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 dimensions, are given respectively by the conformal and the bms3 Contents algebras. A central extension of the two-dimensional conformal 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]

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Pa: null boundary, we discuss the BMS-like solution of the theory. We provide boundary conditions allowing a well-defined [Pa, Pb]=Mab, (1.1) action principle. In Sect. 3, we show that the asymptotic symmetry 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 compared 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 stationary 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). The former the flat limit →∞leads to the non-vanishing components has been largely studied in [31,44,70,71] whose asymptotic of the invariant tensor for the Maxwell algebra [29]. symmetry is described by three copies of the Virasoro algebra Considering the one-form gauge potential (2.5) and the [31,32], while the latter has only been approached through non-vanishing components of the invariant tensor (2.8), one a deformation process [66,69]. In the present work, using can rewrite the CS action (2.4)as the basis {J , P , M }, we find asymptotic symmetry of the a a a k 1 CS gravity theory based on the iso(2, 1) ⊕ so(2, 1) algebra. S = α ωa ω + abcω ω ω CS π 0 d a a b c The motivation to use such basis is twofold. First, it allows 4 M 3 1 1 us to recover the Maxwell CS gravity theory in a particular + α 2R ea + abce e e + T ae (2.9) 1 a 2 a b c a limit. Second, as we shall see, it reproduces the Maxwell field 3 equations with a non-vanishing torsion. a a 1 abc + α T ea + 2R fa + eaebec A three-dimensional gravity can be formulated as a CS 2 3 theory described by the action up to a surface term. One can see that the CS action is proportional to three independent sectors each one with its respec- [ ]= k + 2 3 , SCS A AdA A (2.4) tive coupling constant α . In particular, the first term is the 4π M 3 i so-called exotic Lagrangian [2]. The second term contains with a given Lie algebra on a manifold M, where A is the the usual Einstein Lagrangian with cosmological constant gauge connection, , denotes the invariant trace and k = term plus a torsional term related to the so-called Nieh–Yan 1/(4G) is the CS level. For the sake of simplicity we have invariant density. omitted writing the wedge product. The gauge connection It is interesting to notice that the CS gravity action (2.9) one-form A for the deformed Maxwell algebra reads can be seen as a Maxwell extension of a particular case of the MB model, which describes a three-dimensional gravity a a a A = e Pa + ω Ja + f Ma, (2.5) model in presence of non-vanishing torsion. The MB action is given by where ea, ωa and f a are the dreibein, the (dualized) spin connection and an auxiliary one-form field, respectively. The IMB = aI1 + I2 + β3 I3 + β4 I4 (2.10) associated field strength F = dA + 1 [A, A] can be written 2 where a,,β3 and β4 are constants and as = a, = a + a + a , I1 2 ea R F K Pa R Ja W Ma (2.6) =−1 a b c, where I2 abce e e 3 a 1 abc 1 I3 = ω dωa + ωaωbωc, Ra = dωa + a ωbωc, (2.7a) 3 2 bc 1 I = e T a. (2.11) K a = T a + a ebec, (2.7b) 4 a 2 bc 1 Particularly, in the absence of the auxiliary field f a in (2.9), W a = D(ω) f a + a ebec. (2.7c) 2 bc the constants appearing in the MB gravity can be identified with those of the deformed Maxwell algebra CS theory as Here, T a = D(ω)ea is torsion two-form, Ra is curvature k k α1 k D(ω)a = a + a ωbc a = α1,=− + α2 ,β3 = α0 two-form, and d bc is the exterior 4π 4π 4π →∞ covariant derivative. Naturally, the flat limit repro- k α β = 1 + α . (2.12) duces the Maxwell field strength [29]. On the other hand, the 4 4π 2 123 967 Page 4 of 10 Eur. Phys. J. C (2020) 80:967

