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# Communications in Number Theory and Physics

## Volume 9 (2015)

### Number 2

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

Pages: 307 – 344

DOI: http://dx.doi.org/10.4310/CNTP.2015.v9.n2.a3

#### Authors

#### 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 $\widehat{f}_n (y) e^{2 \pi \mathit{inx}}$, 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 $f(x + iy)$ 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.