Three-dimensional fractional-spin gravity

Using Wigner-deformed Heisenberg oscillators, we construct 3D Chern--Simons models consisting of fractional-spin fields coupled to higher-spin gravity and internal non-abelian 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(N,N +/- 1) or gl(N|N +/- 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.


Anyons and statistical gauge fields
An interesting feature of quantum physics in three-dimensional spacetime is the presence of identical particles with exotic statistics. The basic notion dates all the way back to Leinaas and Myrheim [1] and later Wilczek [2], who provided specific models realizing such particles, which he referred to as Anyons, as flux-charge quanta. Subsequently, quantum field theories in flat spacetime containing Anyons were constructed in [3,4,5]; see also [6] for an early review. Later, a group-theory approach based on wave-functions was developed in 1 [9,10,11,12]; see [13] for a summary and extensions of these models. More recently, the flux-charge realization has been 1 Note also that equations of motion for massless, fractional-spin particles in four-dimensional Minkowski spacetime have been studied in [7,8]. Although this formalism is covariant under infinitesimal Lorentz transformations, the four-dimensional Poincaré symmetry is violated for finite transformations [8].
generalized to models with non-Abelian gauge fields [14]. Anyons can also be realized without integrating out any statistical gauge fields either as clusters of non-relativistic particles in two spatial dimensions kept together by external one-body potentials, such as a simple harmonic potential, and interacting with each other only via boundary conditions imposed on the multibody wave functions [15], or as vertex operators in two-dimensional conformal field theories [16]. For a more recent axiomatic treatise without gauge fields, see [17].
The key idea is that at fixed time the configuration space of a collection of massive particles whose trajectories cannot coincide as the result of their interactions has a non-trivial first homotopy group that is represented non-trivially on the multi-body wave-functions or correlation functions involving point-like operators [17]. These representations thus furnish representations of the braid group, which is why Anyon statistics is synonymous to braid statistics. The wave functions transform under rotations with phase factors which can be identified with the statistical phases under exchange of identical particles. Hence, one and the same phase characterizes the statistics of the particles as well as the representation of the spatial rotation group, which is the essence of the generalized spin-statistics theorem for massive particles in three-dimensional Minkowski space with exotic statistics and fractional spin 2 [17]. Thus, in 2+1 dimensions, the spin of a massive particle can be an arbitrary real number, thereby providing and interpolation between bosons and fermions.
In the realization of anyons in quantum field theories, their fractional quantum numbers are typically quantum effects due to the presence of Chern-Simons fields, usually referred to as statistical gauge fields. Their realizations as charged vortices [2] and Hopf-interacting massive particles arise in effective descriptions of matter-coupled Abelian Chern-Simons systems [3].
Integrating out the statistical Chern-Simons gauge field produces effective topological non-local Hopf interactions among the matter fields that transmute their statistics; see also [19,5,6] and [20,21,22] for related works using the CP 1 formalism. As for non-Abelian generalizations, the conformal Chern-Simons-scalar [23] and Chern-Simons-fermion [24] vector models exhibiting level-rank type dualities providing examples of three-dimensional Bose-Fermi transmutation [19]. In [23] it is suggested that these models contain Anyons at finite couplings. Moreover, as proposed by Itzhaki [14], the statistical gauge fields can be taken to be non-minimally coupled Yang-Mills fields by using Wilson lines for connections shifted by the Hodge dual of the field strength to generate the flux-charge bound states.

Coupling of anyons to background fields
On general grounds, one may ask whether Anyons can be described by any quantum-effective field theory that facilitates their coupling to ordinary tensorial and tensor-spinorial particles and fields, including gravity. In an arbitrary curved background the description of Anyons requires the introduction of a Lorentz connection valued in non-(half-)integer spin representations of the Lorentz algebra, which are infinite dimensional. As such representations admit oscillator realizations, it seems natural to incorporate them into Vasiliev's general framework for higherspin gravity [25,26,27]. The aim of this paper is to take a first step 3 in this direction.
Vasiliev's equations provide a fully non-linear and background-independent description of a large class of higher-spin gravities in various dimensions, including models with internal symmetry algebras [29,30] and fermions [31,29,30,32,33,34], of which some exhibit standard spacetime supersymmetry; for a recent review in the case of four-dimensional higher-spin gravities, see [35]. As far as spin-statistics relations are concerned, with notable exceptions in the presence of a positive cosmological constant [36,37,35] 4 or in Kleinian spacetime signature [35], Vasiliev's higher-spin gravities have so far been assumed to consist of fields that are either bosonic Lorentz tensors or fermionic tensor-spinors 5 .
