Small representations, string instantons, and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa)

Abstract

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 D4R4 interactions in the low energy expansion of scattering amplitudes in maximally supersymmetric string theory reduced to D=10-d dimensions on a torus T^d, d<8. For each d these amplitudes are automorphic functions on the rank d+1 symmetry group E_d+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 nonperturbative corrections to string perturbation theory, while in a third limit they describe nonperturbative 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 ones. The corresponding mathematics involves showing that the wavefront sets of the Eisenstein series are supported on only certain 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 are exotic theta-functions.

v3: 127 pp. Minor changes. Final version to appear in the Special Issue in honor of Professor Steve Rallis

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• Branes and the Kraft-Procesi transition: classical case
• Loops in exceptional field theory
• $D^6{R}^4$ amplitudes in various dimensions
• D-instantons, String Field Theory and Two Dimensional String Theory
• Normalization of Type IIB D-instanton Amplitudes
• Minimal unitary representations from supersymmetry
• Fluxes and Warping for Gauge Couplings in F-theory
• Tetrahedral modular graph functions
• On supersymmetric E11 exceptional field theory
• The two D6R4 type invariants and their higher order generalisation
• Higher Derivative Supergravities in Diverse Dimensions
• Infrared divergences and harmonic anomalies in the two-loop superstring effective action
• Exact ${\nabla }^4{R}^4$ couplings and helicity supertraces
• Cancellation of divergences up to three loops in exceptional field theory
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• $D^6{R}^4$ curvature corrections, modular graph functions and Poincaré series
• Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• $\mathcal{E}{\nabla }^4{R}^4$ type invariants and their gradient expansion
• The E10 Wheeler-DeWitt operator at low levels
• Black Holes and Higher Composition Laws
• Entirety of cuspidal Eisenstein series on loop groups
• Perturbative higher-derivative terms in N=6 asymmetric orbifolds
• Global Iwasawa-decomposition of SL($n$, ${\mathbb{A}}_{\mathbb{Q}}$)
• Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
• Fourier coefficients attached to small automorphic representations of ${\mathrm{SL}}_n(\mathbb{A})$
• Saturating unitarity bounds at U-duality symmetric points
• Kac--Moody groups and automorphic forms in low dimensional supergravity theories
• Symmetries of non-maximal supergravities with higher-derivative corrections