Unfolding Mixed-Symmetry Fields in AdS and the BMV Conjecture: I. General Formalism

Abstract

We present some generalities of unfolded on-shell dynamics that are useful in analysing the BMV conjecture for mixed-symmetry fields in constantly curved backgrounds. In particular we classify the Lorentz-covariant Harish-Chandra modules generated from primary Weyl tensors of arbitrary mass and shape, and in backgrounds with general values of the cosmological constant. We also discuss the unfolded notion of local degrees of freedom in theories with and without gravity and with and without massive deformation parameters, using the language of Weyl zero-form modules and their duals.

Figures (4)

• Figure 1: An unfolded module of the form $\mathfrak{R}=\mathfrak{R}'\subsetplus \widetilde{\mathfrak{R}}_2$ where (i) $\mathfrak{R}'=\mathfrak{C}^{\bf 0}\subsetplus \widetilde{\mathfrak{R}}_1$ is a submodule consisting of a Weyl zero$\,$-form module $\mathfrak{C}^{\bf 0}$ with primary Weyl tensor $C$ and dual subcycle $\widetilde{\mathfrak{R}}_1$ ("potential module") with dynamical field $\varphi_{_1}$; and (ii) $\widetilde{\mathfrak{R}}_2$ is a dual cycle ("dual potential module") with dynamical field $\varphi_{_2}$ ("dual potential'). The dashed lines indicate "gluings" by non-trivial generators in $\sigma^-_{_0}$ (see Section \ref{['Sec:sminus']}) whose existence conditions depend on the nature of the underlying symmetry Lie algebra $\mathfrak{g}$ (see Section \ref{['Sec:cycles']}).
• Figure 2: By means of the integration lemma, the primary Weyl tensor $C(\overline{\Theta}_{_2})$ with Bianchi identity $\mathbb{B}_{_{2,1}}(\overline{\Theta}_{_2})$ is shown to correspond to a massless gauge field $\varphi_{_2}(\Theta)$ whose shape is obtained from $\overline{\Theta}_{_2}$ by cutting off one row from its second block and by adding one to its third block. It possesses a one-derivative gauge symmetry with parameter $\epsilon_{_2}(\Theta')$, obtained from $\Theta$ by deleting one cell in the second block.
• Figure 3: Through the integration lemma explained above, the primary Weyl tensor $C(\overline{\Theta}_{_2})$ with second block of height one and Bianchi identity $\mathbb{B}_{_{2,1}}(\overline{\Theta}_{_2})$ is shown to correspond to a partially massless gauge field $\varphi_{_2}(\Theta)$ whose shape is obtained from $\overline{\Theta}_{_2}$ by cutting off its second block and by adding one row to its third block. It possesses a higher-derivative gauge symmetry with parameter $\epsilon_{_2}(\Theta')$, obtained from $\Theta$ by deleting $\overline{s}_{_{12}}+1$ cells in the second block.
• Figure 4: Some entries of the bi-graded triangular module for the massive spin-1 field in flat spacetime. The $\sigma^-$-cohomology contains the massive gauge field $\blacksquare$ at $g=1$, the massive gauge condition $\bullet$ at $g=1$, the Proca equation $\blacksquare$ at $g=2$ and the Noether identity $\bullet$ at $g=2$.