D4R4term in type IIB string theory onT2and U-duality

Abstract

We propose a manifestly U-duality invariant modular form for the ${D}^{4}{\mathcal{R}}^{4}$ interaction in type IIB string theory compactified on ${T}^{2}$. It receives perturbative contributions up to two loops, and nonperturbative 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 11 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.