D6R4term in type IIB string theory onT2and U-duality

Abstract

We propose a manifestly U-duality invariant modular form for the ${D}^{6}{\mathcal{R}}^{4}$ interaction in the effective action of type IIB string theory compactified on ${T}^{2}$. It receives perturbative contributions up to genus three, as well as nonperturbative 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 11 dimensional supergravity compactified on ${T}^{3}$ at one loop. We also obtain parts of the genus one and genus $k$ amplitudes for the ${D}^{2k}{\mathcal{R}}^{4}$ interaction for arbitrary $k\ensuremath{\ge}4$. We enhance a part of this amplitude to a U-duality invariant modular form.