A higher-spin Chern-Simons theory of anyons

Abstract

We propose Chern-Simons models of fractional-spin fields interacting with ordinary tensorial higher-spin fields and internal color gauge fields. For integer and half-integer values of the fractional spins, the model reduces to finite sets of fields modulo infinitedimensional ideals. We present the model on-shell using Fock-space representations of the underlying deformed-oscillator algebra.Quantum mechanics in 2 + 1 dimensions admits anyons: fractional-spin particles [1] with correlated generalized statistics properties governed by the braid group [2]. Anyons arise in a number of systems; for example, as non-relativistic charged vortices [3], relativistic Hopf-interacting massive particles in matter-coupled Chern-Simons theories [4,5] and vertex operators in two-dimensional conformal field theories [6]. In this letter, we present topological models of Chern-Simons type that describe fractional-spin gauge fields coupled to higherspin gravities (HSGRA) and internal gauge fields, which we will refer to as fractional-spin gravities. Diffeomorphism invariance as well as higher-spin symmetries are indeed natural in theories of anyons, essentially due to the topological character of the local rotations and translations underlying the generalized spin-statistics relations [2] and the fact that local constructs built out of fractional-spin fields decompose under the Lorentz algebra into infinite towers of higher-spin Lorentz tensors and tensor-spinors.Our construction stands on the observation that the Prokushkin-Vasiliev system [7], which provides the only known fully non-linear description of three-dimensional mattercoupled HSGRA, admits several inequivalent embeddings of the Lorentz algebra into its higher-spin algebra [8], besides the standard embedding leading to tensor-spinorial HSGRA. The Prokushkin-Vasiliev system consists of a connection one-form A and matter zero-form B living on a base manifold given locally by the direct product of a commutative spacetime M and non-commutative twistor space Z with a closed and central two-form J . These master fields are valued in associative algebras consisting of functions on a fiber manifold Y × I , the product of an additional twistor space Y and an internal manifold I whose coordinates generate a matrix algebra; for further details, see [8]. The Prokushkin-Vasiliev field equations, viz.

Three-dimensional fractional-spin gravity

“…In section 2, which can be skipped at first reading, we collect further background material, general remarks on higher-spin gravities in three dimensions and how our fractional-spin 3 See also the conference proceeding [28]. 4 As observed by Vasiliev, in Lorentzian signature and in the presence of a positive cosmological constant, supergravities [36] and linearized higher-spin supergravities [37] admit twisted reality conditions compatible with Z2 × Z2 graded quantum algebras; for a recent review and the extension to fully non-linear dS4 higherspin supergravities, see [35].…”

“…The aim of this paper is to demonstrate within a simple class of models, namely topological models of Chern-Simons type which we refer to as fractional-spin gravities, how standard tensorial higher-spin gravities can be extended by fractional-spin fields by including additional sets of fiber functions that form Lorentz representations characterized by arbitrary real-valued 3 See also the conference proceeding [28]. 4 As observed by Vasiliev, in Lorentzian signature and in the presence of a positive cosmological constant, supergravities [36] and linearized higher-spin supergravities [37] admit twisted reality conditions compatible with Z2 × Z2 graded quantum algebras; for a recent review and the extension to fully non-linear dS4 higher-spin supergravities, see [35].…”

From phase space to multivector matrix models

“…The theory of Vasiliev extends gauge gravities with generalized spin connections and frame fields which carry not only spin two but all possibles spin. Besides standard statistics [50][51][52][53][54], in 2`1 dimensions the Chern-Simons higher spin gravity [34,55] can be extended with fractional spin gauge fields [56][57][58]. The gauge algebras of these theories are referred as to (fractional) higher spin algebras.…”

Notes on higher-spin algebras: minimal representations and structure constants

“…The latter algebra, also known as Aq(2; ν) [14], has been investigated by many authors -see [71,72] for recent works. In this subsection, we show how the known structures of hs[λ] can be derived from the results obtained in this paper for hs λ (sl N ).…”