SL(2,Z)-invariance and D-instanton contributions to the D6R4 interaction

Michael B. Green, Stephen D. Miller, Pierre Vanhove

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


The modular invariant coefficient of the D6R4 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 fn(y)e2πinx, where the mode coefficients, fn(y) are bilinear in K- Bessel functions. Invariance under SL(2,Z) requires these modes to satisfy the nontrivial boundary condition fn(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.

Original languageEnglish (US)
Pages (from-to)307-344
Number of pages38
JournalCommunications in Number Theory and Physics
Issue number2
StatePublished - 2015

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Mathematical Physics
  • Physics and Astronomy(all)


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