On the Maxwell Supergravity and Flat Limit in 2+1 Dimensions

Physics Letters B 785 (2018) 247–253

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Physics Letters B

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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 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction three-dimensional Maxwell CS gravity [43]. Such asymptotic symmetry 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 supergravity, 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 peralgebra and the invariant tensor applying the expansion method 4π 3 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 gravitational 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. The gauge connection one-form for the Maxwell algebra reads a 1 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 recovered in the limit α = 0. Such interesting feature is proper of where ωa corresponds to the spin connection, ea is the vielbein 2 the Maxwell like symmetry B [50,62] which corresponds to the and σ a is the gravitational Maxwell gauge field [43]. On the other k Maxwell algebra for k = 4. As was shown in [32–35], GR can be hand, the non-vanishing components of the invariant tensor for the recovered under a particular limit from a CS and BI gravity theory Maxwell algebra are given by invariant under the Maxwell like groups. J J = , a b α0ηab 3. Minimal Maxwell Chern–Simons supergravity in

Ja Pb = α1ηab , three-dimensional spacetime = Ja Zb α2ηab , (2.4) In this section, following a similar procedure used in [60], we present a novel way of ﬁnding the so called minimal Maxwell Pa Pb = α2ηab , superalgebra sM. Moreover the construction of the CS action in- where α0, α1 and α2 are real constants. In particular, the Maxwell variant under this superalgebra is presented. The minimal Maxwell algebra has the following quadratic Casimir [29,42], superalgebra in D = 4dimensions was ﬁrst introduced in [52]. P. Concha et al. / Physics Letters B 785 (2018) 247–253 249

1 Then, in D = 3 dimensions, a generalized minimal Maxwell su- β [Pa, Q α] = ( a) α β , peralgebra was derived as a semigroup expansion (S-expansion) 2 of the osp(2|1) ⊗ sp(2), and it was used in order to study the 1 a Q α, Q β =− C Pa , corresponding CS theory [60]. Although the procedure was useful 2 αβ to derive a three-dimensional CS formalism, the generalized min- 1 a Q α, β =− C Za . imal Maxwell superalgebra obtained there contains a large set of 2 αβ bosonic generators J , P , Z , Z˜ , Z˜ . In particular the bosonic ab a ab ab a The (anti-)commutation relations (3.5) correspond to the minimal ˜ ˜ Maxwell superalgebra sM, introduced in [64]in three spacetime content Zab, Za implies a large number of extra terms in the CS dimensions. This supersymmetric extension of the Maxwell sym- action whose presence is due to the considered starting superalge- metry is characterized by the introduction of a new Majorana bra. spinor charge which is required to satisfy the Jacobi identity Here we show that using a smaller structure as the original al- (P, Q , Q ). The presence of a second abelian spinorial generators gebra, the minimal Maxwell superalgebra can be obtained through is not new in the literature and has already been studied in the the S-expansion [45]. As we shall see, the extra bosonic genera- context of D = 11 supergravity [65] and superstring theory [66]. tors do not appear if a super-Lorentz algebra is considered as the The non-vanishing components of the invariant tensor of the starting superalgebra. In addition, such method allows not only to Maxwell superalgebra can be obtained in terms of the super- recover the complete set of (anti-)commutation relations of the minimal Maxwell superalgebra but also the invariant tensor which Lorentz ones using