Partially-massless higher spin algebras in four dimensions

Abstract

We propose a realisation of partially-massless higher spin algebras in four dimensions in terms of bosonic and fermionic oscillators, using Howe duality between $sp(4,\mathbb{R})\cong so(2,3)$ and $osp(1|2(\ell -1),\mathbb{R})$. More precisely, we show that the centraliser of $osp(1|2(\ell -1),\mathbb{R})$ in the Weyl--Clifford algebra generated by $4$ bosonic and $8(\ell -1)$ fermionic symbols, modulo $osp(1|2(\ell -1),\mathbb{R})$ generators, is isomorphic to the higher spin algebra of the type-A${}_{\ell }$ theory whose spectrum contains partially-massless fields of all spins and depths $t=1,3,\dots ,2\ell -1$. We also discuss the possible existence of a deformation of this algebra, which would encode interaction for the type-A${}_{\ell }$ theory.