Small representations, string instantons, and Fourier modes of Eisenstein series

Michael B. Green, Stephen D. Miller, Pierre Vanhove

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


Text: This paper concerns some novel features of maximal parabolic Eisenstein series at certain special values of their analytic parameter, s. These series arise as coefficients in the R4 and ∂4R4 interactions in the low energy expansion of the scattering amplitudes in maximally supersymmetric string theory reduced to D=10-d dimensions on a torus, Td (0≤d≤7). For each d these amplitudes are automorphic functions on the rank d+1 symmetry group Ed+1. Of particular significance is the orbit content of the Fourier modes of these series when expanded in three different parabolic subgroups, corresponding to certain limits of string theory. This is of interest in the classification of a variety of instantons that correspond to minimal or "next-to-minimal" BPS orbits. In the limit of decompactification from D to D+1 dimensions many such instantons are related to charged 1/2-BPS or 1/4-BPS black holes with euclidean world-lines wrapped around the large dimension. In a different limit the instantons give non-perturbative corrections to string perturbation theory, while in a third limit they describe non-perturbative contributions in eleven-dimensional supergravity. A proof is given that these three distinct Fourier expansions have certain vanishing coefficients that are expected from string theory. In particular, the Eisenstein series for these special values of s have markedly fewer Fourier coefficients than typical maximal parabolic Eisenstein series. The corresponding mathematics involves showing that the wavefront sets of the Eisenstein series in question are supported on only a limited number of coadjoint nilpotent orbits - just the minimal and trivial orbits in the 1/2-BPS case, and just the next-to-minimal, minimal and trivial orbits in the 1/4-BPS case. Thus as a byproduct we demonstrate that the next-to-minimal representations occur automorphically for E6, E7, and E8, and hence the first two nontrivial low energy coefficients in scattering amplitudes can be thought of as exotic θ-functions for these groups. The proof includes an appendix by Dan Ciubotaru and Peter E. Trapa which calculates wavefront sets for these and other special unipotent representations. Video: For a video summary of this paper, please click here or visit

Original languageEnglish (US)
Pages (from-to)187-309
Number of pages123
JournalJournal of Number Theory
Issue numberC
StatePublished - 2015

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


  • Automorphic forms
  • BPS states
  • Charge lattice
  • Coadjoint nilpotent orbits
  • Eisenstein series
  • Fourier expansions
  • Scattering amplitudes
  • Small representations
  • String theory
  • Unipotent representations

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