However, our key observation is: Vasiliev's higher-spin gravities are not formulated a priori in terms of Lorentz tensors and tensor-spinors; rather they are formulated in terms of master fields living on products of space-time and fiber manifolds. The latter contain non-commutative twistor or twistor-like spaces whose coordinates generate the higher-spin and internal symmetry algebras. The full specification of a Vasiliev-type higher-spin gravity model thus requires the choice of a set of fiber functions that form an associative algebra. Hence, the incorporation of fractional-spin fields into the higher-spin framework can be reduced to the technical problem of in which ways Vasiliev's theory admits non-standard embeddings of the Lorentz connection leading to fractional-spin representations.
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]. 5 To our best understanding, this assumption on spin and statistics is required for consistency only within the context of relativistic quantum field theories in flat spacetimes of dimension four or higher; see e.g. [38].
Extensions to curved backgrounds of the spin-statistics correspondence are given in [39] and references therein.
Lorentz spins. As we shall see, the fractional spin fields appear within a bi-module of one-forms acted upon by one-sides actions of the higher-spin algebra and an internal color gauge group of infinite rank. In doing so, a particular set of technical problems that we shall have to address concerns the nature of infinite-dimensional representations and how it is affected by different choices of bases. To this end, we are going to focus on the on-shell formulation of a class of Blencowe-Vasiliev models [40,41] that arise within the Prokushkin-Vasiliev system [30] as a consistent truncation.

Outline of the paper
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 gravity models can be embedded into this context. We then summarize our main results, including material from a work in progress. In Section 3, we then proceed with the main analysis of anyon representations in AdS 3 and their realizations using the Wigner-deformed Heisenberg oscillators. In this section we shall stress details concerning the infinite-dimensional nature of these representations and in particular the importance of keeping track of their indecomposable structures in critical limits and related choices of bases will be stressed in Section 3.5. In Section 4, the fractional-spin Chern-Simons theory is formulated and some of its truncations are presented. We conclude in Section 5.

Preliminary remarks and summary
In this section we review some features of higher-spin gravity that are of conceptual interest and of importance for generalizations of our models. We then summarize our results including some material concerning mainly the off-shell formulation to be presented elsewhere. As the contents of this section are not crucial for the main analysis in the coming sections, to which the reader may therefore skip immediately if so desired.

Preliminary remarks on three-dimensional higher-spin gravities
Three-dimensional higher-spin gravity landscape. Three-dimensional topological higher-spin gravities with Lorentz-tensorial and tensor-spinorial gauge fields are described semiclassically by the Fradkin-Vasiliev-inspired Blencowe actions [40]. These theories are of Chern-Simons type and based on Lie algebras generated by ordinary Heisenberg oscillators, or equivalently, area preserving diffeomorphisms of two-spheres and two-hyperboloids [41]. As pointed out by Vasiliev [42], these algebras admit deformations based on Wigner-deformed Heisenberg oscillators [43,44], or equivalently, algebras of symplectomorphisms of fuzzy two-hyperboloids and two-spheres.
These topological models sit inside a larger landscape of matter-coupled higher-spin gravities described by the Prokushkin-Vasiliev equations [45,30]; see also [46]. Although their structure resembles that of the higher-dimensional Vasiliev equations [47,27], the three-dimensional higher-spin gravities exhibits a proper feature: its dynamical Weyl zero-forms are necessarily accompanied by topological Weyl zero-forms 6 while the corresponding sectors can be consistently truncated in four and higher dimensions.
