Extension of the Poincar\'e group with half-integer spin generators: hypergravity and beyond

An extension of the Poincar\'e group with half-integer spin generators is explicitly constructed. We start discussing the case of three spacetime dimensions, and as an application, it is shown that hypergravity can be formulated so as to incorporate this structure as its local gauge symmetry. Since the algebra admits a nontrivial Casimir operator, the theory can be described in terms of gauge fields associated to the extension of the Poincar\'e group with a Chern-Simons action. The algebra is also shown to admit an infinite-dimensional non-linear extension, that in the case of fermionic spin-$3/2$ generators, corresponds to a subset of a contraction of two copies of WB$_2$. Finally, we show how the Poincar\'e group can be extended with half-integer spin generators for $d\geq3$ dimensions.


I. INTRODUCTION
Nowadays, we have the good fortune of witnessing the era in which the simplest minimal realistic versions of supersymmetric field theories are about to be either tested or falsified by the LHC. The underlying geometric structure of these kind of theories, as well as most of their widely studied extensions, relies on the super-Poincaré group (see, e.g., [1], [2], [3]).
According to the Haag-Lopuszański-Sohnius theorem [4], this is a consistent extension of the Poincaré group that includes fermionic generators of spin 1/2. Indeed, in flat spacetimes of dimension greater than three, the addition of fermionic generators of spin s ≥ 3/2 would imply that the irreducible representations necessarily contained higher spin fields, which are known to suffer from inconsistencies (see, e.g., [5], [6], [7], [8], [9], [10], [11]). However, in three spacetime dimensions, higher spin fields do not possess local propagating degrees of freedom, and as a consequence, it is possible to describe them consistently [12], [13], [14], [15], [16], [17], [18] even on locally flat spacetimes [19], [20], [21], [22]. Hence, in the latter context, since no-go theorems about massless higher spin fields can be circumvented, it is natural to look for an extension of the Poincaré group with fermionic half-integer spin generators. Results along these lines have already been explored in [23]. In what follows, we begin with the construction of the searched for extension of the Poincaré group in the case of spin 3/2 generators, that for short, hereafter we dub it the "hyper-Poincaré" group.
It is shown that the algebra admits a nontrivial Casimir operator and, as an application, we explain how the hypergravity theory of Aragone and Deser [24] can be formulated so as to incorporate the hyper-Poincaré group as its local gauge symmetry. Concretely, we show how hypergravity can be described in terms of hyper-Poincaré-valued gauge fields with a Chern-Simons action. The results are then extended to the case of fermionic generators of spin n+ 1 2 , as well as to the minimal coupling of General Relativity with gauge fields of spin n+ 3 2 , so that super-Poincaré group and supergravity are recovered for n = 0. The hyper-Poincaré algebra is also shown to admit an infinite-dimensional nonlinear extension that contains the BMS 3 algebra, which in the case of spin-3/2 generators, reduces to a subset of a suitable contraction of two copies of WB 2 . We conclude explaining how the hyper-Poincaré group is extended to the case of d ≥ 3 dimensions.

II. FERMIONIC SPIN-3/2 GENERATORS
In three spacetime dimensions, the nonvanishing commutators of the Poincaré algebra can be written as The additional fermionic generators are assumed to transform in an irreducible spin-3/2 representation of the Lorentz group, so that they are described by "Γ-traceless" vectorspinors that fulfill Q a Γ a = 0, where Γ a stand for the Dirac matrices. Their corresponding commutation rules with the Lorentz generators are then given by Therefore, requiring consistency of the closure as well as the Jacobi identity, implies that the only remaining nonvanishing (anti-) commutators of the algebra read where C is the charge conjugation matrix 1 . It is then simple to verify that apart from I 1 = P a P a , the algebra admits another Casimir operator given by which implies the existence of an invariant (anti-) symmetric bilinear form, whose only nonvanishing components are of the form It is worth highlighting that the inclusion of the higher spin generators Q a α does not jeopardize the causal structure, since there is no need to enlarge the Lorentz group. 1 In our conventions, the Minkowski metric η ab is assumed to follow the "mostly plus" convention, and the Levi-Civita symbol fulfills ε 012 = 1. Round brackets stand for symmetrization of the enclosed indices, without the normalization factor, e. g., It is also useful to keep in mind the Fierz expansion of the product of three Dirac matrices, given by Γ a Γ b Γ c = ε abc +η ab Γ c +η bc Γ a −η ac Γ b . Afterwards, the presence of the imaginary unit "i" in the product of real Grassmann variables is because we assume that (θ 1 θ 2 ) * = −θ 1 θ 2 .

