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On the Maxwell Supergravity and Flat Limit in 2+1 Dimensions Physics Letters B 785 (2018) 247–253 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb On the Maxwell supergravity and flat limit in 2 + 1dimensions ∗ Patrick Concha a, , Diego M. Peñafiel a, Evelyn Rodríguez b a Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaiso, Chile b Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Viña del Mar, Chile a r t i c l e i n f o a b s t r a c t Article history: The construction of the three-dimensional Chern–Simons supergravity theory invariant under the Received 4 July 2018 minimal Maxwell superalgebra is presented. We obtain a supergravity action without cosmological Received in revised form 23 August 2018 constant term characterized by three coupling constants. We also show that the Maxwell supergravity Accepted 27 August 2018 presented here appears as a vanishing cosmological constant limit of a minimal AdS–Lorentz supergravity. Available online 30 August 2018 The flat limit is applied at the level of the superalgebra, Chern–Simons action, supersymmetry Editor: M. Cveticˇ transformation laws and field equations. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction three-dimensional Maxwell CS gravity [43]. Such asymptotic sym- metry corresponds to a deformed bms3 algebra and has been first The three-dimensional (super)gravity is considered as an in- introduced in [44]by using the semigroup expansion procedure teresting toy model for approaching richer (super)gravity models, [45]. Furthermore, it has been shown that the vacuum energy and which in general are non-trivial. Furthermore, the three-dimen- angular momentum are influenced by the presence of the gravita- sional theory shares many interesting properties with higher- tional Maxwell field [43]. dimensional theories such as black hole solutions [1]. Remarkably, The Maxwell algebra also appears as an Inönü–Wigner (IW) supergravity theory in three spacetime dimensions [2,3]can be ex- contraction [46,47]of an enlarged algebra known as AdS–Lorentz pressed as a gauge theory using the Chern–Simons (CS) formalism algebra. The AdS–Lorentz algebra, also known as the semisim- for the AdS [4]or Poincaré supergroup [5]. In the last decades, di- ple extended Poincaré algebra, was introduced in [48,49] and has verse three-dimensional supergravity models have been studied in been studied in the context of gravity in diverse dimensions [40, [6–24]. In particular, there has been a growing interest to extend 50,51]. Such symmetry can be written as the semidirect sum AdS and Poincaré supergravity theories to other symmetries. s0(d − 1, 2) ⊕ s0(d − 1, 1) which motivates the name of AdS– The Maxwell symmetry has become an interesting alternative Lorentz. for generalizing Einstein gravity. The Maxwell algebra has been in- At the supersymmetric level, a minimal extension of the troduced to describe a particle moving in a four-dimensional back- Maxwell algebra appears to describe the geometry of a four- ground in the presence of a constant electromagnetic field [25–27]. dimensional superspace in presence of a constant abelian super- Further applications of the Maxwell group have been developed in symmetric gauge field background [52]. Subsequently in [53,54], [28–30]. The Maxwell symmetry and its generalizations have been the Maxwell superalgebra has been used to construct a pure su- useful to recover General Relativity (GR) as a particular limit us- pergravity action in four dimensions using geometric methods. ing the CS and Born–Infeld (BI) gravity formalism [31–35]. In four Further applications of the Maxwell superalgebra in the context dimensions, Maxwell gravity models have been successfully con- of supergravity can be found in [55,56]. Such superalgebra has the structed in [36–38]. More recently, there has been a growing inter- particularity of having more than one spinor charge and has been est in studying the three-dimensional CS gravity theory invariant obtained in [57,58]through algebraic expansion mechanisms [45, under the Maxwell group [39–43]. This theory, as the Poincaré one, 59]. More recently, aCS gravity action for a generalized Maxwell is asymptotically flat and does not contain a cosmological constant superalgebra has been presented in [60,61]in three spacetime di- term. Interestingly, anovel asymptotic structure appears for the mensions. Although the CS supergravity theory is appropriately constructed, it requires to consider a large amount of gauge fields whose origin is due to the methodology. To our knowledge, the * Corresponding author. E-mail addresses: [email protected] (P. Concha), [email protected] construction of a CS supergravity action for the minimal Maxwell (D.M. Peñafiel), [email protected] (E. Rodríguez). superalgebra in absence of extra fields has not been presented yet. https://doi.org/10.1016/j.physletb.2018.08.050 0370-2693/© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 248 P. Concha et al. / Physics Letters B 785 (2018) 247–253 a a a a A well-defined minimal Maxwell supergravity theory would allow C = α0 J Ja + α1 P Ja + α2 P Pa + J Za . (2.5) to study more features of the three-dimensional Maxwell super- gravity, like its asymptotic structure and non-relativistic limit. Then considering the connection one-form (2.3) and the invari- In this paper, we present the minimal three-dimensional ant tensor (2.4), the CS gravity action invariant under the Maxwell Maxwell CS supergravity theory without considering a large symmetry reads amount of extra fields. In particular, we present two alternative k a 1 a b c a procedures to obtain it. We first derive the minimal Maxwell su- ICS = α0 ω dωa + abcω ω ω + 2α1 Rae 4 3 peralgebra and the invariant tensor applying the expansion method π M to a supersymmetric extension of the Lorentz algebra. Then, we a a present the corresponding CS action, supersymmetry transforma- + α2 T ea + 2R σa , (2.6) tions and field equations. In the second part, we recover the minimal Maxwell CS supergravity as a flat limit of a minimal AdS– where the Lorentz curvature and torsion two-forms are given re- Lorentz CS supergravity. spectively by The present work is organized as follows: In Section 2, we give 1 a brief review of Maxwell CS gravity theory in three spacetime a a abc R = dω + ωbωc , dimensions. In Section 3, we present a minimal Maxwell CS super- 2 a a abc gravity theory in three dimensions. The Section 4 is devoted to the T = de + ωbec . (2.7) obtention of the Maxwell supergravity theory through a flat limit = 1 of a minimal AdS–Lorentz CS supergravity theory. In Section 5, we Here k 4G is the CS level of the theory and it is related to the end our work with some comments about future possible develop- gravitational constant G. Note that the term proportional to α0 ments. is the exotic Lagrangian also known as the Lorentz Lagrangian. The second term corresponds to the Einstein–Hilbert (EH) term 2. Maxwell Chern–Simons gravity theory in 2 + 1dimensions while the term proportional to α2 contains the explicit gravita- tional Maxwell field σa. Note that each term is invariant under the In this section, we briefly review the three-dimensional CS grav- Maxwell symmetry. In particular, the local gauge transformations a a a ity theory invariant under the Maxwell algebra [39–43]. The ex- δ A = d + [A, ], with gauge parameter = ρ Ja + ε Pa + γ Za, plicit commutators of the Maxwell algebra can be obtained as a are given by deformation and enlargement of the Poincaré one. In particular, a = a the Maxwell algebra is spanned by the set { Ja, Pa, Za} whose gen- δω Dρ , erators satisfy the following non-vanishing commutation relations: a a abc δe = Dε − ρbec , (2.8) c c a = a + abc − [ Ja, Jb] = abc J , [ Ja, Pb] = abc P , δσ Dγ (ebεb ρbσc) , c c a = a + abc [ Ja, Zb] = abc Z , [Pa, Pb] = abc Z , (2.1) where Du du ωbuc is the Lorentz covariant derivative. The equations of motion derived from the action (2.6)read where a, b, ···=0, 1, 2are raised and lowered with the Minkowski metric ηab and abc is the Levi-Civita tensor. Note that, unlike the a 1 b c δω : α0 Ra + α1 Ta + α2 Dσa + abce e = 0 , Poincaré algebra, the commutator of the translational generators 2 Pa is no longer zero but proportional to the new abelian genera- a δe : α1 Ra + α2 Ta = 0 , (2.9) tor Za. In order to construct a CS action δσ a : α R = 0 . 2 a k 2 3 In particular, when α = 0, the field equations are given by the ICS = AdA + A , (2.2) 2 4π 3 vanishing of every curvature two-form M invariant under the Maxwell group, we require the Maxwell Ra = 0 , connection one-form A = A dxμ and the corresponding non- μ T a = 0 , (2.10) vanishing components of the invariant tensor. 1 The gauge connection one-form for the Maxwell algebra reads a b c F = Dσa + abce e = 0 . 2 = a + a + a A ω Ja e Pa σ Za , (2.3) On the other hand, let us note that the EH dynamics is recov- a a ered in the limit α2 = 0. Such interesting feature is proper of where ω corresponds to the spin connection, e is the vielbein a the Maxwell like symmetry Bk [50,62] which corresponds to the and σ is the gravitational Maxwell gauge field [43].
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