The D^4 R^4 term in type IIB string theory on T^2 and U-duality

TL;DR

This paper constructs a U-duality invariant modular form for the D^4 R^4 interaction in type IIB string theory on T^2, incorporating perturbative and non-perturbative effects, and matches known perturbative results.

Contribution

It introduces a new modular form for the D^4 R^4 term that is invariant under U-duality and matches perturbative and non-perturbative string theory calculations.

Findings

Perturbative contributions up to two loops are captured.

Non-perturbative effects from D-instantons and (p,q) string instantons are included.

The modular form matches known string and supergravity results at various limits.

Abstract

We propose a manifestly U-duality invariant modular form for the D^4 R^4 interaction in type IIB string theory compactified on T^2. It receives perturbative contributions upto two loops, and non-perturbative contributions from D-instantons and (p,q) string instantons wrapping T^2. We provide evidence for this modular form by showing that the coefficients at tree level and at one loop precisely match those obtained using string perturbation theory. Using duality, parts of the perturbative amplitude are also shown to match exactly the results obtained from eleven dimensional supergravity compactified on T^3 at one loop. Decompactifying the theory to nine dimensions, we obtain a U-duality invariant modular form, whose coefficients at tree level and at one loop agree with string perturbation theory.