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

Abstract

The modular invariant coefficient of the $D^6{R}^4$ 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 $SL(2,\mathbb{Z})$-invariant solution $f(x+iy)$ to this differential equation satisfying an appropriate moderate growth condition as $y\to \infty$ (the weak coupling limit). The solution is presented as a Fourier series with modes ${\hat{f}}_n(y)e^{2\ddot{I}€inx}$, where the mode coefficients, ${\hat{f}}_n(y)$ are bilinear in $K$-Bessel functions. Invariance under $SL(2,\mathbb{Z})$ requires these modes to satisfy the nontrivial boundary condition ${\hat{f}}_n(y)=O(y^{-2})$ 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-D-instantons and D-instanton/anti-D-instanton pairs.

Latex. 31 pages. v3 : a few clarifications. Version to be published