The D⁶ R⁴ term in type IIB string theory on T² and U-duality

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.