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

TL;DR

The paper constructs a manifestly U-duality invariant modular form for the D^4 R^4 coupling in type IIB string theory on T^2, built from SL(3,Z) and SL(2,Z) Eisenstein series as $E_{5/2}(M)^{SL(3,Z)} - 8 E_2(M^{-1})^{SL(3,Z)} E_2(U)^{SL(2,Z)}$, and shows that it yields perturbative contributions up to two loops and non-perturbative D-instantons and (p,q) instantons wrapping T^2. It provides evidence by matching eight-dimensional string perturbation theory at tree level and one loop to the predicted coefficients, and by reproducing parts of the one-loop eleven-dimensional supergravity amplitude on T^3, consistent with the modular form. In the decompactification limit to nine dimensions, the form remains U-duality invariant and reproduces known results, reinforcing the role of modular forms in encoding protected higher-derivative couplings across dimensions.

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.