Scalar Fields in 3D Asymptotically Flat Higher-Spin Gravity

Abstract

In this work we construct a novel associative algebra and use it to define a theory of higher-spin gravity in (2+1)-dimensional asymptotically flat spacetimes. Our construction is based on a quotient of the universal enveloping algebra (UEA) of $\mathfrak{isl}(2,\mathbb{R})$ with respect to the ideal generated by its Casimir elements, the mass squared ${\mathcal{M}}^2$ and the three-dimensional analogue of the square of the Pauli-Lubanski vector $\mathcal{S}$ and propose to call the resulting associative algebra $\mathfrak{ihs}({\mathcal{M}}^2,\mathcal{S})$. We provide a definition of its generators and even though we are not yet able to provide the complete set of multiplication rules of this algebra our analysis allows us to study many interesting and relevant sub-structures of $\mathfrak{ihs}({\mathcal{M}}^2,\mathcal{S})$. We then show how to consistently couple a scalar field to an $\mathfrak{ihs}({\mathcal{M}}^2,\mathcal{S})$ higher-spin gauge theory.

arXiv comment29 pages, 4 figures; v2: modified the title, added references as well as a Mathematica file containing the multiplication rules of $\mathfrak{ihs}({\mathcal{M}}^2,\mathcal{S})$ as an ancillary file

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• Fractons, dipole symmetries and curved spacetime
• Carrollian and Galilean conformal higher-spin algebras in any dimensions
• Carrollian manifolds and null infinity: A view from Cartan geometry
• Massless Scalars and Higher-Spin BMS in Any Dimension
• Carrollian conformal scalar as flat-space singleton
• Higher-spin gauge theories in three spacetime dimensions
• Unfolded Fierz-Pauli Equations in Three-Dimensional Asymptotically Flat Spacetimes
• Conformal Carrollian Spin-3 Gravity in 3d
• Conformal Galilean Spin-3 Gravity in 3d
• Asymptotic symmetries in Carrollian theories of gravity with a negative cosmological constant