Higher Spin Conformal Symmetry for Matter Fields in 2+1 Dimensions

TL;DR

The paper realizes conformal higher spin symmetry on 3d massless matter fields by using an auxiliary Fock space dual to a singleton representation, making the symmetry manifest in unfolded, covariant-constancy form $d|\Phi\rangle - \omega * |\Phi\rangle = 0$ on flat and AdS$_3$ backgrounds. It identifies the global symmetry algebra as $hu(1;1|4)$ and derives the covariant transformations and field equations for scalar and spinor within this higher-spin framework. A central result is the explicit Bogolyubov transform relating the non-unitary Fock-module realization to the unitary singleton representation, clarifying the field–singleton duality and connecting to the Flato–Fronsdal spectrum for AdS$_4$ higher spin theories. The work provides a concrete, unfolded description of how higher spin conformal symmetries act on matter in 3d, with potential implications for boundary CFTs and nonlinear extensions in AdS$_4$ HS theories.

Abstract

A simple realization of the conformal higher spin symmetry on the free $3d$ massless matter fields is given in terms of an auxiliary Fock module both in the flat and $AdS_3$ case. The duality between non-unitary field-theoretical representations of the conformal algebra and the unitary (singleton--type) representations of the $3d$ conformal algebra $sp(4,\R)$ is formulated explicitly in terms of a certain Bogolyubov transform.