In any dimension, there exists a special topological zero-form (which is a singlet) that can acquire an expectation value, ν say, that deforms the higher-spin symmetries. However, it is only in three dimensions that ν does not deform the anti-de Sitter vacuum. 7 The expansion around this AdS 3 -vacuum, with its expectation value ν, yields the aforementioned Chern-Simons models based on deformed oscillators as consistent truncations (upon setting all fluctuations in the Weyl zero-form to zero). In particular, for critical values of ν , given conventionally by ν = −2ℓ − 1 with ℓ = 0, 1, 2, . . . , the higher-spin algebras contain gl(2ℓ + 1) subalgebras [42], and the Chern-Simons models can be reduced further down to sl(N |N ± 1) and pure bosonic sl(N ) models studied in [41]. Working within the fully non-linear system its constructors observed that models with sufficiently elaborate internal algebra admit AdS 3 -vacuum expectation values [30] B = ν , (2.1) 6 As pointed out to us by D. Jatkar, it is natural to think of these topological degrees of freedom in higher-spin gravity as corresponding two-dimensional conformal field theory defects. 7 In four dimensions, the maximal finite sub-algebra of the higher-spin algebra that is preserved by ν is so (1,3) or so(2, 2) depending on the choice of signature. This suggests that four-dimensional fractional-spin gravities correspond holographically to three-dimensional massive quantum field theories with anyons, and that these models are integrable in a suitable sense, as the higher-spin symmetries are deformed rather than broken. and that the perturbative expansions around these vacua yield parity-invariant threedimensional higher-spin gravities containing massive scalars. 8 After a suitable redefinition, the perturbatively-defined master fields become valued in associative algebras where I refers to sets of internal generators (including the Z 2 -generator Γ used to double sl(2) to sl(2) + ⊕ sl(2) − ), consisting of sectors A Σ of suitable non-polynomial extensions of the universal enveloping algebra Aq(2; ν) [42] of the Wigner-deformed Heisenberg oscillator algebra [43,42,48] Thus, as the fully non-linear formulation rests on associative differential algebras, one may ask whether these can be extended by adding sectors of composite operators and refining correspondingly the star-product composition rule as to retain associativity, thus allowing the formal structure of the full master-field equations to remain intact, with the aim of facilitating modified embeddings of the Lorentz algebra into the gauge algebra that produces perturbatively-defined field contents containing fractional-spin fields.
Indeed, as we shall outline next, this can be done in a relatively straightforward fashion by adding sectors of non-polynomial operators corresponding to Fock-space endomorphisms.
These operators are given essentially by star-product versions of vacuum-to-vacuum projectors dressed by left and right multiplications by arbitrary polynomials. The extended associative star-product rules can then be defined using a matrix structure.

Outline of fractional-spin gravities
Matrix fusion rules . The fractional-spin gravities that we shall consider are based on Z 2graded associative algebras that are formed by extending the enveloping algebra Aq(2; ν) by sectors of operators acting in the Fock space F consisting of states with distinct eigenvalues of the spatial spin generator J 0 . More formally, we define where the injective homomorphism, or monomorphism, ρ F : End(F) ֒→ Aq(2; ν) (2.5) 8 These scalars behave as massive higher-spin fields for critical values of ν; whether parity invariance can be broken within the Prokushkin-Vasiliev formalism remains an open issue. 9 Blencowe's construction [40] makes use of the undeformed algebra Aq(2; 0)+ ⊕ Aq(2; 0)− .
The monomorphism ρ F is defined by the rule where the generalized projectors P m|n ∈ Aq(2; ν) obey P m|n P k|l = δ nk P m|l , (N ν − m)P m|n = 0 = P m|n (N ν − n) , (2.11) with number operator N ν := ρ F (Ň ) related to J 0 by 12) expressed in terms of the deformed oscillators (a − , a + ) obeying [a − , a + ] = 1 + k ν . The relationship [48] between deformed and undeformed oscillators (a − , a + ) and (b − , b + ), respectively, is presented in Subsection 3.2. In defining A(2; ν|o(2) J 0 ; F) we have also used The associative composition rule in A(2; ν|o(2) J 0 ; F) is defined as follows: Let be two elements in A(2; ν|o(2) J 0 ; F) with A i ∈ Aq(2; ν) being finite polynomials anď idemČ i andĎ i (i = 1, 2). Then M 1 M 2 is defined by standard matrix multiplication followed by star-product compositions and expansions of the results in the appropriate bases. In particular, the quantity ρ F (B 1 )ρ F (Č 2 ) = K m,n,p=0 M mp 1 M pn 2 P m|n is given by its expansion in the monomial basis of Aq(2; ν), while A 1 ρ F (B 2 ) and ρ F (Č 1 )A 2 are expanded in the matrix basis of End(F). This composition rules is then extended to End(F) by allowing the degree of A i and K to be arbitrarily large. Thus, the fractional-spin algebra has a product rule that combines star-product compositions in initial bases followed by expansions of the results in a final bases, which on may refer to as a fusion rule. We note that in the case at hand, the fusion rule does not require any expansion of P ∈ Aq(2; ν) in the matrix basis of End(F) .
