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Asymptotic Symmetries of Maxwell Chern–Simons Gravity with Torsion

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Asymptotic Symmetries of Maxwell Chern–Simons Gravity with Torsion Eur. Phys. J. C (2020) 80:967 https://doi.org/10.1140/epjc/s10052-020-08537-z Regular Article - Theoretical Physics Asymptotic symmetries of Maxwell Chern–Simons gravity with torsion H. Adami1,a, P. Concha2,b , E. Rodríguez3,c, H. R. Safari1,d 1 School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran 2 Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile 3 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Viña del Mar, Chile Received: 26 June 2020 / Accepted: 10 October 2020 © The Author(s) 2020 Abstract We present a three-dimensional Chern–Simons share many properties with higher-dimensional gravity mod- gravity based on a deformation of the Maxwell algebra. This els which, in general, are more difficult to study. Three dimen- symmetry allows introduction of a non-vanishing torsion to sional General Relativity (GR) with and without cosmolog- the Maxwell Chern–Simons theory, whose action recovers ical constant can be described through a CS action based on the Mielke–Baekler model for particular values of the cou- the AdS and Poincaré algebra, respectively [1–3]. Nowadays, pling constants. By considering suitable boundary condi- there is a growing interest in exploring bigger symmetries in tions, we show that the asymptotic symmetry is given by the order to study more interesting and realistic physical models. bms3 ⊕ vir algebra with three independent central charges. Well-known infinite-dimensional enhancements of the AdS and Poincaré symmetries, in three spacetime dimen- sions, are given respectively by the conformal and the bms3 Contents algebras. A central extension of the two-dimensional confor- mal algebra, which can be written as two copies of the Vira- 1 Introduction and motivations .............. soro algebra, appears as the asymptotic symmetry of three- 2 Maxwell Chern–Simons gravity theory with torsion . dimensional GR with negative cosmological constant [4]. In 2.1 BMS-like solution ................. the asymptotically flat case, the three-dimensional version of 3 Asymptotic symmetries ................ the Bondi–Metzner–Sachs (BMS) algebra [5–8], denoted as 3.1 Boundary conditions ............... bms3, corresponds to the asymptotic symmetry of GR [9]. 3.2 Residual gauge transformations .......... bms3 can be alternatively obtained as a flat limit of the confor- 3.3 Canonical surface charges and asymptotic sym- mal one, in a similar way as the Poincaré symmetry appears metry algebra ................... as a vanishing cosmological constant limit of AdS. The study 3.4 Change of basis .................. of richer boundary dynamics could offer a better understand- 4 Discussion and outlook ................. ing of the bulk/boundary duality beyond the AdS/CFT cor- References ......................... respondence [10]. Thus, the exploration of new asymptotic symmetries of CS gravity theories based on enlarged global symmetries could be worth studying. In particular, extensions 1 Introduction and motivations and generalizations of the conformal and the bms3 algebras have been subsequently developed in diverse contexts in [11– Three-dimensional Chern–Simons (CS) gravity theories are 34]. considered as interesting toy models since they allow us A particular extension and deformation of the Poincaré to approach diverse aspects of the gravitational interaction algebra is given by the Maxwell algebra. Such symmetry and underlying laws of quantum gravity. Furthermore, they appears to describe a particle moving in a four-dimensional Minkowski background in presence of a constant electro- a e-mail: [email protected] magnetic field [35–37]. This algebra is characterized by the b e-mail: [email protected] (corresponding author) non-vanishing commutator of the four-momentum generator c e-mail: [email protected] d e-mail: [email protected] 0123456789().: V,-vol 123 967 Page 2 of 10 Eur. Phys. J. C (2020) 80:967 Pa: null boundary, we discuss the BMS-like solution of the the- ory. We provide boundary conditions allowing a well-defined [Pa, Pb]=Mab, (1.1) action principle. In Sect. 3, we show that the asymptotic sym- metry algebra for the Maxwell CS gravity with torsion is which is proportional to a new Abelian generator Mab.The given by an infinite enhancement of a deformed Maxwell Maxwell algebra and its generalizations have been useful to algebra, which can be written as the direct sum bms3 ⊕ vir. recover standard GR without cosmological constant from CS Finally, in Sect. 4 we discuss the obtained results and possible and Born–Infeld gravity theories in a particular limit [38– future developments. 