The D^{10} R^4 term in type IIB string theory
  The modular invariant coefficient of the D^{2k} {\cal{R}}^4 term in the effective action of type IIB superstring theory is expected to satisfy Poisson equation on the fundamental domain of SL(2,Z). Under certain assumptions, we obtain the equation satisfied by D^{10} {\cal{R}}^4 using the tree level and one loop results for four graviton scattering in type II string theory. This leads to the conclusion that the perturbative contributions to D^{10} {\cal{R}}^4 vanish above three loops, and also predicts the coefficients at two and three loops. 
Anirban Basu