Hermitian conjugation . Defining the hermitian conjugation operation † in Aq(2; ν) by it follows that the Fock-space realizationq α of the deformed oscillators [48,49] witȟ where † refers to the standard hermitian conjugation operation in End(F) and the charge conjugation matrixČ is given by the identity in the unitary regime εν −1 and a non-trivial matrix in the non-unitary regime εν < −1. Assuming furthermore thatČ 2 = 1 and (Č) † =Č , Master gauge fields . Starting from a Prokushkin-Vasiliev model with B = ν and fiber algebra where Cliff 1 (Γ) and Cliff 1 (ξ) denote, respectively, a bosonic Clifford algebra and a fermionic Clifford algebra with respective generators obeying and where ǫ s denotes the Grassmann parity, we may consider the consistent truncation B = ν, leaving the flat connection We demand the master fields to be Grassmann-even, i.e. ǫ s (W, ψ ± , ψ ± , U ) = (0, 0, 0, 0) , (2.23) and to have intrinsic parities where σ is defined on polynomials f (q, k, ξ) of definite degrees in q α and ξ by where we used the automorphisms Taking into account the Π ± -projections and assigning the following Grassmann parity it follows that (W, U ) and (ψ − , ψ − ) are ξ-independent, hence consisting of Grassmann-even component fields, while (ψ + , ψ + ) are linear in ξ, hence consisting of Grassmann-odd component fields. We note that ψ ± and ψ ± , respectively, transform under the left actions of Aq(2; ν) ++ ⊗ Cliff 1 (Γ) and ρ F (End(F)) −− ⊗ Cliff 1 (Γ) and under the right actions of The reality conditions on A σ will be chosen such that W belongs to a non-compact real form of Aq(2; ν) ++ ⊗Cliff 1 (Γ) containing the Lorentz generators Λ αβ Π + q (α q β) Π + with (Λ αβ ) † = Λ αβ , while U ∈ u(∞) ⊗ Cliff 1 (Γ) . We note that for generic ν, the model may be level truncated such that for K = 1, 2, . . . , ∞, but that the more interesting truncations arise spontaneously as ν assumes critical values.
Embedding of Lorentz algebra . A standard space-time formulation of the Chern-Simons field theory requires the choice of a canonical Lorentz connection ω ∈ sl(2) Lor associated with a principal Lorentz bundle over M 3 . In general, the Lorentz algebra can be embedded into the gauge algebra in several inequivalent ways leading to physically distinct models. In particular, one has • the diagonal embedding which yields standard higher-spin (super)gravities consisting of Lorentz tensors (and tensor spinors); • the alternative non-diagonal embedding which yields the fractional-spin (super)gravities in which the canonical Lorentz connection ω is thus embedded in W such that ψ andψ, respectively, transform in left-and rightmodules with fractional Lorentz spin.
Supertrace and action . The non-polynomial completion Aq(2; ν) of the enveloping algebra Aq(2; ν) admits the trace operation where the supertrace operation STr ν is fixed uniquely by its defining properties . Using the Weylordered basis (2.9), one has [42] STr Upon including the Clifford algebras, we define and equip A ± with the trace operation where thus σ(B ± ) = σ(C ± ) = ±1, which obeys 10 The Chern-Simons action as can be seem using the Π ± projections and (2.40) On ρ F (End(F)) the operation STr ν reduces to the standard Fock-space supertrace, viz.
Thus, the level of the internal gauge algebra is proportional to the ν-dependent quantity On-shell formulation in the Fock space . The equations of motion take the form Assuming that W lies in the image of ρ F , one can thus equivalently work on-shell with the Fock-space presentation of the equations of motion, viz.
Fractional Lorentz spin . By the construction outlined so far, and working in conventions where the deformed ladder operators a ± are linear combinations of q α obeying the Lorentz spin of (ψ,ψ), say α, defined to be the lowest weight of the generator is one of the roots of the quadratic Lorentz Casimir Taking into account k = ε(−1) Nν , one has The Lorentz spin of (ψ,ψ) is thus fractional and hence (ψ,ψ) transform in an infinitedimensional irreducible representation of sl(2) Lor except for critical values of ν. In the following, we will implicitly assume that ε = +1 unless explicitly mentioned otherwise.
Generalized h 1−sided . Finally, the fractional-spin gravity admits a natural generalization based on the Fock space F, in whichJ 0 is diagonal, and an additional state spacẽ whereȞ is a Hamiltonian with normalizable (bound) states. If there exists a -product implementation with fusion rules corresponding to where ρF : End(F ) → Aq(2; ν), then we propose a Chern-Simons action based on the Killing form STr ρF ( where λ,λ ′ |λ λ ′ |f λλ ′ ∈ End(F ) and P λ|λ is the star-product algebra element corresponding to |λ λ|.