the definitions of the S-expansion method, is required for the construction of a CS action. J J = μ M M L = μ η , The most natural minimal supersymmetric extension of the a b 0 a b s 0 ab μ2 μ2 Lorentz algebra in three spacetime dimensions is spanned by J P = M M L = η , a b a b s ab Lorentz generators Ma and Majorana fermionic generators Q α (α = μ4 μ4 1, 2) [63]. The (anti-)commutation relations read Ja Z = Ma M L = η , b 2 b s 2 ab = c μ4 μ4 [Ma, Mb] abc M , P P = M M L = η , (3.6) a b 2 a b s 2 ab 1 β [Ma, Q α] = ( a) α Q β , (3.1) μ2 μ2 Q α Q β = Q α Q β = Cαβ , 2 sL 1 Q , Q =− C a M , μ4 μ4 α β αβ a Q αβ = Q α Q β = Cαβ , 2 2 sL 2 where a, b, ···= 0, 1, 2are the Lorentz indices, a are the Dirac where μ0, μ2 and μ4 are arbitrary constants. For convenience, we matrices in three dimensions, and C represents the charge conju- will redefine the μ’s as follows gation matrix, 2 μ0 → α0, μ2 → α1, μ4 → α2 , αβ 0 −1 Cαβ = C = . (3.2) 10 and the invariant tensor takes de the form One can easily check the consistency of the (anti-)commutation re- Ja Jb = α0ηab , Pa Pb = α2ηab , lations of the super-Lorentz algebra sL (3.1)by checking the Jacobi T a a T identities, and by considering that C =−C and C = (C ) . Ja Pb = α1ηab , Q α Q β = α1Cαβ , (3.7) (4) Let S = {λ0,λ1,λ2,λ3,λ4,λ5} be the abelian semigroup E Ja Zb = α2ηab , Q αβ = α2Cαβ . whose elements satisfy the multiplication law The connection one-form reads + ≤ = λα+β , when α β 5 , λαλβ + (3.3) a a a ¯ ¯ λ5 , when α β>5 , A = ω Ja + e Pa + σ Za + ψ Q + ξ , (3.8) a a where λ5 = 0s is the zero element of the semigroup. where ω is the spin connection one-form, e corresponds to the Following the definitions of [45], after extracting a resonant vielbein one-form, σ a is the Maxwell gravity field one-form while (4) × L ψ and ξ are fermionic fields. subalgebra of S E s and applying its 0s-reduction, one finds a new superalgebra whose generators { Ja, Pa, Za, Q α, α} are re- The corresponding curvature two-form is given by lated to the super-Lorentz ones as a a a ¯ ¯ F = R Ja + T Pa + F Za +∇ψ Q +∇ξ , (3.9) 2 Ja = λ0 Ma ,Pa = λ2 Ma ,Za = λ4 Ma , 1/2 3/2 (3.4) with Q α = λ1 Q α , α = λ3 Q α . a a 1 abc Using the multiplication law of the semigroup and the super- R = dω + ωbωc , 2 Lorentz (anti-)commutators, one can see that the expanded gen- a a abc 1 ¯ a erators satisfy the following non-vanishing (anti-)commutation re- T = de + ωbec + ψ ψ, (3.10) 4 lations 1 1 Fa = a + abc + abc + ¯ a c c dσ ωbσc ebec ψ ξ. [ Ja, Jb] = abc J , [ Ja, Pb] = abc P , 2 2 c c Here, the covariant derivative ∇=d +[A, ·] acting on spinors read [ Ja, Zb] = abc Z , [Pa, Pb] = abc Z ,

1 β 1 a [ Ja, Q α] = ( a) α Q β , ∇ψ = dψ + ω aψ, 2 2 1 β 1 a 1 a [ Ja, α] = ( a) α β , (3.5) ∇ξ = dξ + ω aξ + e aψ. (3.11) 2 2 2 250 P. Concha et al. / Physics Letters B 785 (2018) 247–253

The CS supergravity action can be written considering the non- cosmological constant. Then, analogously to the Poincaré CS super- vanishing component of the invariant tensor (3.7) and the gauge gravity which appears as a flat limit of the AdS one, it is natural to connection one-form (3.8), expect that the Maxwell CS supergravity obtained here can also be found as a flat limit of a particular supergravity theory. The next k a 1 a b c section is devoted to the obtention of the Maxwell supergravity IsM = α0 ω dωa + abcω ω ω 4π 3 theory as a flat limit of an AdS–Lorent supergravity. a ¯ + α1 2e Ra − ψ∇ψ 4. Maxwell supergravity theory as a flat limit of the AdS–Lorentz a a ¯ ¯ + α2 2R σa + e Ta − ψ∇ξ − ξ∇ψ , (3.12) supergravity where T a = Dea is the usual torsion two-form. It has been shown that the Maxwell symmetry can be alterna- Note that the term proportional to α0 contains just the exotic tively obtained as an IW contraction of an enlarged symmetry