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We consider a class of eight derivative interactions in the effective actionof type IIB string theory compactified on T^2. These 1/2 BPS interactions havemoduli dependent couplings. We impose the constraints of supersymmetry to showthat each of these couplings satisfy a first order differential equation onmoduli space which relate it to other couplings in the same supermuliplet.These equations can be iterated to give second order differential equations forthe various couplings. The couplings which only depend on the SO(2)\SL(2,R)moduli satisfy Laplace equation on moduli space, and are given by modular formsof SL(2,Z). On the other hand, the ones that only depend on the SO(3)\SL(3,R)moduli satisfy Poisson equation on moduli space, where the source terms aregiven by other couplings in the same supermultiplet. The couplings of theinteractions which are charged under SU(2) are not automorphic forms ofSL(3,Z). Among the interactions we consider, the R^4 coupling depends on allthe moduli.

Supersymmetry constraints on the R^4 multiplet in type IIB on T^2