A bstract The local higher-derivative interactions that enter into the low-energy expansion of the effective action of type IIB superstring theory with constant complex modulus generally violate the U(1) R-symmetry of IIB supergravity by q ~( U )units. These interactions have coefficients that transform as non-holomorphic modular forms under SL(2, ?) transformations with holomorphic and anti-holomorphic weights ( w, ? w ), where q ~( U )= ?2 w . In this paper SL(2, ?)-covariance and supersymmetry are used to determine first-order differential equations on moduli space that relate the modular form coefficients of classes of BPS-protected maximal U(1)-violating interactions that arise at low orders in the lowenergy expansion. These are the moduli-dependent coefficients of BPS interactions of the form d _( 2p ) P $$ mathcal{P} $$ ~( n )in linearised approximation, where P $$ mathcal{P} $$ ~( n )is the product of n fields that has dimension = 8 with q ~( U )= 8 ? 2 n , and p = 0, 2 or 3. These first-order equations imply that the coefficients satisfy SL(2, ?)-covariant Laplace eigenvalue equations on moduli space with solutions that contain information concerning perturbative and non-perturbative contributions to superstring amplitudes. For p = 3 and n ≥ 6 there are two independent modular forms, one of which has a vanishing tree-level contribution. The analysis of super-amplitudes for U(1)-violating processes involving arbitrary numbers of external fluctuations of the complex modulus leads to a diagrammatic derivation of the first-order differential relations and Laplace equations satisfied by the coefficient modular forms. Combining this with a SL(2, ?)-covariant soft axio-dilaton limit that relates amplitudes with different values of n determines most of the modular invariant coefficients, leaving a single undetermined constant.
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