Asymptotically flat structure of hypergravity in three spacetime dimensions

Published for SISSA by Springer Received: August 21, 2015 Accepted: September 9, 2015 Published: October 2, 2015 Asymptotically flat structure of hypergravity in three spacetime dimensions a Centro de Estudios Cientı́ficos (CECs), Av. Arturo Prat 514, Valdivia, Chile b Departamento de Fı́sica, Universidad de Concepción, Casilla 160-C, Concepción, Chile E-mail: , , Abstract: The asymptotic structure of three-dimensional hypergravity without cosmological constant is analyzed. In the case of gravity minimally coupled to a spin-5/2 field, a consistent set of boundary conditions is proposed, being wide enough so as to accommodate a generic choice of chemical potentials associated to the global charges. The algebra of the canonical generators of the asymptotic symmetries is given by a hypersymmetric nonlinear extension of BMS3 . It is shown that the asymptotic symmetry algebra can be recovered from a subset of a suitable limit of the direct sum of the W(2,4) algebra with its hypersymmetric extension. The presence of hypersymmetry generators allows to construct bounds for the energy, which turn out to be nonlinear and saturate for spacetimes that admit globally-defined “Killing vector-spinors”. The null orbifold or Minkowski spacetime can then be seen as the corresponding ground state in the case of fermions that fulfill periodic or antiperiodic boundary conditions, respectively. The hypergravity theory is also explicitly extended so as to admit parity-odd terms in the action. It is then shown that the asymptotic symmetry algebra includes an additional central charge, being proportional to the coupling of the Lorentz-Chern-Simons form. The generalization of these results in the case of gravity minimally coupled to arbitrary half-integer spin fields is also carried out. The hypersymmetry bounds are found to be given by a suitable polynomial of degree s + 21 in the energy, where s is the spin of the fermionic generators. Keywords: Higher Spin Symmetry, Conformal and W Symmetry, Gauge-gravity correspondence, Classical Theories of Gravity ArXiv ePrint: 1508.04663 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP10(2015)009 JHEP10(2015)009 Oscar Fuentealba,a,b Javier Matulicha and Ricardo Troncoso,a Contents 1 2 General Relativity minimally coupled to a spin-5/2 field 2 3 Unbroken hypersymmetries: Killing vector-spinors 3.1 Cosmological spacetimes and solutions with conical defects 4 4 4 Asymptotically flat behaviour and the hyper-BMS3 algebra 4.1 Flat limit of the asymptotic symmetry algebra from the case of negative cosmological constant 6 9 5 Hypersymmetry bounds 11 6 Hypergravity reloaded 13 7 General Relativity minimally coupled to half-integer spin fields 7.1 Killing tensor-spinors 7.2 Asymptotically flat structure and hypersymmetry bounds 15 16 17 8 Final remarks 20 A Conventions 24 B Killing vector-spinors from an alternative approach 24 C Hyper-Poincaré algebra with fermionic generators of spin n + D Asymptotic hypersymmetry algebra D.1 Spin-3/2 fields (supergravity) D.2 Spin-7/2 fields D.3 Spin-9/2 fields 1 1 2 26 26 26 27 29 Introduction It has been shown that the inconsistencies arising in the minimal coupling of a massless spin5/2 field to General Relativity [1–4] can be successfully surmounted in three-dimensional spacetimes [5]. This theory is known as hypergravity, and it has been recently reformulated as a Chern-Simons theory of a new extension of the Poincaré group with fermionic generators of spin 3/2 [6]. In the case of negative cosmological constant, additional spin-4 fields are required by consistency [7–9], and it can be seen that the anticommutator of the –1– JHEP10(2015)009 1 Introduction 2 General Relativity minimally coupled to a spin-5/2 field It has been recently shown that the hypergravity theory of Aragone and Deser [5] can be reformulated as a gauge theory of a suitable extension of the Poincaré group with fermionic spin-3/2 generators [6]. The action is described by a Chern-Simons form, so –2– JHEP10(2015)009 generators of the asymptotic hypersymmetries, associated to fermionic spin-3/2 parameters, leads to interesting nonlinear bounds for the bosonic global charges of spin 2 and 4 [9]. The bounds saturate provided the bosonic configurations admit globally-defined “Killing vector-spinors”. One of the main purposes of this paper is to show how these results extend to the case of asymptotically flat spacetimes in hypergravity, also in the case of arbitrary half-integer spin fields. In the next section we briefly summarize the formulation of hypergravity as a Chern-Simons theory for the hyper-Poincaré group in the simplest case of fermionic spin-5/2 fields, while section 3 is devoted to explore the global hypersymmetry properties of cosmological spacetimes and solutions with conical defects. In the case of fermions that fulfill periodic boundary conditions, it is shown that the null orbifold possesses a single constant Killing vector-spinor. Analogously, for antiperiodic boundary conditions, Minkowski spacetime is singled out as the maximally (hyper)symmetric configuration, and the explicit expression of the globally-defined Killing vector-spinors is found. The asymptotically flat structure of hypergravity in three spacetime dimensions is analyzed in section 4, where a precise set of boundary conditions that includes “chemical potentials” associated to the global charges is proposed. The algebra of the canonical generators of the asymptotic symmetries is found to be given by a suitable hypersymmetric nonlinear extension of the BMS3 algebra. It is also shown that this algebra corresponds to a subset of a suitable Inönü-Wigner contraction of the direct sum of the W(2,4) algebra with its hypersymmetric extension W(2, 5 ,4) . The hypersymmetry bounds that arise from the 2 anticommutator of the fermionic generators are found to be nonlinear, and are shown to saturate for spacetimes that admit unbroken hypersymmetries, like the ones aforementioned. This is explicitly carried out in section 5. In section 6, the previous analysis is performed in the case of an extension of the hypergravity theory that includes additional parity-odd terms in the action. It is found that the asymptotic symmetry algebra admits an additional central charge along the Virasoro subgroup. The results are then extended to the case of General Relativity minimally coupled to half-integer spin fields in section 7, including the asymptotically flat structure, and the explicit expression of the Killing tensor-spinors. The hypersymmetry bounds are shown to be described by a polynomial of degree s + 1/2 in the energy, where s is the spin of the fermionic generators. We conclude in section 8 with some final remarks, including the extension of these results to the case of hypergravity with additional parity-odd terms and fermions of arbitrary half-integer spin. The coupling of additional spin-4 fields is also address (...truncated)