A. Hypergravity
In order to describe a massless spin-5 2 field minimally coupled to General Relativity, let us consider a connection 1-form that takes values in the hyper-Poincaré algebra described above, which reads where e a , ω a and ψ α a stand for the dreibein, the dualized spin connection (ω a = 1 2 ε abc ω bc ), and the Γ-traceless spin- 5 2 field (Γ a ψ a = 0), respectively. The components of the field strength F = dA + A 2 are then given by where the covariant derivative of the spin-5 2 field can be written as with T a = de a + ε abc ω b e c , andψ aα = ψ β a C βα is the Majorana conjugate. Note that under an infinitesimal gauge transformation δA = dλ + [A, λ], spanned by a hyper-Poincaré-valued zero-form given by λ = λ a P a + σ a J a + ǫ α a Q a α , the components of the gauge field transform according to The invariant bilinear form (5) then allows to construct a Chern-Simons action for the gauge field (6), given by which up to a boundary term, reduces to It is worth pointing out that, despite the action (11) is formally the same as the one considered by Aragone and Deser in [24], it does possess a different local structure. Indeed, note that under local hypersymmetry transformations spanned by λ = ǫ α a Q a α , the nonvanishing transformation rule for the spin connection considered in [24], agrees with ours only on-shell.
Actually, by construction, as in the case of supergravity [25], here the algebra of the local gauge symmetries (9) closes off-shell according to the hyper-Poincaré group, without the need of auxiliary fields.
In the case of negative cosmological constant, it can be seen that hypergravity requires the presence of additional spin-4 fields [26], [27], [28].
where χ a 1 ...an is Γ-traceless and completely symmetric in the vector indices, which can be seen as a flat connection that fulfills Note that the Jacobi identity now translates into the consistency of the nilpotence of the exterior derivative (d 2 = 0), which for the algebra (13) is clearly satisfied.
The nontrivial Casimir operator now reads It is also worth pointing out that the super-Poincaré algebra corresponds to the case of n = 0, while the hyper-Poincaré algebra described above is recovered for n = 1.

