The D^6 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^6 R^4 interaction in type IIB string theory on T^2, incorporating perturbative and non-perturbative effects, and relates it to supergravity results.

Contribution

It introduces a new U-duality invariant modular form for the D^6 R^4 term, including contributions up to genus three and instantons, extending previous perturbative and non-perturbative analyses.

Findings

Perturbative contributions up to genus three identified.

Non-perturbative D-instanton and (p,q) string effects incorporated.

Matching with supergravity results at one loop achieved.

Abstract

We propose a manifestly U-duality invariant modular form for the D^6 R^4 interaction in the effective action of type IIB string theory compactified on T^2. It receives perturbative contributions upto genus three, as well as non-perturbative contributions from D-instantons and (p,q) string instantons wrapping T^2. Our construction is based on constraints coming from string perturbation theory, U-duality, the decompactification limit to ten dimensions, and the equality of the perturbative part of the amplitude in type IIA and type IIB string theories. 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. We also obtain parts of the genus one and genus k amplitudes for the D^{2k} R^4 interaction for arbitrary k > 3. We enhance a part of this amplitude to a U-duality invariant modular form.