Three-dimensional fractional-spin gravity

Abstract

Using Wigner-deformed Heisenberg oscillators, we construct 3D Chern-Simons models consisting of fractional-spin fields coupled to higher-spin gravity and internal nonabelian gauge fields. The gauge algebras consist of Lorentz-tensorial Blencowe-Vasiliev higher-spin algebras and compact internal algebras intertwined by infinite-dimensional generators in lowest-weight representations of the Lorentz algebra with fractional spin. In integer or half-integer non-unitary cases, there exist truncations to gl(ℓ, ℓ ± 1) or gl(ℓ|ℓ ± 1) models. In all non-unitary cases, the internal gauge fields can be set to zero. At the semi-classical level, the fractional-spin fields are either Grassmann even or odd. The action requires the enveloping-algebra representation of the deformed oscillators, while their Fock-space representation suffices on-shell.

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 ).…”

A higher-spin Chern-Simons theory of anyons

“…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 .…”

“…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.…”

Higher Spin Matrix Models

“…Note that, considering formal Taylor expansions of the sections Ypy; xq in terms of the basis generators ys, we can extend the universal enveloping algebra U pYq to non-polynomial classes of functions/distributions (see [16,17] for their use in fractional spin gravity). If ys are finite dimensional, the expansion (10) will be truncated.…”

“…where, compared to Labels (11) and (12), the kinetic constraints are given by Labels (15) and the rigid ones by Labels (16), while the zero forms are given by Y " tB, S α u. Here α is a spinor index in three dimensions.…”