$SL(2, \mathbb{Z})$-invariance and D-instanton contributions to the $D^6 R^4$ interaction

Abstract

The modular invariant coefficient of theD 6 R 4 interaction in the low energy expansion of type IIB string theory has been conjec- tured to be a solution of an inhomogeneous Laplace eigenvalue equa- tion, 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 differ- ential equation satisfying an appropriate moderate growth condition as y ! 1 (the weak coupling limit). The solution is presented as a Fourier series with modes b fn(y)e 2�inx , where the mode coefficients,b fn(y) are bi- linear in K-Bessel functions. Invariance under SL(2,Z) requires these modes to satisfy the nontrivial boundary condition b fn(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 contribu- tions that have the form expected of D-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs.