English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The modular invariant coefficient of the $D^6R^4$ interaction in the lowenergy expansion of type~IIB string theory has been conjectured to be asolution of an inhomogeneous Laplace eigenvalue equation, obtained byconsidering the toroidal compactification of two-loop Feynman diagrams ofeleven-dimensional supergravity. In this paper we determine the exact$SL(2,\mathbb Z)$-invariant solution $f(x+iy)$ to this differential equationsatisfying an appropriate moderate growth condition as $y\to \infty$ (the weakcoupling 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 boundarycondition $ \widehat{f}_n(y) =O(y^{-2})$ for small $y$, which uniquelydetermines the solution. The large-$y$ expansion of $f(x+iy)$ contains theknown perturbative (power-behaved) terms, together with precisely-determinedexponentially decreasing contributions that have the form expected ofD-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs.

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