de- Lagrangian. The piece along α1 contains the EH and the Rarita– noted as the AdS–Lorentz algebra [50]. Such algebra and its gener- Schwinger terms while the term proportional to α2 is of particular alizations have been useful to relate diverse (pure) Lovelock gravity interest since it contains the Maxwell gravitational field σ a plus theories [70–72]. At the supersymmetric level, numerous studies a torsional term. Let us note that the fermionic terms contribute have been done in four and three dimensions leading to interest- only to the α1 and α2 sectors of the bosonic action (2.6). In ad- ing AdS–Lorentz supergravities [73–76]. Recently, it was shown in dition, the CS action (3.12)reproduces the pure three-dimensional three dimensions that a generalized Maxwell supergravity can be supergravity action when the exotic CS terms are neglected (α0 = obtained as an IW contraction of a generalized AdS–Lorentz super- α2 = 0). Such feature also appears on four-dimensional supergravity model [61]. Nevertheless, as was discussed in the previous gravity action based on the Maxwell supergroup [54]using the MacDowell–Mansouri formalism [67]. section, the field content of such theories is large and the physical interpretation of the extra fields remains unknown. In particular, By construction, the CS action (3.12)is invariant under the ˜ ˜ gauge transformation δ A = d + [A, ]. In particular, the action the extra fields related to the set of generators Zab, Za appear is invariant under the following local supersymmetry transforma- as a consequence of the procedure used to construct the super- tion laws gravity theories. In this section we show that the minimal Maxwell supergrav- δωa = 0 , ity theory presented previously can be recovered as a flat limit of a 1 minimal AdS–Lorentz supergravity. To this purpose, let us first con- δea = ¯ aψ, 2 sider a novel supersymmetric extension of the AdS–Lorentz algebra 1 1 in three dimensions. We extend the AdS–Lorentz algebra generated a = ¯ a + ¯ a δσ ξ ψ, (3.13) by { J , P , Z } with the fermionic generators {Q , }, and we get 2 2 a a a α α the following superalgebra 1 a δψ = d + ω a , 2 [ J , J ] = J c , [ J , P ] = P c , 1 1 a b abc a b abc = + a + a δξ d ω a e a , = c = c 2 2 [Pa, Pb] abc Z , [ Ja, Zb] abc Z , (4.1) a a a 1 1 where the gauge parameter is = ρ Ja +ε Pa +γ Za +¯ Q +¯ . c c [Za, Zb] = abc Z , [Pa, Zb] = abc P , The equations of motion derived from (2.6)are 2 2 1 1 a β β δω : α R + α T + α F = 0 , [ Ja, Q α] = ( a) α Q β , [ Ja, α] = ( a) α β , 0 a 1 a 2 a 2 2 a δe : α1 Ra + α2Ta = 0 , 1 β 1 β [Pa, Q α] = ( a) α β , [Pa, α] = ( a) α Q β , a 2 22 δσ : α2 Ra = 0 , (3.14) 1 β 1 β ¯ : ∇ + ∇ = [Za, Q α] = ( a) α Q β , [Za, α] = ( a) α β , δψ α1 ψ α2 ξ 0 , 22 22 ¯ 1 1 δξ : α2∇ψ = 0 . a a Q α, Q β =− C Pa , Q α, β =− C Za , 2 αβ 2 αβ As in the bosonic case, the field equations reduce to the vanishing 1 a of the curvature two-forms provided α = 0, α, β =− C Pa . 2 22 αβ Ra = 0 , T a = 0 , Fa = 0 , (4.2) (3.15) ∇ψ = 0 , ∇ξ = 0 . We will refer to this superalgebra as the minimal AdS–Lorentz su- As was recently shown in the Maxwell gravity case [43], σ a is not peralgebra in three dimensions. Note that the supersymmetric ex- simply an extra field but modifies the asymptotic charges of the tension of the AdS–Lorentz algebra is not unique. However, to our solutions. It would be interesting to extend the results obtained knowledge, this corresponds to the minimal supersymmetric ex- in [43]to the Maxwell supergravity theory presented here. As in tension of the AdS–Lorentz algebra containing two spinor charges. the bosonic theory, one could expect a deformation of the super One can see that the limit →∞ leads appropriately to the bms3 algebra [68,69]as the corresponding asymptotic symmetry. minimal Maxwell