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

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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 $y\to \infty$ (the weak coupling limit). The solution is presented as a Fourier series with modes $\widehat{f}_n(y) e^{2\pi i n x}$, 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.

Dates and versions

hal-00976540 , version 1 (10-04-2014)

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M. B. Green, S. D. Miller, P. Vanhove. SL(2,Z)-invariance and D-instanton contributions to the D^6R^4 interaction. 2014. ⟨hal-00976540⟩
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