The Wigner-deformed Heisenberg oscillator algebra
This section describes the concrete explicit realization of the fractional-spin algebras using Wigner-deformed Heisenberg oscillators.
3.1 The enveloping algebra Aq(2; ν) and its derived Lie (super)algebras The universal enveloping algebra Aq(2; ν) of the Wigner-deformed Heisenberg oscillator algebra is the associative algebra spanned by arbitrary polynomials in the deformed oscillators q α and the Kleinian k modulo their relations (2.3). It contains two associative subalgebras given by its these algebras turn into Lie algebras, which we denote by lq(2; ν) and lq ±± (2; ν), in their turn containing the Lie subalgebras slq(2; ν) and slq(2; ν) ±± , respectively, obtained by factoring out C ⊗ 1 and C ⊗ Π ± . The algebra Aq(2; ν) can also be endowed with the structure of a Z 2 -graded Lie algebra, denoted by q(2; ν), with graded commutator Factoring out the identity from q(2; ν) yields a superalgebra, which we denote by sq(2; ν). The Lie algebras lq(2; ν) and slq(2; ν) as well as their graded counter parts q(2; ν) and sq(2; ν) contain sl(2) subalgebras generated by using conventions where the matrices (γ a ) αβ = ǫ βγ (γ a ) α γ are normalized such that and the spinor indices are raised and lowered using the conventions together with the realization One has where we have defined [a − , a + ] = 1 + νk , {k, a ± } = 0 , k 2 = 1 . (3.10) In the Z 2 -graded case, the sl(2) algebra can be extended further to osp(2|2) by taking the supercharges Q i α (i = 1, 2) and so(2) generator T 12 to be given by [50] using conventions in which osp(N |2) has the following graded commutation rules (i = 1, . . . , N ) 12 : (3.14) The quadratic Casimir operators For N = 0, 1, 2, the oscillator realization gives rise to one-sided representations in various leftor right-modules, as we shall discuss below, with Casimirs The sl(2) subalgebras can be extended to sl(2)⊕sl(2) by taking translations 13 P a to be realized as P a = J a k. Instead, by tensoring with the bosonic Clifford algebra Cliff 1 (Γ) one can take 12 The structure coefficients of osp(N |2) can be found using the realization M αβ = q (α q β) , Q i α = ξ i qα and T ij = iξ i ξ j where qα obey (2.3) with ν = 0 and ξ i are external operators that obey {ξ i , ξ j } = 2δ ij . 13 The Lorentz generators We shall use the latter realization in the construction of the anyonic models, as Γ commutes to the projectors Π ± = 1 2 (1 ± k) used to define the tensorial, fractional-spin and Lorentz-singlet representations making up the fractional-spin gravity model.
For ν = 0 (and all τ ) one hasǎ ± =b ± and the representation of Aq ( such that where the conjugation matrixČ ∈ End(F) depends on ν, or rather, as we shall see, the integral part [ν] . We may further require that (f †) † =f for anyf ∈ End(F) ⇔Č † =Č . is invariant under similarity transformations generated by the elementsf ∈ End(F) that satisfy the reality conditionf † viz.
II. ν = −1: In this hyper-critical case, which is also unitary, one haš  and that F decomposes into two irreducible representations of the deformed oscillators, Indeed, the projectors commute with (ǎ ± ,ǩ) iff ν is critical or hyper-critical, and hencě 56) and the hermicity conditionš In terms of the bra-ket basis, one haš  Thus, (ǎ ± f ,ǩ f ) provide a finite-dimensional non-unitary representation of the Wigner-Heisenberg algebra with deformation parameter ν = −2ℓ − 1 whose enveloping algebra is isomorphic to gl(2ℓ + 1), while (ǎ ± ∞ ,ǩ ∞ ) provide an infinite-dimensional unitary representation of the Wigner-Heisenberg algebra with deformation parameter 2ℓ + 1, as can be seen fromǩ which implies that the redefinitionǩ ∞ → −ǩ ∞ yields a representation of Aq(2; 2ℓ + 1) on F ∞ . Thus, at critical ν one haš Aq(2; −2ℓ − 1; 0) ∼ = gl(2ℓ + 1) ⊕Ǎq(2; 2ℓ + 1; 0) .  Since the Klein operator take place in this expression, the value of the Casimir operator does not take a fixed value [42]. The Fock space thus decomposes into two invariant eigenspaces of k, whereΠ ± are the projectors defined in (2.13). The projected Lorentz generators and spins are given by The spins of the odd and even representations differ by half a unit, The classification of the unitary irreducible representations of SL(2, R) was first done by Bargmann [51]. Comparing with the unitary irreducible representations of sl (2) in Fock space by Barut and Fronsdal [52] and adapting the notation to this paper, we see that the unitary irreducible representations appearing in the non-critical case ν > −1 above furnish the discrete series D + (j ± ) . For ν < −1, but non-critical, these representations are still of a discrete type, but non-unitary. In reference [13] it was shown that the latter were essential for the construction of anyon wave equations possessing standard boson/fermion limits.  .  For the alternative choice of the Lorentz generators (3.69), the ket-bra products transform as follows: It follows that if n and m have different parity then |m n| will transform under either the left or right action of the rotation group, and hence their spin will have a ν-dependent fractional component. This observation indicates that, in order to include particles with fractional spin into the higher-spin connection, one needs to identify the Lorentz connection, which activates the local rotation symmetry, with the generators J ± a . The AdS algebra so(2, 2) ∼ = sl(2) ⊕ sl(2) is obtained by doubling the algebra as in (3.19).