superalgebra (3.5) considered in the previous In particular, as in the finite Maxwell superalgebra, two infinite- section. As we shall see, the flat limit can also be directly per- dimensional fermionic generators should appear in the asymptotic formed at the level of the action, supersymmetry transformations structure. and field equations. As an ending remark, the CS theory presented here offers us an In order to write down a CS action for the minimal AdS–Lorentz alternative three-dimensional minimal supergravity with vanishing superalgebra, let us consider the connection one-form P. Concha et al. / Physics Letters B 785 (2018) 247–253 251

a a a ¯ ¯ A = ω Ja + e Pa + σ Za + ψ Q + ξ , (4.3) absent in the super Maxwell theory. This is due to the presence of the component Pa Zb of the invariant tensor which vanishes and the corresponding curvature two-form in the Maxwell case. Interestingly, from the minimal AdS–Lorentz supergravity action (4.9)we see that the vanishing cosmological F = Ra J + T a P + Fa Z +∇ψ¯ Q +∇ξ¯ , (4.4) a a a constant limit →∞ leads us to the minimal Maxwell supergrav- where ity action (3.12)introduced in the previous section. This is very similar to the flat behavior present in AdS supergravity and in the a a 1 abc bosonic AdS–Lorentz gravity theory. For completeness we apply the R = dω + ωbωc , 2 flat limit at the level of the supersymmetry transformations and a a abc 1 abc 1 ¯ a 1 ¯ a equations of motion. T = de + ωbec + σbec + ψ ψ + ξ ξ, 2 4 42 The CS action (4.9)is invariant by construction under the gauge transformation δ A = d + [A, ]. In particular, the supersymmetry a a abc 1 abc 1 abc 1 ¯ a F = dσ + ωbσc + σbσc + ebec + ψ ξ. 22 2 2 transformation laws read (4.5) δωa = 0 , Furthermore, 1 1 δea = ¯ aψ + ¯ aξ, 1 1 1 2 22 ∇ψ = dψ + a ψ + a ψ + ea ξ, ω a 2 σ a 2 a 1 1 2 2 2 δσ a = ¯ aξ + ¯ aψ, (4.11) 1 1 1 2 2 ∇ξ = dξ + a ξ + a ξ + ea ψ. (4.6) ω a 2 σ a a 1 1 1 2 2 2 δψ = d + a + a + ea , ω a 2 σ a 2 a Note that the curvatures reduce to the Maxwell ones (3.10) and 2 2 2 (3.11)by performing the limit →∞. 1 a 1 a 1 a δξ = d + ω a + e a + σ a , The non-vanishing components of the invariant tensor are given 2 2 22 a a a by (3.7)along with with gauge parameter = ρ Ja + ε Pa + γ Za + ¯ Q + ¯ . Note that the supersymmetry transformations (4.11)reduce to the = α1 Pa Zb ηab , Maxwell ones of eq. (3.13)in the limit →∞. 2 The field equations are given by α2 Za Zb = ηab , (4.7) 2 a : + T + F = δω α0 Ra α1 a α2 a 0 , α1 αβ = Cαβ . 1 2 δea : α R + F + α T = 0 , 1 a 2 a 2 a Thus, the most general quadratic Casimir of the minimal AdS– α1 1 Lorentz superalgebra is δσ a : T + α R + F = 0 , (4.12) 2 a 2 a 2 a a a 1 a ¯ 1 ¯ ¯ C = α0 J Ja + α1 P Ja + P Za + QQ+ δψ : α ∇ψ + α ∇ξ = 0 , 2 2 1 2 α1 δξ¯ : ∇ψ + ∇ξ = 0 . 1 α2 2 + α P a P + J a Z + Za Z + Q¯ . (4.8) 2 a a 2 a When α0, α1 and α2 are independent, the equations of motion Let us note that the flat limit →∞ at the level of the invari- reduce to the vanishing of the curvature two-forms, ant tensor of the AdS–Lorentz superalgebra leads us to the non- Ra = 0 , T a = 0 , Fa = 0 , degenerate invariant tensor of the minimal Maxwell superalgebra (4.13) ∇ψ = 0 , ∇ξ = 0 . (3.7). Then, using the connection one-form (4.3) and the invariant tensors (3.7) and (4.7)in the CS action (2.2), we get It is simple to verify that the vanishing cosmological constant limit →∞ in (4.12)leads to the super Maxwell field equations (3.14). k 1 a a b c It is important to clarify that an alternative supersymmetric IsAdS−L = α0 ω dωa + abcω ω ω 4π 3 extension of the AdS–Lorentz algebra can be obtained with one ˜ 2 1 spinor generator Q α such that + 2ea R + ea F + eaebec α1 a 2 a 2 abc 3 ˜ ˜ ˜ c ˜ ˜ ˜ c Ja, Jb = abc J , Ja, Pb = abc P , ¯ 1 ¯ a 2 a a − ψ∇ψ − ξ∇ξ + α2 2R σa + F σa + e Ta 2 2 P˜ , P˜ = Z˜ c , ˜J , Z˜ = Z˜ c , a b abc a b abc 1 + a b c − ¯ ∇ − ¯∇ ˜ ˜ 1 ˜ c ˜ ˜ 1 ˜ c abce σ e ψ ξ ξ ψ , (4.9) Z , Z = Z , P , Z = P , 2 a b 2 abc a b 2 abc 1 where T a = dea + abc e is the torsion two-form and ˜ ˜ β ˜ ωb c Ja, Q α = ( a) α Q β , (4.14) 2 1 a a abc abc 1 β F = dσ + ωbσc + σbσc . (4.10) ˜ ˜ = ˜ 22 Pa, Q α ( a) α Q β , 2 Although the field content of the AdS–Lorentz supergravity the- ˜ ˜ 1 β ˜ Za, Q α = ( a) α Q β , ory is the same as in the super Maxwell one, the CS action is 22 much richer. As in the four-dimensional case [74], this symmetry ˜ ˜ 1 a ˜ 1 a ˜ allows the presence of a cosmological constant term which was Q α, Q β =− C Za − C Pa . 2 αβ 2 αβ 252 P. Concha et al. / Physics Letters B 785 (2018) 247–253

This superalgebra can be seen as a supersymmetric extension of Acknowledgements a semisimple generalization of the Poincaré algebra [48]. Although this superalgebra contains the bosonic AdS–Lorentz subalgebra, its This work was supported by the Chilean FONDECYT Projects supersimmetrization is quite different from the minimal one pre- No. 3170437 (P.C.) and No. 3170438 (E.R.). D.M.P. acknowledges DI- sented previously (see eqs. (4.1)–(4.2)). One can see that there is VRIEA for financial support through Proyecto Postdoctorado 2018 no redefinition of the generators allowing to relate them. More- VRIEAPUCV. The authors would like to thank to L. Ravera for valu- over, a reinterpretation of the Pa and Za generators would modify able discussions and comments. drastically the bosonic algebra. More recently, a three-dimensional CS supergravity action invariant under this super AdS–Lorentz al- References gebra has been presented in [73] allowing to introduce a generalized cosmological term to the EH term. [1] M. Banados, C. Teitelboim, J. Zanelli, The black hole in three-dimensional spacetime, Phys. Rev. Lett. 69 (1992) 1849, arXiv:hep -th /9204099. In particular, a non-standard Maxwell superalgebra [29,77]can [2] S. Deser, J.H. Kay, Topologically massive supergravity, Phys. Lett. B 120 (1983) be obtained applying the IW contraction to (4.14). Although a su- 97. pergravity CS action can be constructed from this alternative super [3] S. Deser, Cosmological Topological Supergravity, Quantum Theory of Gravity: Essays in Honor of the 60th Birthday of Bryce S. DeWitt, Adam Hilger Ltd., AdS–Lorentz version, the IW contraction does not reproduce asu- Bristol, 1984. pergravity theory. Indeed, in the non-standard Maxwell superalge- [4] A. Achucarro, P.K. Townsend, A Chern–Simons action for three-dimensional bra we have anti-De Sitter supergravity theories, Phys. Lett. B 180 (1986) 89. = + [5] A. Achucarro, P.K. Townsend, Extended supergravities in d (2 1) as Chern– 1 Simons theories, Phys. Lett. B 229 (1989) 383. ˜ ˜ =− a ˜ Q α, Q β C αβ Za . (4.15) [6] N. Marcus, J.H. Schwarz, Three-dimensional supergravity theories, Nucl. Phys. B 2 228 (1983) 145. ˜ [7] P. van Nieuwenhuizen, Three-dimensional conformal supergravity and Chern– Here, the four-momentum generators Pa are no more expressed as ˜ Simons terms, Phys. Rev. D 32 (1985) 872. bilinear expressions of the fermionic generators Q which leads to [8] M. Rocek, P. van Nieuwenhuizen, N ≥ 2 supersymmetric Chern–Simons terms an exotic supersymmetric action. as d = 3extended conformal supergravity, Class. Quantum Gravity 3 (1986) 43. [9] H. Nishino, S.J. Gates Jr., Chern–Simons theories with supersymmetries in three-dimensions, Mod. Phys. A 8 (1993) 3371. 