Thus, the CS gravity action (2.9) can be interpreted as the we have that the gauge transformations δ A = d +[A,] Maxwellian version of a particular case of the MB gravity of the theory are given by action when the MB’s constants satisfy (2.12). a a abc 1 abc δe = D(ω)ε − ρ e + e ε , It is important to point out that one can accommodate a b c b c generalized cosmological constant in the Maxwell gravity a a δω = D(ω)ρ , (2.16) theory using another deformation of the Maxwell algebra, a a abc abc known as AdS–Lorentz algebra [72]. However, in the AdS– δ f = D(ω)χ + ebεc − ρb fc. Lorentz case, besides the Einstein–Hilbert term there is an In this work, we analyze the consequences of this particu- α additional gauge field fa, while there is no torsion term 1 lar deformation of the Maxwell symmetry at the level of the [31,44,70,71]. In the present case, the deformed Maxwell asymptotic structure. In the Maxwell case, as was shown in symmetry allows us to introduce both a cosmological con- [29], the presence of the additional gauge field f a leads to →∞ stant and a torsion term. In the flat limit the CS action new effects compared to GR and the asymptotic symmetries reproduces the Maxwell CS gravity action which contains is found to be a deformed bms3, denoted as Max3 in [66]. pure GR as sub-case. Dynamics of the fa gauge field is com- Here, we explore the implications of deforming the Maxwell pletely determined by the last term with coupling constant algebra as in (2.1). α 2. In particular, the equations of motion appear by consid- Let us recall that given an action there are two ways to ren- ering the variation of the action (2.9) under the respective der it having a well-posed variation principle. One of them is gauge fields: to add boundary terms to the action and the other is imposing 1 suitable boundary conditions on fields. Let us consider the δea : = α Ra + K a + α K a, 0 1 2 variation of the action (2.4), a a a a δω : 0 = α0 R + α1 K + α2W , k k δSCS[A]= δ AF+ δ AA, (2.17) a a π π δ f : 0 = α2 R . (2.13) 2 M 4 ∂M where ∂M is the boundary of M. The field equations require Then, when α2 = 0 we find the curvature two-forms (2.6) vanishing of the field strength and the on-shell boundary con- should vanish, tribution to the action is the surface term Ra = 0, K a = 0, W a = 0. (2.14) δ [ ] =− k δ . SCS A bdy A A (2.18) Indeed, from last equation in (2.13), we find Ra = 0. Never- 4π ∂M theless, it is important to emphasize that α =−α , which In our case this term reads as 1 2 solves the first equation of (2.13) and would imply a relation k δ = δωa (α ω + α + α ) between K a and W a, cannot be considered as a solution of SCS 0 a 1ea 2 fa bdy 4π ∂M the theory. As was previously mentioned, the non-degeneracy α a 1 a α = α =−α +δe α1ωa + + α2 ea + α2δ f ωa . (2.19) of the invariant tensor implies 2 0 and 1 2.In l particular, in the three-dimensional CS formalism, the non- We will see later that for spacetimes with null boundary, degeneracy of the bilinear form ensures that the CS action where the boundary is located at r = const →∞, the action involves a kinematical term for each gauge field and the equa- principle is satisfied without addition of boundary terms. tions of motion imply that all curvature two-forms vanish as in (2.14). Note that the CS gravity theory (2.9) describes the 2.1 BMS-like solution Maxwell CS gravity theory in presence of a non-vanishing torsion T a = 0. In particular, the first two equations Ra = 0 In this section we analyze the field equations (2.14). We con- and T a =−1 a ebec correspond to the three-dimensional 2 bc sider spacetimes with null boundary, which can be described teleparallel theory in which the cosmological constant can in the three-dimensional BMS gauge. We parametrize space- be seen as a source for the torsion. On the other hand, the time by the local coordinates xμ = (u, r,φ), where −∞ < vanishing of W a implies that the exterior covariant derivative u < ∞ is the retarded time coordinate, φ ∼ φ + 2π is the of the auxiliary field f is constant. In particular, in the flat a angular coordinate and the boundary is located at r = const. limit →∞the field equation for f remains untouched a Then, the metric can be written as [18] and is analogue to the constancy of the electromagnetic field in flat spacetime. ds2 = Mdu2 − 2dudr + N dφdu + r 2dφ2, (2.20) One can see that each term of the action (2.9) is invariant where M and N are two arbitrary functions of u,φ.As under the gauge transformation laws of the algebra (2.1). was previously discussed, the deformed Maxwell symme- Indeed, considering try (2.1) can be written in a certain basis as the direct sum a a a = ε Pa + ρ Ja + χ Ma, (2.15) iso(2, 1) ⊕ so(2, 1). Following the same trick considered in 123 Eur. Phys. J. C (2020) 80:967 Page 5 of 10 967

