An Action for Matter Coupled Higher Spin Gravity in Three Dimensions

TL;DR

This paper introduces a covariant Hamiltonian action for matter-coupled higher spin gravity in three dimensions, unifying various models and truncations within a comprehensive framework involving noncommutative twistor space.

Contribution

It formulates a new covariant Hamiltonian action for matter-coupled higher spin gravity in three dimensions, encompassing multiple models and truncations, including those with non-zero vacuum expectation values.

Findings

Derived a covariant Hamiltonian action on ${\ m X}_4 \times {\ m Z}_2$.

Connected the action to known models like Blencowe's and Prokushkin--Segal--Vasiliev.

Presented alternative models with gauge fields in the Weyl algebra for generic vacuum values.

Abstract

We propose a covariant Hamiltonian action for the Prokushkin and Vasiliev's matter coupled higher spin gravity in three dimensions. The action is formulated on ${\cal X}_4 \times {\cal Z}_2$ where ${\cal X}_4$ is an open manifold whose boundary contains spacetime and ${\cal Z}_2$ is a noncommutative twistor space. We examine various consistent truncations to models of BF type in ${\cal X}_4$ and ${\cal Z}_2$ with B$^2$ terms and central elements. They are obtained by integrating out the matter fields in the presence of a vacuum expectation value $\nu\in \mathbb{R}$ for the zero-form master field. For $\nu=0$, we obtain a model on ${\cal X}_4$ containing Blencowe's action and a model on ${\cal Z}_2$ containing the Prokushkin--Segal--Vasiliev action. For generic $\nu$ (including $\nu=0$), we propose an alternative model on ${\cal X}_4$ with gauge fields in the Weyl algebra of Wigner's deformed oscillator algebra and Lagrange multipliers in the algebra of operators acting in the Fock representation space of the deformed oscillators.