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



Michael B. Green (Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom)

Stephen D. Miller (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Pierre Vanhove (Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France; Institut de Physique Théorique, CEA, IPhT, Gif-sur-Yvette, France; and CNRS, URA 2306, Gif-sur-Yvette, France)


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.

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Published 12 June 2015