Minimal representations, spherical vectors, and exceptional theta series I

Abstract

Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group $G$ is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of $G$, generalizing the Schrödinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic theta series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the $p$-adic number fields provides the summation measure in the theta series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born-Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.

41 pages, uses JHEP.cls, form and mathematica files at http://www.lpthe.jussieu.fr/~pioline/minrep/; v2: discussion of p-adic spherical vector and adelic formulation of theta series in sec 2.3, discussion of standard minimal rep improved in sec 3.1, complex spherical vector obtained in sec 4.5, plus various cosmetic changes. Final version to appear in CMP

References in corpus (2)

• U-duality from Matrix Membrane Partition Function
• M-theory and automorphic scattering

Cited by in corpus (46)

• Universal BPS structure of stationary supergravity solutions
• Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
• BPS black holes, quantum attractor flows and automorphic forms
• $R^4$ couplings and automorphic unipotent representations
• Small representations, string instantons, and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa)
• Quantum Attractor Flows
• BPS Black Hole Degeneracies and Minimal Automorphic Representations
• Topological wave functions and heat equations
• Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups
• Quantum Cosmology and Conformal Invariance
• Eisenstein series and automorphic representations
• Loops in exceptional field theory
• Minimal unitary representation of SU(2,2) and its deformations as massless conformal fields and their supersymmetric extensions
• Minimal unitary representations from supersymmetry
• Quasi-conformal actions, quaternionic discrete series and twistors: SU(2,1) and G_2(2)
• The two D6R4 type invariants and their higher order generalisation
• A Unified Approach to the Minimal Unitary Realizations of Noncompact Groups and Supergroups
• Deformed Twistors and Higher Spin Conformal (Super-)Algebras in Six Dimensions
• Minimal Unitary Realizations of Exceptional U-duality Groups and Their Subgroups as Quasiconformal Groups
• Aspects of higher curvature terms and U-duality
• The Automorphic Membrane
• Higher-derivative gauge field terms in the M-theory action
• Comments on the topological open membrane
• Minimal representations: spherical vectors and automorphic functionals
• Eisenstein series for infinite-dimensional U-duality groups
• U-duality covariant membranes
• The double cover of odd general spin groups, small representations and applications
• Minimal unitary representation of SO*(8) = SO(6,2) and its SU(2) deformations as massless 6D conformal fields and their supersymmetric extensions
• Automorphic Instanton Partition Functions on Calabi-Yau Threefolds
• The automorphic NS5-brane
• Arithmetic and Hyperbolic Structures in String Theory
• Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories
• Minimal unitary representation of D(2,1;λ) and its SU(2) deformations and d=1, N=4 superconformal models
• SU(2) deformations of the minimal unitary representation of OSp(8*|2N) as massless 6D conformal supermultiplets
• Heterotic-type II duality in twistor space
• $\mathcal{E}{\nabla }^4{R}^4$ type invariants and their gradient expansion
• The E10 Wheeler-DeWitt operator at low levels
• Perturbative higher-derivative terms in N=6 asymmetric orbifolds
• Quasiconformal Group Approach to Higher Spin Algebras, their Deformations and Supersymmetric Extensions
• Minimal representations and reductive dual pairs in conformal field theory
• Conformally invariant differential operators on Heisenberg groups and minimal representations
• Representations distinguished by pairs of exceptional representations and a conjecture of Savin
• Existence of supersingular representations of p-adic reductive groups
• Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
• The Minimal Unitary Representation of E_8(8)
• Saturating unitarity bounds at U-duality symmetric points