[31], we can find solutions of the present theory by first work- conditions on the fields are imposed. The radial dependence ing in the direct sum basis and then, go to the basis we are of the connection A can be gauged away by the gauge trans- interested in. In the direct sum basis, the fields (ω˜ a, eã) asso- formation A = h−1dh + h−1ah, where the asymptotic field ciated to the iso(2, 1) generators obey the very well-known a = au(u,φ)du + aφ(u,φ)dφ does not depend on r and GR boundary conditions, and the field fã associated to the h = e rM0 . Then, at the boundary r = const. →∞,the so(2, 1) generator can be set as a flat connection. on-shell action (2.18) takes the form Furthermore, in the aforementioned direct sum basis, the M N k functions and are given for the known results in asymp- δSCS =− aδa bdy 4π ∂M totically flat gravity in three dimensions k α = φ δ a α ω + 1 + α M = M(φ), N = J (φ) + uM (φ). (2.21) dud eu 1 aφ 2 eaφ 4π ∂M α The spacetime line element can be written in terms of the a 1 − δeφ α1ωau + + α2 eau 2 a b dreibein as ds = ηabe˜ e˜ , where ⎛ ⎞ + δωa α ω + α + α u 0 aφ 1eaφ 2 faφ 010 a ⎝ ⎠ − δωφ (α0ωau + α1eau + α2 fau) ηab = 100 (2.22) 001 +α δ aω − α δ aω . 2 fu aφ 2 fφ au (2.27) is the Minkowski metric in null coordinate system. Then, the dreibein and the torsionless spin connection one-forms are Furthermore, from (2.26) we find the following boundary written as conditions for the gauge fields a = ,ωa = ,ωa =− a, 1 1 eu 0 u 0 φ fu (2.28) e˜0 =−dr + Mdu + N dφ, e˜1 = du, e˜2 = rdφ, 2 2 upon the first two, the variation of the action reduces to ω˜ 0 = 1M φ, ω˜ 1 = φ, ω˜ 2 = . α d d 0 k 2 a a δS = dudφ δω f − δ f ω φ . (2.29) 2 CS bdy π φ au u a (2.23) 4 ∂M ωa =− a Finally, applying the last condition φ fu ,wearriveat The field fã can then be chosen as a Lorentz flat connection, δ = , SCS bdy 0 (2.30) ˜0 1 ˜1 ˜2 f = Ldφ, f = dφ, f = 0, (2.24) α 2 for any value of 2. Thus, in space-time with boundary conditions (2.28), the action principle is well-posed. where L = L(φ). In this way, we have found the solutions in the BMS gauge for the fields (eã, ω˜ a, fã). As mentioned before, we are interested in the basis where 3 Asymptotic symmetries the Poincaré–Lorentz symmetry appears as a deformation of the Maxwell algebra. From (2.2), it is possible to show that The aim of this section is to find the asymptotic symmetry the fields (ea,ωa, f a) are related to those in the direct sum algebra for the Maxwell CS gravity with torsion which was basis (eã, ω˜ a, fã) as follows: previously constructed. To start with, we provide the suit- ea = fã −˜ωa ,ωa =˜ωa, able fall-off conditions for the gauge fields at infinity and the gauge transformations which preserve our boundary condi- a 2 ã a a f = f −˜ω − e˜ . (2.25) tions. Then, the charge algebra is found using the Regge– Teitelboim method [73]. Consequently, the field equations (2.14) are solved by the following components of the gauge fields 3.1 Boundary conditions 1 1 1 e0 = Pdφ, ω0 = Mdφ, f 0 = dr + Fdφ − Mdu, 2 2 2 2 Inspired by the results obtained in the previous section, we e1 = 0,ω1 = dφ, f 1 =−du, consider the gauge connection evaluated in the BMS gauge as follows e2 = 0,ω2 = 0, f 2 =−rdφ, (2.26) 1 1 A = M (u,φ) dφ J0 + dφ J1 + P (u,φ) dφ P0 where, for later convenience, we have defined the functions 2 2 P = (L − M) and F = (P − N ). 1 (3.1) + dr + F (u,φ) dφ − M (u,φ) du M0 As discussed we need to ensure vanishing of the bound- 2 2 ary term in the variation of the action when suitable boundary − duM1 − rdφM2. 123 967 Page 6 of 10 Eur. Phys. J. C (2020) 80:967

