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Eisenstein Series and String Thresholds

N. A. Obers, B. Pioline

TL;DR

The paper develops a unified automorphic framework for representing G(Z)-invariant string theory amplitudes using Eisenstein series on symmetric spaces associated with duality groups (e.g., SL(d, Z), SO(d,d, Z), E_{d+1(d+1)}(Z)). By matching Laplacian eigenvalues and asymptotics with perturbative string thresholds, it derives manifestly T-duality invariant expressions for R^4-type couplings and conjectures non-perturbative U-duality completions, including predictions for D-brane and NS5-brane instantons. The authors further extend these methods to higher genus, linking genus-g thresholds and higher derivative couplings to Eisenstein series in larger duality groups, and propose a non-perturbative, U-duality invariant structure for R^4 H^{4g-4} terms. Collectively, the work provides a mathematical and physical bridge between automorphic forms on symmetric spaces and exact non-perturbative information in maximal supergravity and M-theory compactifications, with implications for instanton physics and duality structures across dimensions.

Abstract

We investigate the relevance of Eisenstein series for representing certain $G(Z)$-invariant string theory amplitudes which receive corrections from BPS states only. $G(Z)$ may stand for any of the mapping class, T-duality and U-duality groups $Sl(d,Z)$, $SO(d,d,Z)$ or $E_{d+1(d+1)}(Z)$ respectively. Using $G(Z)$-invariant mass formulae, we construct invariant modular functions on the symmetric space $K\backslash G(R)$ of non-compact type, with $K$ the maximal compact subgroup of $G(R)$, that generalize the standard non-holomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincaré upper half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in one- and $g$-loop string amplitudes, we obtain a manifestly T-duality invariant representation of the latter, conjecture their non-perturbative U-duality invariant extension, and analyze the resulting non-perturbative effects. This includes the $R^4$ and $R^4 H^{4g-4}$ couplings in toroidal compactifications of M-theory to any dimension $D\geq 4$ and $D\geq 6$ respectively.

Eisenstein Series and String Thresholds

TL;DR

The paper develops a unified automorphic framework for representing G(Z)-invariant string theory amplitudes using Eisenstein series on symmetric spaces associated with duality groups (e.g., SL(d, Z), SO(d,d, Z), E_{d+1(d+1)}(Z)). By matching Laplacian eigenvalues and asymptotics with perturbative string thresholds, it derives manifestly T-duality invariant expressions for R^4-type couplings and conjectures non-perturbative U-duality completions, including predictions for D-brane and NS5-brane instantons. The authors further extend these methods to higher genus, linking genus-g thresholds and higher derivative couplings to Eisenstein series in larger duality groups, and propose a non-perturbative, U-duality invariant structure for R^4 H^{4g-4} terms. Collectively, the work provides a mathematical and physical bridge between automorphic forms on symmetric spaces and exact non-perturbative information in maximal supergravity and M-theory compactifications, with implications for instanton physics and duality structures across dimensions.

Abstract

We investigate the relevance of Eisenstein series for representing certain -invariant string theory amplitudes which receive corrections from BPS states only. may stand for any of the mapping class, T-duality and U-duality groups , or respectively. Using -invariant mass formulae, we construct invariant modular functions on the symmetric space of non-compact type, with the maximal compact subgroup of , that generalize the standard non-holomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincaré upper half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in one- and -loop string amplitudes, we obtain a manifestly T-duality invariant representation of the latter, conjecture their non-perturbative U-duality invariant extension, and analyze the resulting non-perturbative effects. This includes the and couplings in toroidal compactifications of M-theory to any dimension and respectively.

Paper Structure

This paper contains 40 sections, 5 theorems, 203 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Toroidal compactification and $SL(d,\mathbb{Z})$ Eisenstein series
  3. Moduli space and Iwasawa gauge
  4. Fundamental and antifundamental Eisenstein series
  5. Higher representations and constrained Eisenstein series
  6. Decompactification and analyticity
  7. Partial Iwasawa decomposition
  8. $SO(d,d,\mathbb{Z})$ Eisenstein series and one-loop thresholds
  9. Moduli space and Iwasawa gauge
  10. Spinor and vector Eisenstein series
  11. One-loop modular integral and method of orbits
  12. A new second order differential operator
  13. Large volume behaviour
  14. Asymmetric thresholds and elliptic genus
  15. Boundary term and symmetry enhancement
  16. ...and 25 more sections

Key Result

Proposition 1

The constrained Eisenstein series geneis associated to the representation of highest weight $\lambda$ is an eigenmode of the Laplacian with eigenvalue

Theorems & Definitions (13)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • Conjecture 5
  • Conjecture 6
  • Conjecture 7
  • Corollary 8
  • Conjecture 9
  • Conjecture 10
  • ...and 3 more