5. Discussion [10]S. P. Howe, J.M. Izquierdo, G. Papadopoulos, P.K. Townsend, New supergravities with central charges and Killing spinors in 2 + 1 dimensions, Nucl. Phys. B 467 (1996) 183, arXiv:hep -th /9505032. The minimal supersymmetric extension of the three-dimen- [11] M. Banados, R. Troncoso, J. Zanelli, Higher dimensional Chern–Simons super- sional Maxwell Chern–Simons gravity has been presented by ex- gravity, Phys. Rev. D 54 (1996) 2605, arXiv:gr-qc /9601003. panding a Lorentz superalgebra. The Maxwell superalgebra ob- [12] A. Giacomini, R. Troncoso, S. Willison, Three-dimensional supergravity tained is the minimal one presented in [52] containing two spinor reloaded, Class. Quantum Gravity 24 (2007) 2845, arXiv:hep -th /0610077. [13] R.K. Gupta, A. Sen, Consistent truncation to three dimensional (super-)gravity, generators Q and . The methodology considered here allows α α J. High Energy Phys. 0803 (2008) 015, arXiv:0710 .4177 [hep -th]. us to construct the CS supergravity action invariant under the min- [14] R. Andringa, E.A. Bergshoeff, M. de Roo, O. Hohm, E. Sezgin, P.K. Townsend, imal superMaxwell which is characterized by three coupling con- Massive 3D supergravity, Class. Quantum Gravity 27 (2010) 025010, arXiv:0907. 4658 [hep-th]. stants. In particular, the gravitational Maxwell field σa appears in [15] R. Andringa, E.A. Bergshoeff, J. Rosseel, E. Sezgin, 3D Newton–Cartan supergrav- the α sector. In addition, the supergravity theory presented here 2 ity, Class. Quantum Gravity 30 (2013) 205005, arXiv:1305 .6737 [hep -th]. does not contain additional bosonic fields as those appearing in [16] D. Butter, S.M. Kuzenko, J. Novak, G. Tartaglino-Mazzucchelli, Conformal supe- the generalized Maxwell superalgebra [60,61]. gravity in three dimensions: off-shell actions, J. High Energy Phys. 1310 (2013) Interestingly, the present Maxwell supergravity theory has also 073, arXiv:1306 .1205 [hep -th]. = been obtained by applying a vanishing cosmological constant limit [17] M. Nishimura, Y. Tanii, N 6conformal supergravity in three dimensions, J. High Energy Phys. 1310 (2013) 123, arXiv:1308 .3960 [hep -th]. to a minimal AdS–Lorentz supergravity theory. The flat limit was [18] G. Alkac, L. Basanisi, E.A. Bergshoeff, M. Ozkan, E. Sezgin, Massive N = 2su- presented not only in the commutation relations but also in the pergravity in three dimensions, J. High Energy Phys. 1502 (2015) 125, arXiv: action, field equations and supersymmetry transformation laws. 1412 .3118 [hep -th]. This flat behavior appears also at the bosonic level, including its [19] O. Fuentealba, J. Matulich, R. Troncoso, Extension of the Poincaré group with half-integer spin generators: hypergravity and beyond, J. High Energy Phys. infinite-dimensional symmetries [43,44] and higher-spin extension 1509 (2015) 003, arXiv:1505 .06173 [hep -th]. of the Maxwell and AdS–Lorentz gravity [41]. [20] O. Fuentealba, J. Matulich, R. Troncoso, Asymptotically flat structure of hyper- Having a well defined Maxwell supergravity theory in three gravity in three spacetime dimensions, J. High Energy Phys. 1510 (2015) 009, spacetime dimensions, it would be interesting to go further and arXiv:1508 .04663 [hep -th]. .[21] E.A Bergshoeff, J. Rosseel, T. Zojer, Newton–Cartan supergravity with torsion analyze the influence of the gravitational Maxwell field in the and Schrödinger supergravity, J. High Energy Phys. 1511 (2015) 180, arXiv: asymptotic symmetry of this supergravity theory. Analogously to 1509 .04527 [hep -th]. the bosonic case, one could expect a deformation and enlargement [22] G. Barnich, L. Donnay, J. Matulich, R. Troncoso, Super-BMS3 invariant boundary theory from three-dimensional flat supergravity, J. High Energy Phys. 1701 of the super bms3 algebra [work in progress]. One could go even further and study the asymptotic symmetry of the AdS–Lorentz su- (2017) 029, arXiv:1510 .08824 [hep -th]. [23] R. Basu, S. Detournay, M. Riegler, Spectral flow in 3D flat spacetimes, J. High pergravity in three dimensions and analyze the existence of a flat Energy Phys. 12 (2017) 134, arXiv:1706 .07438 [hep -th]. limit. [24] O. Fuentealba, J. Matulich, R. Troncoso, Asymptotic structure of N = 2su- Another important aspect that deserves further investigation is pergravity in 3D: extended super-BMS3 and nonlinear energy bounds, J. High the study of the non-relativistic (NR) limit of the minimal Maxwell Energy Phys. 09 (2017) 030, arXiv:1706 .07542 [hep -th]. [25] H. Bacry, P. Combe, J.L. Richard, Group-theoretical analysis of elementary par- CS supergravity presented here. It has been pointed out that the ticles in an external electromagnetic fields. 