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

Abstract

The $sp(8,R)$ invariant formulation of free field equations of massless fields of all spins in $Ad{S}_4$ 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 $Ad{S}_4$ 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 ${\sigma }_{-}$ cohomology.

arXiv comment66 pages, V2: minor corrections, references and acknowledgments added, coefficients and typos corrected, V3: typos corrected, version to appear in Nucl. Phys. B

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