The Fock space F, viewed as an sl(2) module, decomposes into two fractional-spin representations in the discrete series [51]. In these representations the spin operator J 0 acts diagonally 14 To our best understanding, the complete classification of all possible representations of Aq(2; ν) is an open problem. Indeed, the classification of infinite-dimensional irreducible representations of finite-dimensional Lie algebras is an active field in pure mathematics [53]. Two key differences between finite-and infinite-dimensional irreps is that the former are completely decomposable and can be labelled by the Casimir operators, while the latter, which can exhibit different branches of indecomposable structures, cannot be labelled faithfully only by Casimir operators. Additional "Langlands parameters" [53] are thus required to distinguish the infinitedimensional irreducible representations, such as the parameter τ introduced in Eqs. (3.26) and (3.27).
with real-valued eigenvalues. The Fock-space module can thus be identified with ρ F (Ǎq(2; ν; τ )) viewed as either a left module or a right module. On the other hand, the separate left and right actions in Aq(2; ν) also give rise to sl(2) modules but of a different type since the only generators of Aq(2; ν) that acts diagonally on itself from one side is the identity 1 and the Kleinian k.
In fact, the above conclusions do not change considerably if one removes the ++ projection.
In a generalization of Feigin's notation [54], we define gl(λ; J; τ ) := Env(sl(2)) I(λ) where I(λ) is the ideal generated by C 2 (sl(2)) + λ(λ − 1); (·) ↓ J indicates that the elements in (·) are given in a basis where the generator J ∈ sl(2) acts diagonally from both sides; and τ parameterizes the indecomposable structure. In particular, Feigin's original construction was performed in the basis of monomials in the generators of sl (2) in which no generator J can be diagonal; we denote this particular basis by gl(λ; −; −). With this notation, it follows that which are thus infinite-dimensional algebras for generic ν with critical limits given by semidirect sums of a finite-dimensional and an infinite-dimensional sub-algebra with ideal structure controlled by τ . Using this notation, one can write Eqs. where β ∈ C (and we note that τ (a ± ) = a ∓ ). As for the associative algebra itself, its real forms require star-maps; the real form Assuming that ((f * ) * ) = f for all f it follows that SS * = 1. Assuming furthermore that S =S 2 and that S S * = 1 it follows that if f * = ϕ(f ) then (Ad S (f )) * = Ad S (f ), that is, Letting (p, q) refer to the signature ofČ, it follows thatǔq(2; ν; τ ) is equivalent to a representation of u(p, q; J 0 ; τ ) whileȟosp(2|2; ν; τ ) is equivalent to a representation of the superalgebra u(p|q; J 0 ; τ ); the list of isomorphisms is given in Table 2. uq(2; ν; τ ) ∼ = u(C) andȟosp(2|2; ν; τ ) ∼ = u(C + |C − ). In the above, u(η) := u(p, q) if η is a diagonal matrix with p positive and q negative entries idem u(η 1 |η 2 ). In the first row, ∞ ± refer to the dimensions of F ± , and in the second row, ∞ ′ ± refer to the dimensions of P ∞ F ± . The real forms in the graded case (second column) are in agreement with [41]. using the Fock-space representation matrix of the higher-spin algebra defined by Q α 1 ···αn t,p,q := q|ǩ tq (α 1 · · ·q αn) |p , (4.3) which one may think of as a generalized Dirac matrix.
