eisenstein series, and rule out cusp form contributions.