Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$R^4$ couplings and automorphic unipotent representations

Boris Pioline

TL;DR

This work proposes that eight-derivative $R^4$ couplings in maximal toroidal compactifications are governed by the minimal theta series of the U-duality group $G_D$, unifying perturbative and non-perturbative physics through automorphic representations. The main approach identifies the infinitesimal character and the 1/2-BPS-restricted Fourier coefficients with the minimal unitary representation, predicting that non-perturbative corrections arise from 1/2-BPS instantons and higher-dimensional particles, with Taub-NUT effects in $D=3$. The paper provides explicit analyses in $D=3$–$6$ across M-theory limits on $T^d$, detailing decompactification and string/M-theory limits via constrained Epstein zeta and minimal theta series, and extends the discussion to potential 1/4-BPS couplings through next-to-minimal representations. If correct, this framework offers a precise, testable bridge between BPS spectra, U-duality, and non-perturbative gravity amplitudes, with decompactification and weak-coupling checks guiding future work.

Abstract

Four-graviton, eight-derivative couplings in the low energy effective action of toroidal type II string compactifications are tightly constrained by U-duality invariance and by supersymmetry. In this note, we revisit earlier proposals for the automorphic form governing these couplings in dimension D=3,4,5,6, and propose that the correct automorphic form is the minimal theta series for the corresponding U-duality group. Evidence for this proposal comes from i) the matching of infinitesimal characters, ii) the fact that the Fourier coefficients have support on 1/2-BPS charges and iii) decompactification limits. In particular, we show that non-perturbative effects can be interpreted as 1/2-BPS instantons, or 1/2-BPS particles in one dimension higher (together with Taub-NUT instantons in the D=3 case). Based on similar considerations, we also conjecture the form of 1/4-BPS saturated couplings such as $\nabla^4 R^4$ couplings in the same dimensions.

$R^4$ couplings and automorphic unipotent representations

TL;DR

This work proposes that eight-derivative couplings in maximal toroidal compactifications are governed by the minimal theta series of the U-duality group , unifying perturbative and non-perturbative physics through automorphic representations. The main approach identifies the infinitesimal character and the 1/2-BPS-restricted Fourier coefficients with the minimal unitary representation, predicting that non-perturbative corrections arise from 1/2-BPS instantons and higher-dimensional particles, with Taub-NUT effects in . The paper provides explicit analyses in – across M-theory limits on , detailing decompactification and string/M-theory limits via constrained Epstein zeta and minimal theta series, and extends the discussion to potential 1/4-BPS couplings through next-to-minimal representations. If correct, this framework offers a precise, testable bridge between BPS spectra, U-duality, and non-perturbative gravity amplitudes, with decompactification and weak-coupling checks guiding future work.

Abstract

Four-graviton, eight-derivative couplings in the low energy effective action of toroidal type II string compactifications are tightly constrained by U-duality invariance and by supersymmetry. In this note, we revisit earlier proposals for the automorphic form governing these couplings in dimension D=3,4,5,6, and propose that the correct automorphic form is the minimal theta series for the corresponding U-duality group. Evidence for this proposal comes from i) the matching of infinitesimal characters, ii) the fact that the Fourier coefficients have support on 1/2-BPS charges and iii) decompactification limits. In particular, we show that non-perturbative effects can be interpreted as 1/2-BPS instantons, or 1/2-BPS particles in one dimension higher (together with Taub-NUT instantons in the D=3 case). Based on similar considerations, we also conjecture the form of 1/4-BPS saturated couplings such as couplings in the same dimensions.

Paper Structure

This paper contains 14 sections, 70 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Constrained zeta series and infinitesimal characters
  3. $R^4$ couplings and minimal representations
  4. Analysis of non-perturbative effects
  5. M-theory on $T^5$
  6. M-theory on $T^6$
  7. M-theory on $T^7$
  8. M-theory on $T^8$
  9. Comments on 1/4-BPS couplings
  10. Decompactification limits
  11. Constrained Epstein Zeta series and degenerate Eisenstein series
  12. Constant terms
  13. Decompactification limits for $R^4$ couplings
  14. Decompactification limits for $\nabla^4 R^4$ couplings