As discussed in the previous section, the connection can be subjected to reality conditions using either complex or hermitian conjugations; for definiteness let us use choose a reality condition of the latter type, namelyW † = −ČWČ , (4.4) whereČ is the charge conjugation matrix in (3.33) chosen such that (3.34) and (3.39) hold. As a result, the multi-spinorial component fields obey As a consequence of (3.42), the representation matrix (4.3) obeys

Internal color gauge fields
The fractional-spin gravity also contains an internal color gauge field U given in the bra-ket It is taken to obey the following reality condition:

Hybrid theory with fractional-spin fields
The higher-spin gravity connectionW given in (4.1) and the internal connectionǓ given in (4.8) can be coupled non-trivially via two intertwining one-forms, that we shall denote by (ψ,ψ), whose gauge symmetries exchange the higher-spin gravity and internal gauge fields.
In what follows, we present a simplified model exhibiting this feature in which the gauge fields are further projected using Π ± as follows: Arranging various master fields into a single two-by-two matrix the equations of motion can be declared to be of the standard form: that is, which form a non-trivial Cartan integrable system by virtue of the assignments that we have made so far. The equations of motion are thus symmetric under the gauge transformationš Thus,W ++ is the connection belonging to the adjoint representation of the non-minimal bosonic higher-spin subalgebra lq(2; ν) ++ ⊕ lq(2; ν) ++ of lq(2; ν) ⊕ lq(2; ν); it consists of all integer spins and has the Fock-space representatioň The action of the global rotationŘ 2π by 2π generated byx ++ = 2πΠ +J 0 on the fields is given by such that the spins and the Grassmann statistics of (ψ,ψ) are correlated in the semi-classical limit for even and odd ℓ, respectively, in the case of Grassmann even and Grassmann odd fields, in agreement with discussion around (2.27).
As discussed in Section 3.5, the key issue is the choice of bases used to expand the various gauge fields. Strictly speaking, the fractional-spin gravity model for which we have an off-shell formulation, is based on a master field where ρ F denotes a morphism from End(F) to the oscillator algebra and Bi(a|b) denotes a bi-module with a left action of a and a right action of b. The higher-spin connection is thus expanded in multi-spinorial basis, in which only the trivial element has a diagonal one-sided action, while the internal gauge field and the intertwiners are expanded in the Fock-space basis, in which the spin operatorJ 0 has diagonal one-sided actions. The role of the map ρ F is to realize the latter basis elements as elements of the oscillator algebra rather than End(F) as to make sense of the source term ψψ in the equation for W ++ .
One can verify that the associative algebra spanned by k t P f q (α 1 · · · q αn) P f with n = 0, ..., 2ℓ ; t = 0, 1 , is isomorphic [42] to Mat 2ℓ+1 , (C) by counting the number of independent generators considering the identity In this way, the hybrid model 4.3 constructed in Fock space, including the correspondent reality conditions (4.26), thus decomposes into a finite-dimensional and an infinite-dimensional for the corresponding algebras (4.30); that iš where Bi(v ⊗ w) denotes a bi-module consisting of a left-module v and a right-module w.

Truncations of color gauge fields
To begin with, for any ν and N ∈ N, it is possible to choose u(N ) subalgebras of End(F) and truncate U ∈ u(N ) and simultaneously take ψ and ψ, respectively, to transform inN and N .
For any given N , there exists an infinite number of such level truncations.
Another type of truncation of the color gauge fields is possible in the non-unitary regime ν < −1. Here one notes that ifψ = |σ c|, where thus σ is a spin and c is a color, theň ψ = −|c σ|C and henceψψ = |σ σ|C whileψψ = |σ c|C|c σ| that can vanish in the nonunitary regime. Thus, the fractional-spin fields necessarily source the tensor-spinorial higherspin gravity field W (c.f. positivity of energy in ordinary gravity) while the internal gauge fielď U can be truncated consistently leading to dW +W 2 +ψψ = 0 , dψ +Wψ = 0 , dψ +ψW = 0 ,ψψ = 0 , (4.36) which defines a quasi-free differential algebra. Thus, the last constraint above possesses nontrivial solutions owing the non-definite signature of the invariant conjugation matrix of the representation of the higher spin algebra carried by the fractional spin fields, whileψψ = 0 does not have non-trivial solutions.

