SL(2,Z)-invariance and D-instanton contributions to the D<sup>6</sup>R<sup>4</sup>interaction
307 - 344
Communications in Number Theory and Physics
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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,Z)-invariant solution f(x + iy) to this differential equation satisfying an appropriate moderate growth condition as y → ∞ (the weak coupling limit). The solution is presented as a Fourier series with modes f n (y)e 2πinx , where the mode coefficients, f n (y) are bilinear in K- Bessel functions. Invariance under SL(2,Z) requires these modes to satisfy the nontrivial boundary condition 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.
AuthorsGreen, MB; Miller, SD; Vanhove, P
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