A. Hypergravity in the generic case
The minimal coupling of General Relativity with a massless fermionic field of spin s = n+ 3 2 , described by a completely symmetric Γ-traceless 1-form ψ a 1 ...an , can then be formulated in terms of a gauge field for the hyper-Poincaré algebra, which now reads A = e a P a + ω a J a + ψ α a 1 ...an Q a 1 ...an α .
The Casimir operator (14) then implies the existence of an (anti-) symmetric tensor of rank 2, which once contracted with the wedge product of two curvatures, gives being an exact form that is manifestly invariant under the hypersymmetry transformations (19). Therefore, as in the case of (super)gravity [29], [30], the action can also be written as a Chern-Simons one (10), which up to a boundary term reduces to R a e a + iψ a 1 ...an Dψ a 1 ...an , so that the field equations now read F = 0, with F given by (16).
Note that the standard supergravity action in [31], [32], [33] is recovered for n = 0; and as it occurs in the spin-5 2 case, the generic theory agrees with the one of Aragone and Deser only on-shell.
We would like to stress that a deeper understanding of the theory cannot be attained unless it is endowed with a consistent set of boundary conditions. In this sense, one of the advantages of formulating hypergravity as a Chern-Simons theory is that the analysis of its asymptotic structure can be directly performed in a canonical form, as in the case of negative cosmological constant [28]. Indeed, in analogy with the case of three-dimensional flat supergravity [34], the mode expansion of the asymptotic symmetry algebra of hypergravity with a spin-5 2 fermionic field is defined through the following Poisson brackets [35] i i {ψ m , ψ n } = 1 4 6m 2 − 8mn + 6n 2 − 9 P m+n + 9 4k q P m+n−q P q +k m 2 − 9 4 n 2 − 1 4 δ m+n,0 , which describe a nonlinear hypersymmetric extension of the BMS 3 algebra [36], [37], [38].
It can also be shown that this algebra corresponds to a subset of a suitable contraction of two copies of the WB 2 algebra [39], [40], [28].
When fermions fulfill antiperiodic boundary conditions, the modes of the fermionic global charges ψ m are labelled by half-integers, so that the wedge algebra of (22) reduces to the one of the hyper-Poincaré group. In fact, dropping the nonlinear terms, and restricting the modes according to |n| < ∆, where ∆ stands for the conformal weight of the generators, the hyper-Poincaré algebra is manifestly recovered provided the modes in (22) are identified with the generators J a , P a , Q αa , according to It is also worth noting that (22) can then be regarded as a hypersymmetric extension of the Galilean conformal algebra in two dimensions [41], [42], which is isomorphic to BMS 3 and turns out to be relevant in the context of non-relativistic holography.
Another advantage of formulating hypergravity in terms of a Chern-Simons action is that, as in case of supergravity [43], [34], the theory can be readily extended to include parity odd terms in the Lagrangian. This can be explicitly performed by a simple modification of the invariant bilinear form, so that it acquires an additional component given by J a , J b = µη ab , followed by a shift in the spin connection of the form ω a → ω a + γe a , so that the constants µ, γ parametrize the new allowed couplings in the action. As a consequence, when hypergravity is extended in this way, the hyper-BMS 3 algebra (22) acquires an additional nontrivial central extension along its Virasoro subgroup.

IV. ENDING REMARKS
The hyper-Poincaré group admits a consistent generalization to the case of d ≥ 3 spacetime dimensions. In the case of fermionic Γ-traceless spin-3 2 generators, the nonvanishing (anti-) commutators of the algebra are given by J ab ,Q αc = 1 2 (Γ ab ) β αQ βc +Q αa η bc −Q αb η ac , whereQ a = Q † a Γ 0 stands for the Dirac conjugate.
In the generic case, the spin-n + 1 2 generators correspond to completely symmetric Γtraceless tensor-spinors that fulfill Γ a 1 Q a 1 ...an = 0. In order to avoid the intricacies related to the latter condition, as well as with the suitable (anti-) symmetrization of the (anti-) commutation rules of the generators, it is better to express the algebra in terms of its Maurer-Cartan form. It is now given by where χ a 1 ...an is Γ-traceless and completely symmetric in the vector indices, so that its This algebra can be easily written in terms of Majorana spinors when they exist, and it reduces to super-Poincaré for n = 0.
Note that there was no need to enlarge the Lorentz group in order to accommodate the higher spin generators, so that the additional symmetries do not seem to interfere with the causal structure. Indeed, as in the case of supersymmetry, the quotient of the hyper-Poincaré group over the Lorentz subgroup now defines a hyperspace which is an extension of Minkowski spacetime with additional Γ-traceless tensor-spinor coordinates. However, as anticipated by Haag, Lopuszański and Sohnius, the irreducible representations, which could be obtained from suitable hyperfields, necessarily contain higher spin fields. Nevertheless, it would be worth to explore whether the hyper-Poincaré algebra may manifest itself through theories or models whose fundamental fields do not transform as linear multiplets, as it would be the case of nonlinear realizations, hyper-Poincaré-valued gauge fields, or extended objects.