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A Fiber Approach to Harmonic Analysis of Unfolded Higher-Spin Field Equations

Carlo Iazeolla, Per Sundell

Abstract

In Vasiliev's unfolded formulation of higher-spin dynamics the standard fields are embedded on-shell into covariantly constant master fields valued in Lorentz-covariant slices of the star-product algebra A of functions on the singleton phase space. Correspondingly, the harmonic expansion is taken over compact slices of A that are unitarizable in a rescaled trace-norm rather than the standard Killing norm. Motivated by the higher-derivative nature of the theory, we examine indecomposable unitarizable Harish-Chandra modules consisting of standard massless particles plus linearized runaway solutions. This extension arises naturally in the above fiber approach upon realizing compact-weight states as non-polynomial analytic functions in A.

A Fiber Approach to Harmonic Analysis of Unfolded Higher-Spin Field Equations

Abstract

In Vasiliev's unfolded formulation of higher-spin dynamics the standard fields are embedded on-shell into covariantly constant master fields valued in Lorentz-covariant slices of the star-product algebra A of functions on the singleton phase space. Correspondingly, the harmonic expansion is taken over compact slices of A that are unitarizable in a rescaled trace-norm rather than the standard Killing norm. Motivated by the higher-derivative nature of the theory, we examine indecomposable unitarizable Harish-Chandra modules consisting of standard massless particles plus linearized runaway solutions. This extension arises naturally in the above fiber approach upon realizing compact-weight states as non-polynomial analytic functions in A.

Paper Structure

This paper contains 46 sections, 371 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Masslessness and compositeness
  3. Interlude: Canonical vs unfolded higher-spin equations
  4. Phase-space quantization
  5. Runaway solutions and extended compositeness
  6. Enveloping-Algebra Realization of Higher-Spin Master Fields
  7. Associative quotient algebra
  8. Adjoint and twisted-adjoint $\mathfrak{so}(D+1;{\mathbb C})$ modules
  9. Adjoining the adjoint and composite-massless representations
  10. Non-composite trace and reflectors
  11. The adjoint and twisted-adjoint higher-spin representations
  12. Adjoint and twisted-adjoint master fields
  13. Harmonic expansion of the Weyl zero-form
  14. Unitarity of the harmonic expansion on $dS_D$
  15. On the Harmonic Expansion on $AdS_D$
  16. ...and 31 more sections

Figures (2)

  • Figure 1: The adjoint module $\mathfrak{D}(-(s-1);(s-1))$ is connected via the conjugate massless module $\mathfrak{D}(-(s-2);(s))$ to the massless module $\mathfrak{D}(s+2\epsilon_0;(s))$.
  • Figure 2: The $(\mathfrak{so}(2)\oplus\mathfrak{so}(D-1))$-types arising in the scalar 2-lineton in $D=9$.