SL(2, Z)-invariance and D-instanton contributions to the D6R4 interaction

The modular invariant coefficient of the (DR4)-R-6 interaction in the low energy expansion of type IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue equation, obtained by considering the toroidal compactification of two-loop Feynman diagrams of eleven-dimensional supergravity. In this paper we determine the exact SL(2, Z)-invariant solution f (x + iy) to this differential equation satisfying an appropriate moderate growth condition as y -> infinity (the weak coupling limit). The solution is presented as a Fourier series with modes (f) over cap (n)(y)e(2 pi inx), where the mode coefficients, (f) over cap (n)(y) are bilinear in K-Bessel functions. Invariance under SL(2, Z) requires these modes to satisfy the nontrivial boundary condition (f) over cap (n)(y) = O(y(-2)) for small y, which uniquely determines the solution. The large-y expansion of f (x + iy) contains the known perturbative (power-behaved) terms, together with precisely-determined exponentially decreasing contributions that have the form expected of D-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs.