On Conformal, SL(4,R) and Sp(8,R) Symmetries of 4d Massless Fields

TL;DR

This work constructs a gauge-potential, $sp(8,\mathbb{R})$-covariant formulation for free massless fields of all spins in 4d, realized via unfolded dynamics and a doubled HS sector with EM duality. It reveals that AdS$_4$ backgrounds support a consistent, conformally extended HS system, while the Minkowski limit induces degeneracies that spoil full conformal or $sp(8,\mathbb{R})$ invariance for spins $s>1$. The analysis leverages Chevalley–Eilenberg cohomology within unfolded FDAs and $\,\sigma_-$ cohomology to classify dynamical fields, gauge symmetries, and equations, and it generalizes the construction to ten-dimensional spaces ${\cal M}_4$ and $Sp(4,\mathbb{R})$-like backgrounds. The results not only clarify the conformal and duality structure of free HS theories but also set a foundation for nonlinear, $sp(8,\mathbb{R})$-invariant HS dynamics and potential connections to matrix-model frameworks and higher-dimensional extensions.

Abstract

The $sp(8, R)$ invariant formulation of free field equations of massless fields of all spins in $AdS_4$ available previously in terms of gauge invariant field strengths is extended to gauge potentials. As a by-product, free field equations for a massless gauge field are shown to possess both $su(2,2)\sim o(4,2)$ and $sl(4,R)\sim o(3,3)$ symmetry. The proposed formulation is well-defined in the $AdS_4$ background but experiences certain degeneracy in the flat limit that does not allow conformal invariant field equations for spin $s>1$ gauge fields in Minkowski space. The basis model involves the doubled set of fields of all spins. It is manifestly invariant under U(1) electric-magnetic duality extended to higher spins. Reduction to a single massless field contains the equations that relate its electric and magnetic potentials which are mixed by the conformal transformations for s>1. We use the unfolded formulation approach recalled in the paper with some emphasis on the role of Chevalley-Eilenberg cohomology of a Lie algebra $g$ in $g$-invariant field equations. This method makes it easy to guess a form of the 4d $sp(8, R)$ invariant massless field equations and then to extend them to the ten dimensional $sp(8,R)$ invariant space-time. Dynamical content of the field equations is analyzed in terms of $σ_-$ cohomology.