The radial dependence can be gauged away by an appropriate where ε, χ and γ are three arbitrary, periodic functions of the gauge transformation on the connection angular coordinate φ. Under the given gauge transformation, − − the dynamical ﬁelds transform as A = h 1dh + h 1ah, (3.2) δ M = M ε + Mε − ε , where the group element is given by h = e rM0 . Then, if we λ 2 2 − use the identity h 1dh = drM , and the Baker–Campbell– χ χ 0 δλP = P + ε + 2P ε + Hausdorff formula, we ﬁnd

− +M χ + Mχ − χ , h 1ah = a − rdφM . (3.3) 2 2 2 δλZ = Z ε + 2Zε + M γ Therefore, once we have dropped out the radial dependence + Mγ + P χ + Pχ − γ . from the gauge ﬁeld A, we are left with the asymptotic ﬁeld 2 2 2 (3.10) a = audu + aφdφ, whose components are given by 3.3 Canonical surface charges and asymptotic symmetry au =− MM0 − M1, algebra 2 (3.4) 1 1 1 aφ = MJ + J + P P + F M , Asymptotic symmetries of the Maxwell gravity theory with 2 0 1 2 0 2 0 non-vanishing torsion can be found in the canonical approach which depend only on time and the angular coordinate. The [73]. In particular, in the case of a three-dimensional Chern– equations of motion, which are required to hold in the asymp- Simons theory, the variation of the canonical generators is totic region, imply that given by [74,75] M = M (φ) , P = P (φ) , F = Z (φ) − uM (φ) . δ [λ]= φλδ . (3.5) Q d aφ (3.11)

3.2 Residual gauge transformations Therefore one can show that surface charge variation associated with (3.8) and (3.9)is Asymptotic symmetries correspond to residual gauge trans- 2π formations δ A = d +[A,] which preserve boundary δQ(ε,χ,γ)= dφ (εδJ + χδP + γδM) (3.12) conditions (3.1). We consider the following gauge parame- 0 ters with −1 a = h λh,λ= λ( ) (u,φ) Ja J k +λa ( ,φ) + λa ( ,φ) . J = (α Z + α M + α P) , (P) u Pa (M) u Ma (3.6) π 2 0 1 (3.13a) 4 Then, gauge transformations of the connection A with gauge k α1 P = + α P + α M , (3.13b) parameter , lead to r-independent gauge transformations 4π 2 1 of the connection a with gauge parameter λ, i.e. k M = α M. (3.13c) 4π 2 δa ≡ δλa = dλ +[a,λ]. (3.7) The gauge transformations that preserve the boundary con- One can take ε, χ and γ to be state-independent and then λa λa ditions (3.1) with (3.5)for (J) and (P) are given by the charge variation (3.12) is integrable on the phase space. There are three independent surface charges, M χ 0 0 1 λ( ) = ε − ε ,λ( ) = P + ε + Mχ − χ , (ε) = (ε, , ), (χ) = ( ,χ, ), J 2 P 2 J Q 0 0 P Q 0 0 λ1 = ε, λ1 = χ, M(γ ) = Q(0, 0,γ), (3.14) (J) (P) λ2 =−ε ,λ2 =−χ , (J) (P) (3.8) associated with three independent symmetry generators ε, χ and γ . It is shown that the algebra among surface charges is λa while for (M) we get given by [73,76,77] M P F M 0 {Q(1), Q(2)}=Q([1,2]) + C(1,2) (3.15) λ( ) = γ + χ + ε − uε + uε − γ , M 2 2 2 2

λ1 = γ − ε , where Dirac bracket is deﬁned as {Q(1), Q(2)}:= (M) u δ Q( ) and C( , ) is central extension term. There- λ2 =−γ + ε , 2 1 1 2 (M) u (3.9) fore, using the transformation laws (3.10), one can show that 123 Eur. Phys. J. C (2020) 80:967 Page 7 of 10 967 the algebra of charges (3.12)is variation be integrable which leads to δG δG δG kα0 ε = ,χ= ,γ= , {J(ε ), J(ε )} = J([ε ,ε ]) − φε ε , (3.20) 1 2 1 2 π d 1 2 δJ δP δM 2 kα1 for some functional {J(ε), P(χ)} = P([ε, χ]) − dφεχ , 2π 2π G[ , , ]= φ ( , , ). kα2 J P M d G J P M (3.21) {J(ε), M(γ )} = M([ε, γ ]) − dφεγ , 0 2π By choosing 1 {P(χ ), P(χ )} = M([χ ,χ ]) + P([χ ,χ ]) 1 2 1 2 1 2 2 2 G =˜ε J − P − M +˜χ P + M − γ˜ M (3.22) k α − 1 + α dφχ χ , 2π 2 1 2 where ε,˜ χ,˜ γ˜ are state-independent functions, one can show {P(χ), M(γ )} = 0, that the charge variation (3.12) can be written as π {M(γ1), M(γ2)} = 0, (3.16) 2 δQ(ε,˜ χ,˜ γ)˜ = dφ (εδ˜ L +˜χδS +˜γδT) (3.23) where [x, y]=xy − yx . The resulting algebra corresponds 0 to an inﬁnite-dimensional lift of the deformed Maxwell alge- with bra. Now we can express the above algebra in Fourier modes, L =J − P − 2M, (3.24a) := ( inφ), := ( inφ), := ( inφ), Jn J e Pn P e Mn M e (3.17) S =P + 2M, (3.24b) =− . which give rise to the following centrally extended algebra T M (3.24c)

