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

The modular invariant coefficient of the D⁶R⁴ 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 → ∞ (the weak coupling limit). The solution is presented as a Fourier series with modes f_n(y)e^2πinx, where the mode coefficients, f_n(y) are bilinear in K- Bessel functions. Invariance under SL(2,Z) requires these modes to satisfy the nontrivial boundary condition f_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-Dinstantons and D-instanton/anti-D-instanton pairs.