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

M. B. Green, S. D. Miller, P. Vanhove

TL;DR

This work determines the exact $SL(2,\mathbb{Z})$-invariant coefficient $f(\Omega)$ of the $D^6R^4$ interaction in type IIB string theory by solving the inhomogeneous Laplace equation $(\Delta_\Omega-12) f(\Omega) = -2\,\zeta(3)\, E_{\frac{3}{2}}(\Omega)^2$ for $\Omega=x+iy$. Through a Fourier expansion $f(x+iy)=\sum_n \hat{f}_n(y)\, e^{2\pi i n x}$ and two boundary conditions—$f(\Omega)=O(y^3)$ as $y\to\infty$ (weak coupling) and $\hat{f}_n(y)=O(y^{-2})$ as $y\to0$ (strong coupling)—the authors derive the full set of mode equations and solve them to obtain explicit forms for all $\hat{f}_n(y)$, which are bilinear in $K$-Bessel functions. The large-$y$ expansion contains the known perturbative terms and a precisely determined spectrum of exponentially suppressed contributions that match D-instanton, anti-D-instanton, and instanton/anti-instanton pair effects; the small-$y$ behavior is constrained by automorphic invariance, enforcing a non-singular strong-coupling limit. The results corroborate string perturbation theory up to genus three and reveal the detailed instanton structure, while the authors also present alternative formulations (Poincaré series, automorphic distributions, spectral methods) with broader implications for higher-order corrections in the IIB effective action.

Abstract

The modular invariant coefficient of the $D^6R^4$ 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 $\widehat{f}_n(y) e^{2πi n x}$, 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 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-D-instantons and D-instanton/anti-D-instanton pairs.

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

TL;DR

This work determines the exact -invariant coefficient of the interaction in type IIB string theory by solving the inhomogeneous Laplace equation for . Through a Fourier expansion and two boundary conditions— as (weak coupling) and as (strong coupling)—the authors derive the full set of mode equations and solve them to obtain explicit forms for all , which are bilinear in -Bessel functions. The large- expansion contains the known perturbative terms and a precisely determined spectrum of exponentially suppressed contributions that match D-instanton, anti-D-instanton, and instanton/anti-instanton pair effects; the small- behavior is constrained by automorphic invariance, enforcing a non-singular strong-coupling limit. The results corroborate string perturbation theory up to genus three and reveal the detailed instanton structure, while the authors also present alternative formulations (Poincaré series, automorphic distributions, spectral methods) with broader implications for higher-order corrections in the IIB effective action.

Abstract

The modular invariant coefficient of the 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 -invariant solution to this differential equation satisfying an appropriate moderate growth condition as (the weak coupling limit). The solution is presented as a Fourier series with modes , where the mode coefficients, are bilinear in -Bessel functions. Invariance under requires these modes to satisfy the nontrivial boundary condition for small , which uniquely determines the solution. The large- expansion of 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.

Paper Structure

This paper contains 13 sections, 6 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Fourier modes of the inhomogeneous Laplace equation
  3. Fourier modes and boundary conditions
  4. Solutions of the equations in distinct sectors of $n_1$ and $n_2$
  5. The complete expression for each Fourier mode, $\widehat{f}_n(y)$
  6. Discussion and connections with string theory
  7. The solution as a Poincaré Series
  8. Sum over translates of one-dimensional solution
  9. Fourier coefficients via a term-by-term analysis
  10. Poisson summation for $\Sigma^{0,1}$ and $\Sigma^{1,0}$
  11. Poisson summation for $\Sigma^{1,1}$
  12. Poincaré series and Eisenstein automorphic distributions
  13. Solution as expressed via spectral theory

Key Result

Theorem 1.6

The unique solution (laplaceeigen) satisfying (largebound) is given in terms of (fouriermodes) by where $\alpha_n$ are constants, and $\tilde{f}$ and $M^{ij}_{n_1,n_2}(z)$ are polynomials in $z$ and $1/z$. The $K$-Bessel functions must be replaced by an appropriate limit when either $n$, $n_1$, or $n_2$ vanishes; see section sec:solutions for complete details.

Theorems & Definitions (9)

  • Theorem 1.6
  • Lemma 2.9
  • proof
  • Lemma A.6
  • Lemma A.41
  • proof
  • Lemma B.7
  • proof
  • Lemma B.12