Journal
COMMUNICATIONS IN NUMBER THEORY AND PHYSICS
Volume 9, Issue 2, Pages 307-344Publisher
INT PRESS BOSTON, INC
DOI: 10.4310/CNTP.2015.v9.n2.a3
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Funding
- European Research Council under the European Community's Seventh Framework Programme (FP7) / ERC [247252]
- ANR [QST ANR 12 BS05 003 01, PICS 6076]
- NSF [DMS-1201362]
- Science and Technology Facilities Council [ST/J000434/1, ST/L000385/1] Funding Source: researchfish
- STFC [ST/L000385/1, ST/J000434/1] Funding Source: UKRI
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The modular invariant coefficient of the (DR4)-R-6 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 -> infinity (the weak coupling limit). The solution is presented as a Fourier series with modes (f) over cap (n)(y)e(2 pi inx), where the mode coefficients, (f) over cap (n)(y) are bilinear in K-Bessel functions. Invariance under SL(2, Z) requires these modes to satisfy the nontrivial boundary condition (f) over cap (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-D-instantons and D-instanton/anti-D-instanton pairs.
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