c1 3 i {Jm, Jn} = (m − n) Jm+n + m δm+n, , Now, by introducing Fourier modes 12 0 c := (ε˜ = inφ, , ), := ( , χ˜ = inφ, ), { , } = ( − ) + 2 3δ , Lm Q e 0 0 Sm Q 0 e 0 i Jm Pn m n Pm+n m m+n,0 φ 12 T := Q(0, 0, γ˜ = ein ), (3.25) c m { , } = ( − ) + 3 3δ , i Jm Mn m n Mm+n m m+n,0 12 one finds the direct sum of the bms3 and Virasoro algebra: 1 c i {Pm, Pn} = (m − n) Mm+n + (m − n) Pm+n LL 3 i {Lm, Ln} = (m − n) Lm+n + m δm+n,0, 12 1 c + 2 + 3δ , cLT 3 c3 m m+n,0 i {Lm, Tn} = (m − n) Tm+n + m δm+n, , 12 12 0 { , } = , i Pm Mn 0 cSS 3 i {Sm, Sn} = (m − n) Sm+n + m δm+n,0, (3.26) i {Mm, Mn} = 0. (3.18) 12 where the central charges are related to those appearing in The central charges c1, c2 and c3 are related to the CS level (3.18)as k and the arbitrary constant appearing in the invariant tensor ≡ − − 2 , ≡ + 2 , ≡− . (2.8)as cLL c1 c2 c3 cSS c2 c3 cLT c3 (3.27) = α . ci 12k i−1 (3.19) The central charges (cLL, cLT, cSS) of the bms3⊕vir algebra are related to the three independent terms of the CS gravity 3.4 Change of basis action for the iso(2, 1) ⊕ so(2, 1) algebra. Let us note that the algebra (3.18) can be obtained alterna- The infinite-dimensional algebra (3.18) can be seen as the tively as a central extension of the deformed Max [66] and infinite-dimensional enhancement of the deformed Maxwell 3 deformed bms4 [67]. It is also worth pointing out that by algebra (2.1). In particular, in the flat limit →∞we recover ignoring the generators Tm and central charge cLT in (3.26), the asymptotic symmetry of the three-dimensional Maxwell which is equivalent to ignoring the Maxwell generators Mm CS gravity theory introduced in [29]. Interestingly, as was and central charge c in (3.18), one obtains two copies of ⊕ 3 shown in [66], the algebra (3.18) is isomorphic to the bms3 Virasoro algebra, which is the asymptotic symmetry of the vir algebra. To observe this at the level of charge algebra, we AdS3 CS gravity (or Teleparallel theory [79]) with two cen- use the change of basis proposed in [78]. tral charges as Suppose ε, χ and γ are now state-dependent, i.e. they are functions of dynamical fields. We require that charge cLL ≡ c1 − c2, cSS ≡ c2. (3.28) 123 967 Page 8 of 10 Eur. Phys. J. C (2020) 80:967

4 Discussion and outlook W(a, b) ⊕ witt algebra in the context of the new CS theory. It would be worthwhile to explore the answer. In this work, we have studied a CS gravity theory with a Another aspect that deserves to be explored is supersym- deformed Maxwell algebra as a gauge group, which allows us metric extension of our analysis and results. Supersymmet- to introduce a non-vanishing torsion to the Maxwell CS grav- ric extensions of three-dimensional gravity with torsion has ity. In particular, the CS action can be seen as a Maxwell gen- been constructed in [61,83]. On the other hand, the three- eralization of a particular case of the MB gravity [54] whose dimensional CS supergravity invariant under the Maxwell field equations correspond to those of the Maxwell gravity algebra has been recently presented in [45,49–51]. However, theory but in presence of a non-vanishing torsion. Motivated to our knowledge, a supersymmetric Maxwell CS gravity by the fact that the deformed Maxwell is isomorphic to the with torsion has not been discussed yet. It would be then iso(2, 1) ⊕ so(2, 1) algebra, we considered asymptotically interesting to explore diverse supersymmetric extensions of flat geometries and discuss the BMS-like solution. We then the deformed Maxwell algebra and study which superalgebra explored the implications of the deformed Maxwell algebra is a good candidate to construct a well-defined supersym- (2.1) at the level of the asymptotic symmetry. In particular, we metric gravity with torsion in three spacetime dimensions. have shown that the asymptotic symmetry for the Maxwell The study of supersymmetric extensions of CS gravity with algebra with torsion is described by an infinite-enhancement torsion and their asymptotic structures could bring a better of the deformed Maxwell algebra which can be written as the understanding of the role of torsion. bms3 ⊕ vir algebra with three independent central charges. It would be interesting to go further in the study of