42]. In three spacetime dimensions, an invariant CS grav- Notation We adopt the same notation as [30,66,67]forthe ity action under Maxwell algebra has been introduced in algebras; for algebras we generically use “mathfrak” fonts, [43] and different aspects of it has been studied [44–53]. like vir, bms3 and Max3. The centrally extended version As Poincaré symmetry, the Maxwell symmetry describes a of an algebra g will be denoted by gˆ, e.g. Virasoro algebra three-dimensional gravity theory whose geometry is Rieman- vir = witt. nian and locally flat. However, the presence of an additional gauge field in the Maxwell case leads to new effects com- pared to GR. In particular, in [29], the authors have shown that the Maxwellian gravitational gauge field modifies not 2 Maxwell Chern–Simons gravity theory with torsion only the vacuum energy and angular momentum of the sta- tionary configuration but also the asymptotic structure. Using the CS formalism, we present the three-dimensional To accommodate a non-vanishing torsion to the Maxwell gravity theory based on a particular deformation of the CS gravity theory it is necessary to deform the Maxwell Maxwell algebra. Unlike the Maxwell case, such deforma- algebra. Here, we show that a particular deformation of the tion leads to a non-vanishing torsion as equation of motion. Maxwell symmetry, which we refer to as “deformed Maxwell The deformed Maxwell algebra is spanned by the genera- algebra”, allows us to introduce not only a torsional but also a tors {Ja, Pa, Ma}, which satisfy the following non-vanishing cosmological constant term along the Einstein–Hilbert term. commutation relations: Then, motivated by the recent results on the Maxwell algebra, [ , ]= c , we explore the effects of deforming the Maxwell symmetry Ja Jb ab Jc both to the bulk and boundary dynamics. At the bulk level, we c [Ja, Pb]= Pc, show that the invariant CS gravity action under the deformed ab [J , M ]= c M , (2.1) Maxwell algebra reproduces the Maxwell field equations a b ab c but with a non-vanishing torsion describing a Riemann– c 1 [Pa, Pb]= Mc + Pc , Cartan geometry. Interestingly, the CS action can be seen ab as a Maxwell version of a particular case of the Mielke– Baekler (MB) gravity theory [54] which describes a three- where abc is the three-dimensional Levi-Civita tensor and dimensional gravity model in presence of non-vanishing tor- a, b = 0, 1, 2 are the Lorentz indices which are lowered sion. Further studies of the MB gravity have been subse- and raised with the Minkowski metric ηab.The parameter quently developed in [55–65]. Here we explore the effects of appearing in the last commutator is related to the cosmolog- having a non-vanishing torsion in Maxwell CS gravity at the ical constant . Then, the vanishing cosmological constant level of the boundary dynamics. In particular, by considering limit →∞reproduces the Maxwell symmetry. Let us note suitable boundary conditions, we show that the asymptotic that the Hietarinta–Maxwell algebra [48,52,68] is recovered →∞ symmetry can be written as the bms3 ⊕ vir algebra. This in the limit when the role of the Pa and Ma genera- infinite-dimensional symmetry was recently obtained as a tors is interchanged. One can see that Ja and Pa are not the deformation of the infinite-dimensional enhancement of the generators of a Poincaré subalgebra. However, as it is pointed ( , ) ⊕ ( , ) Maxwell algebra, denoted as Max3 algebra [66]. We also out in [66], (2.1) can be rewritten as the iso 2 1 so 2 1 show that the vanishing cosmological constant limit →∞ algebra. This can be seen by a redefinition of the generators, can be applied not only at the CS gravity theory level but 2 also at the asymptotic algebra, leading to the Maxwell CS La ≡ Ja − Pa − Ma, gravity and its respective asymptotic symmetry previously S ≡ P + 2 M , (2.2) introduced in [29]. a a a ≡− , The paper is organized as follows. In Sect. 2, we present Ta Ma the three-dimensional CS gravity theory which is invariant under a particular deformation of the Maxwell algebra. Fur- where La and Ta are the respective generators of the iso(2, 1) thermore, considering asymptotically flat geometries with algebra, while Sa is a so(2, 1) generator. Then, the Lie algebra 123 Eur. Phys. J. C (2020) 80:967 Page 3 of 10 967 (2.1) can be rewritten as non-degenerate bilinear form of the algebra (2.1) reads α [ , ]= c , J J =α η , P P = 1 + α η , La Lb ab Lc a b 0 ab a b 2 ab c [La, Tb]= Tc, (2.3) (2.8) ab Ja Pb=α1ηab, Pa Mb=0, c [Sa, Sb]= Sc. ab Ja Mb=α2ηab, Ma Mb=0, It is important to point out that (2.1) is not the unique way where α0,α1 and α2 are arbitrary constants satisfying α2 = 0 of deforming the Maxwell algebra. It is shown in [69] that and α1 =−α2. Both conditions are required to ensure the the Maxwell algebra can be deformed into two different alge- non-degeneracy of the invariant tensor (2.8). One can see that bras: so(2, 2)⊕so(2, 1) and iso(2, 1)⊕so(2, 1).
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