Publication : t14/009

Abstract:
The modular invariant coefficient of the $D^6R^4$ 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 its exact $SL(2,mathbb Z)$-invariant solution $f(Omega)$ as a function of the complex modulus, $Omega=x+iy$, satisfying an appropriate moderate growth condition as $yto infty$ (the weak coupling limit). The solution is presented as a Fourier series with modes $widehat{f}_n(y) e^{2pi i n x}$, where the mode coefficients, $widehat{f}_n(y)$ are bilinear in $K$-Bessel functions. Invariance under $SL(2,mathbb Z)$ requires these modes to satisfy the nontrivial boundary condition $ widehat{f}_n(y) =O(y^{-2})$ for small $y$, which uniquely determines the solution. The large-$y$ expansion of $f(Omega)$ 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.