the solu- Acknowledgements The authors would like to thank N. Merino and M. M. Sheikh-Jabbari for the correspondence and comments. This work tions of the present theory. In particular, one expects the solu- was funded by the National Agency for Research and Development tions to describe a Maxwell version of the so-called Telepar- (ANID) CONICYT – PAI grant No. 77190078 (P.C.) and FONDECYT allel theory in which the cosmological constant is a source for Project No. 3170438 (E.R.). H.A. and H.R.S. acknowledge the partial the torsion. As was shown in [57,79], both GR with cosmo- support of Iranian NSF under grant no. 950124. H.A. acknowledges the support by the Saramadan grant no. ISEF/M/98204. H.R.S. wishes logical constant and three-dimensional Teleparallel gravity to thank to M. Henneaux and S. Detournay for their kind hospitality have the same asymptotic symmetry and dynamics. Further- at the Physique Théorique et Mathématique department of the Uni- more, three-dimensional gravity with torsion possesses BTZ versité Libre de Bruxelles (ULB) which part of this work was done [80] black hole solution [56,81,82] whose thermodynamics and acknowledges the support by F.R.S.-FNRS fellowship “Bourse de séjour scientifique IN”. P.C. would like to thank to the Dirección de properties have been discussed in [59]. Then, it seems natu- Investigación and Vice-rectoria de Investigación of the Universidad ral to expect to find a BTZ type solution for our theory. The Católica de la Santísima Concepción, Chile, for their constant support. approach to this last point and thermodynamics remains as an interesting open issue to explore. In particular, one can study Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The results obtained the effects of the present deformation in the vacuum energy and presented in the manuscript did not require any data.] and vacuum angular momentum of the stationary configuration and their relations to the GR and Maxwell ones. Open Access This article is licensed under a Creative Commons Attri- Although the three-dimensional Maxwell algebra can be bution 4.0 International License, which permits use, sharing, adaptation, ( , ) ⊕ ( , ) ( , ) ⊕ ( , ) distribution and reproduction in any medium or format, as long as you deformed into so 2 2 so 2 1 and iso 2 1 so 2 1 give appropriate credit to the original author(s) and the source, pro- algebras, the former can be obtained as a deformation of vide a link to the Creative Commons licence, and indicate if changes the latter. The same argument is true for their corresponding were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indi- infinite enhancements, meaning that the bms3 ⊕witt algebra ⊕ ⊕ cated otherwise in a credit line to the material. If material is not can be deformed into the witt witt witt algebra. Since included in the article’s Creative Commons licence and your intended both infinite-dimensional algebras are obtained as asymptotic use is not permitted by statutory regulation or exceeds the permit- symmetry algebras of different three-dimensional CS gravity ted use, you will need to obtain permission directly from the copy- theories, it might be interesting to consider the deformation right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. relation between these theories at the level of their solutions, Funded by SCOAP3. phase spaces and physical quantities. It should be pointed out that the bms3 ⊕ witt algebra can also be deformed into another algebra which is known as W(a, b) ⊕ witt,aswas proved in [30]. Regarding the recent work [78], where it was References shown that the W(0, b) algebra can be obtained as near horizon symmetries of three-dimensional black holes, a natural 1. A. Achucarro, P. Townsend, A Chern–Simons action for three- dimensional anti-de Sitter supergravity theories. Phys. Lett. B 180, question that one can ask is what is the interpretation of the 89 (1986) 2. E. Witten, (2 + 1)-Dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46 (1988) 123 Eur. Phys. J. C (2020) 80:967 Page 9 of 10 967

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