Higher Spins in D=2+1

TL;DR

This work analyzes higher-spin interactions in $D=2+1$ using a Chern-Simons (CS) formulation based on enlarged gauge algebras, providing a tractable setting where non-linear deformations of free higher-spin dynamics can be explored. It shows how the Einstein–Hilbert action emerges as a CS theory with a suitable invariant form and explains how truncations to $ ext{sl}(N, ext{R})$ or infinite-dimensional algebras realize finite or Vasiliev-like spectra, with asymptotic symmetries governed by nonlinear $ ext{W}$-algebras. The paper also investigates topologically massive extensions by adding a gravitational CS term and torsion constraints, highlighting gauge-invariance challenges in preserving extended translations at the non-linear level and outlining the open question of fully consistent massive higher-spin theories in three dimensions. Overall, the work provides a coherent, tractable framework for studying higher-spin interactions, holographic dualities through $ ext{W}$-algebras, and potential massive deformations in a lower-dimensional setting.

Abstract

We give a brief overview of some three-dimensional toy models for higher-spin interactions. We first review the construction of pure higher-spin gauge theories in terms of Chern-Simons theories. We then discuss how this setup could be modified along the lines of the known topologically massive theories.