1. The relativistic particle in a con- NR gravities could play an important role to the understanding stant and uniform field, Nuovo Cimento A 67 (1970) 267. of non-relativistic coupled systems in the boundary. Recently, the [26] R. Schrader, The Maxwell group and the quantum theory of particles in classi- NR limit of a three-dimensional Maxwell CS gravity has been con- cal homogeneous electromagnetic fields, Fortschr. Phys. 20 (1972) 701. [27] J. Gomis, A. Kleinschmidt, On free Lie algebras and particles in electro-magnetic sidered in [42] where a generalization of the Extended Bargmann fields, J. High Energy Phys. 07 (2017) 085, arXiv:1705 .05854 [hep -th]. gravity has been obtained. Therefore, it would be interesting to ex- [28] J. Beckers, V. Hussin, Minimal electromagnetic coupling schemes. II. Relativistic tend the approach of [42]to our supergravity case. and nonrelativistic Maxwell groups, J. Math. Phys. 24 (1983) 1295. P. Concha et al. / Physics Letters B 785 (2018) 247–253 253

[29] D.V. Soroka, V.A. Soroka, Tensor extension of the Poincaré algebra, Phys. Lett. B [53]. J.A de Azcarraga, J.M. Izquierdo, Minimal D = 4supergravity from super- 607 (2005) 302, arXiv:hep -th /0410012. Maxwell algebra, Nucl. Phys. B 885 (2014) 34, arXiv:1403 .4128 [hep -th]. [30] S. Bonanos, J. Gomis, Infinite sequence of Poincare group extensions: structure [54]K. P. Concha, E.K. Rodríguez, N = 1supergravity and Maxwell superalgebras, and dynamics, J. Phys. A 43 (2010) 015201, arXiv:0812 .4140 [hep -th]. J. High Energy Phys. 1409 (2014) 090, arXiv:1407.4635 [hep -th]. [31] J.D. Edelstein, M. Hassaine, R. Troncoso, J. Zanelli, Lie-algebra expansions, [55] D.M. Peñafiel, L. Ravera, On the hidden Maxwell superalgebra underlying D = 4 Chern–Simons theories and the Einstein–Hilbert Lagrangian, Phys. Lett. B 640 supergravity, Fortschr. Phys. 65 (2017) 1700005, arXiv:1701.04234 [hep -th]. (2006) 278, arXiv:hep -th /0605174. [56] L. Ravera, Hidden role of Maxwell superalgebras in the free differential algebras [32] F. Izaurieta, E. Rodríguez, P. Minning, P. Salgado, A. Perez, Standard general of D = 4and D = 11 supergravity, Eur. Phys. J. C 78 (2018) 211, arXiv:1801. relativity from Chern–Simons gravity, Phys. Lett. B 678 (2009) 213, arXiv: 08860 [hep -th]. 0905.2187 [hep-th]. .[57] J.A de Azcarraga, J.M. Izquierdo, J. Lukierski, M. Woronowicz, Generalizations K.[33] P. Concha, D.M. Peñafiel, E.K. Rodríguez, P. Salgado, Even-dimensional general of Maxwell (super)algebras by the expansion method, Nucl. Phys. B 869 (2013) relativity from Born–Infeld gravity, Phys. Lett. B 725 (2013) 419, arXiv:1309. 303, arXiv:1210 .1117 [hep -th]. 0062 [hep -th]. [58]K. P. Concha, E.K. Rodríguez, Maxwell superalgebras and Abelian semigroup ex- K.[34] P. Concha, D.M. Peñafiel, E.K. Rodríguez, P. Salgado, Chern–Simons and Born– pansion, Nucl. Phys. B 886 (2014) 1128, arXiv:1405 .1334 [hep -th]. Infeld gravity theories and Maxwell algebras type, Eur. Phys. J. C 74 (2014) .[59] J.A de Azcarraga, J.M. Izquierdo, M. Picon, O. Varela, Extensions, expansions, 2741, arXiv:1402 .0023 [hep -th]. Lie algebra cohomology and enlarged superspaces, Class. Quantum Gravity 21 [35]K. P. Concha, D.M. Peñafiel, E.K. Rodríguez, P. Salgado, Generalized Poincaré (2004) S1375, arXiv:hep-th/0401033. algebras and Lovelock–Cartan gravity theory, Phys. Lett. B 742 (2015) 310, arXiv:1405 .7078 [hep .th]. K.