Conclusions
In this paper, we have presented a new class of three-dimensional Chern-Simons higher-spin gravities that we refer to as fractional-spin gravities. These theories are extensions of or- The fundamental representations of the higher-spin algebras are infinite-dimensional and characterized by a deformation parameter ν ∈ R: For non-critical ν they remain irreducible under the Lorentz sub-algebra with spin 1 4 (1 + ν); for critical ν = −1, −3, . . . they decompose into a finite-dimensional tensor or tensor-spinor and an infinite-dimensional representation with spin −ν. The color indices, on the other hand, can be chosen to be finite-dimensional by level truncation, and if the fractional-spin representation is non-unitary, that is, if ν < −1 then the internal gauge fields can be truncated; the theory then consists only of the higher-spin gravity fields and the fractional-spin fields.
Denoting the Blencowe-Vasiliev connection by W , which thus consists of a collection of Lorentz-tensorial gauge fields making up the adjoint representation of the higher-spin algebra, and the fractional-spin fields and internal connection by (ψ, ψ) and U , respectively, we have proposed to describe the fractional-spin gravities on-shell using the following integrable system of equations: dW + W 2 + ψψ = 0 , dψ + W ψ + ψU = 0 , (5.1) dψ + ψW + U ψ = 0 , dU + U U + ψψ = 0 , The underlying fractional-spin algebra carries a Z 2 -grading similar to that of ordinary superalgebras: The fractional-spin generators close onto higher-spin and internal generators, while the higher-spin and internal generators rotate the fractional-spin charges into themselves. Thus, the fractional-spin fields transform under one-sided actions of the higher-spin and internal Lie, and the fractional-spin transformations can send higher-spin gauge fields into internal gauge fields and vice versa.
We would like to stress that the simple appearance of the construction is due to the fact that it relies on the consistent fusion of two sectors of the enveloping algebra of the Wigner-Heisenberg deformed oscillators: The sector of arbitrary polynomials in deformed oscillators can be combined with the sector of Fock-space endomorphisms into an associative algebra by realizing the latter as elements of the enveloping algebra. In this paper, we have demonstrated this algebraic structure at the level of Fock-space representations, which are sufficient for the on-shell formulation. The off-shell formulation requires, however, the implementation using enveloping algebra techniques, as to realize the bi-linear form going into the definition of the Chern-Simons action; we leave a more detailed description of the off-shell formulation as well as the construction of non-topological fractional-spin models for forthcoming works.
In terms of sl(2) representation theory, the fractional-spin representations belong to the discrete series [51] which are lowest-weight representations in the compact basis, labeled by the lowest eigenvalue of the spatial rotation generator J 0 of so(2, 1) ∼ = sl (2). Generic values of the lowest spin imply irreducibility, while negative integer or negative half-integer lowest spins, respectively, imply decomposability with finite-dimensional invariant tensor or tensor-spinorial subspaces. Hence, finite-dimensional higher-spin models can be singled out; by combining various reality conditions and working with fractional-spin fields that are either bosons or fermions one may arrive at models based on sl(N ), su(p, q) or su(p|q).
The fact that the fractional-spin fields (ψ, ψ) are constructed from tensor-spinor higher-spin fields by a change of basis, can be interpreted as that the latter condense into the former in a new vacuum of the Prokushkin-Vasiliev system where color interactions emerge. This phenomena is reminiscent of how new phases can be reached in strongly correlated systems by means of large gauge transformation, as for example in the confined phase of QCD according to t'Hooft's mechanism [55]. It is thus inspiring to entertain the idea that the new vacua of Blencowe-Vasiliev theory studied arise in a similar fashion, namely, via a large gauge transformation of the Blencowe-Vasiliev vacuum formed by tensor and tensor-spinor fields. This physical picture also resembles the fractional quantum Hall effect [2,56,57,58,59] where many-electron systems exposed to strong magnetic fields become confined giving rise to quasi-particle anyons.
As mentioned already, anyons can be obtained in the form of a Wilson line coming from infinite and attached in its extreme to a charged particle [14], yielding the transmutation to braided statics. Although we have not discussed these aspects in this paper, it suggests by analogy that we may be in a similar picture, namely that the fractional spin fields should correspond to Wilson lines attached to the AdS boundary and to some higher spin particles with usual boson or fermion statistics, although in the present states of our theory the latter particles must also be located at the boundary, as it happens in Chern-Simons theory where the dynamical degrees of freedom are confined to the boundary.
It is worth to mention that open higher-spin Wilson-lines have been analyzed recently [60,61] and their insertions have been argued to be dual to sending the dual conformal field theory to phases with finite entanglement entropies. One problem that one can investigate, starting from our model, is a particular type of classical solutions in fractional-spin gravity that may have an interpretation as entanglement entropy. Along the same lines, a suitable approach could be suggested by the considerations made in the work [62].