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Automorphic properties of low energy string amplitudes in various dimensions

Michael B. Green, Jorge G. Russo, Pierre Vanhove

TL;DR

The paper develops a unifying automorphic-framework for the moduli-dependent coefficients of higher-derivative interactions in the low-energy expansion of four-graviton amplitudes of maximal string theory on toroidal backgrounds. By enforcing consistency across decompactification, perturbative string, and M-theory limits, it derives explicit forms for the R^4, ∂^4R^4, and ∂^6R^4 coefficients in dimensions D=9,8,7 (and partial results in D=6), expressing them as linear combinations of Eisenstein series for maximal parabolic subgroups. The approach reveals that these coefficients satisfy Laplace eigenvalue equations with (inhomogeneous) source terms, whose structure encodes ultraviolet behaviours of supergravity and the finiteness of string theory, including intricate logarithmic terms tied to massless thresholds. The results are cross-validated against known genus expansions, one- and two-loop supergravity calculations, and M-theory interpretations, providing a coherent, high-rank generalization of earlier SL(2)–based analyses. The framework highlights how maximal supersymmetry and dualities tightly constrain low-energy effective actions across dimensions, with implications for higher-rank automorphic forms and potential extensions to more derivatives and higher D.

Abstract

This paper explores the moduli-dependent coefficients of higher derivative interactions that appear in the low-energy expansion of the four-graviton amplitude of maximally supersymmetric string theory compactified on a d-torus. These automorphic functions are determined for terms up to order D^6R^4 and various values of d by imposing a variety of consistency conditions. They satisfy Laplace eigenvalue equations with or without source terms, whose solutions are given in terms of Eisenstein series, or more general automorphic functions, for certain parabolic subgroups of the relevant U-duality groups. The ultraviolet divergences of the corresponding supergravity field theory limits are encoded in various logarithms, although the string theory expressions are finite. This analysis includes intriguing representations of SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and two-loop string and supergravity amplitudes.

Automorphic properties of low energy string amplitudes in various dimensions

TL;DR

The paper develops a unifying automorphic-framework for the moduli-dependent coefficients of higher-derivative interactions in the low-energy expansion of four-graviton amplitudes of maximal string theory on toroidal backgrounds. By enforcing consistency across decompactification, perturbative string, and M-theory limits, it derives explicit forms for the R^4, ∂^4R^4, and ∂^6R^4 coefficients in dimensions D=9,8,7 (and partial results in D=6), expressing them as linear combinations of Eisenstein series for maximal parabolic subgroups. The approach reveals that these coefficients satisfy Laplace eigenvalue equations with (inhomogeneous) source terms, whose structure encodes ultraviolet behaviours of supergravity and the finiteness of string theory, including intricate logarithmic terms tied to massless thresholds. The results are cross-validated against known genus expansions, one- and two-loop supergravity calculations, and M-theory interpretations, providing a coherent, high-rank generalization of earlier SL(2)–based analyses. The framework highlights how maximal supersymmetry and dualities tightly constrain low-energy effective actions across dimensions, with implications for higher-rank automorphic forms and potential extensions to more derivatives and higher D.

Abstract

This paper explores the moduli-dependent coefficients of higher derivative interactions that appear in the low-energy expansion of the four-graviton amplitude of maximally supersymmetric string theory compactified on a d-torus. These automorphic functions are determined for terms up to order D^6R^4 and various values of d by imposing a variety of consistency conditions. They satisfy Laplace eigenvalue equations with or without source terms, whose solutions are given in terms of Eisenstein series, or more general automorphic functions, for certain parabolic subgroups of the relevant U-duality groups. The ultraviolet divergences of the corresponding supergravity field theory limits are encoded in various logarithms, although the string theory expressions are finite. This analysis includes intriguing representations of SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and two-loop string and supergravity amplitudes.

Paper Structure

This paper contains 51 sections, 375 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Degeneration limits and Eisenstein series for parabolic subgroups
  3. Parabolic subgroups
  4. Eisenstein series for maximal parabolic subgroups and their constant terms.
  5. The expansion parameters.
  6. The $\mathcal{R}^4$ interaction
  7. Nine dimensions
  8. Eight dimensions
  9. Seven dimensions
  10. Six dimensions
  11. The $\partial^4\mathcal{R}^4$ interaction
  12. Nine dimensions
  13. Eight dimensions
  14. Seven dimensions
  15. The $\partial^6\mathcal{R}^4$ interaction
  16. ...and 36 more sections

Figures (1)

  • Figure 1: The Dynkin diagrams relevant to: (i) the $E_{d(d)}$ ($d\le 8$) type II duality groups of type II string theory compactified to $D=11-d$ dimensions on a $(d-1)$-torus. Successive decompactifications to higher dimensions are obtained by deleting the nodes $\alpha_d, \alpha_{d-1}\ \dots$ in (i); (ii) The T-duality groups $SO(10-D,10-D)$ obtained by deleting the left node $\alpha_1$ of (i) are the symmetries of string perturbation theory in $D$ dimensions; (iii) The $SL(11-D)$ groups obtained by deleting node $\alpha_2$ in (i) are associated with the geometric compactification of eleven-dimensional supergravity on a $(11-D)$-torus.