[60] P. Concha, O. Fierro, E.K. Rodríguez, P. Salgado, Chern–Simons supergravity [36]. J.A de Azcarraga, K. Kamimura, J. Lukierski, Generalized cosmological term in D = 3and Maxwell superalgebra, Phys. Lett. B 750 (2015) 117, arXiv:1507. from Maxwell symmetries, Phys. Rev. D 83 (2011) 124036, arXiv:1012 .4402 02335 [hep -th]. [hep -th]. K.[61] P. Concha, O. Fierro, E.K. Rodríguez, Inönü–Wigner contraction and D = 2 + 1 [37] R. Durka, J. Kowalski-Glikman, M. Szczachor, Gauges AdS–Maxwell algebra and supergravity, Eur. Phys. J. C 77 (2017) 48, arXiv:1611.05018 [hep -th]. gravity, Mod. Phys. Lett. A 26 (2011) 2689, arXiv:1107.4728 [hep -th]. [62]K. P. Concha, R. Durka, N. Merino, E.K. Rodríguez, New family of Maxwell like .[38] J.A de Azcarraga, K. Kamimura, J. Lukierski, Maxwell symmetries and some algebras, Phys. Lett. B 759 (2016) 507, arXiv:1601.06443 [hep -th]. applications, Int. J. Mod. Phys. Conf. Ser. 23 (2013) 01160, arXiv:1201.2850 [63] J. Lukierski, A. Nowicki, Superspinors and graded Lorentz groups in three, four [hep -th]. and five dimensions, Fortschr. Phys. 30 (1982) 75. [39] P. Salgado, R.J. Szabo, O. Valdivia, Topological gravity and transgression holog- [64] S. Bonanos, J. Gomis, K. Kamimura, J. Lukierski, Deformations of Maxwell su- raphy, Phys. Rev. D 89 (2014) 084077, arXiv:1401.3653 [hep-th]. peralgebras and their applications, J. Math. Phys. 51 (2010) 102301, arXiv: [40] S. Hoseinzadeh, A. Rezaei-Aghdam, (2 + 1)-dimensional gravity from Maxwell 1005 .3714 [hep -th]. and semisimple extension of the Poincaré gauge symmetric models, Phys. Rev. [65] R. D’Auria, P. Fré, Geometric supergravity in d = 11 and its hidden supergroup, D 90 (2014) 084008, arXiv:1402 .0320 [hep -th]. Nucl. Phys. B 201 (1982) 101. [41] R. Caroca, P. Concha, O. Fierro, E. Rodríguez, P. Salgado-Rebolledo, Generalized [66] M.B. Green, Supertranslations, superstrings and Chern–Simons forms, Phys. Chern–Simons higher-spin gravity theories in three dimensions, Nucl. Phys. B Lett. B 223 (1989) 157. 934 (2018) 240, arXiv:1712 .09975 [hep -th]. [42] L. Avilés, E. Frodden, J. Gomis, D. Hidalgo, J. Zanelli, Non-relativistic Maxwell [67]. S.W MacDowell, F. Mansouri, Unified geometric theory of gravity and super- Chern–Simons gravity, arXiv:1802 .08453 [hep -th]. gravity, Phys. Rev. Lett. 38 (1977) 739. [43] P. Concha, N. Merino, O. Miskovic, E. Rodríguez, P. Salgado-Rebolledo, O. Val- [68] A. Bagchi, I. Mandal, Supersymmetric extension of Galilean conformal algebras, divia, Extended asymptotic symmetries of three-dimensional gravity in flat Phys. Rev. D 80 (2009) 086011, arXiv:0905 .0580 [hep -th]. space, arXiv:1805 .08834 [hep -th]. [69] G. Barnich, L. Donnay, J. Matulich, R. Troncoso, Asymptotic symmetries and dy- [44] R. Caroca, P. Concha, E. Rodríguez, P. Salgado-Rebolledo, Generalizing the bms3 namics of three-dimensional flat supergravity, J. High Energy Phys. 1408 (2014) and 2D-conformal algebras by expanding the Virasoro algebra, Eur. Phys. J. C 78 071, arXiv:1407.4275 [hep -th]. (2018) 262, arXiv:1707.07209 [hep -th]. [70]K. P. Concha, R. Durka, C. Inostroza, N. Merino, E.K. Rodríguez, Pure Lovelock [45] F. Izaurieta, E. Rodríguez, P. Salgado, Expanding Lie (super)algebras through gravity and Chern–Simons theory, Phys. Rev. D 94 (2016) 024055, arXiv:1603 . Abelian semigroups, J. Math. Phys. 47 (2006) 123512, arXiv:hep -th /0606215. 09424 [hep -th]. [46] E. Inönü, E.P. Wigner, On the contraction of groups and their representations, K.[71] P. Concha, N. Merino, E.K. Rodríguez, Lovelock gravity from Born–Infeld grav- Proc. Natl. Acad. Sci. USA 39 (1953) 510. ity theory, Phys. Lett. B 765 (2017) 395, arXiv:1606 .07083 [hep -th]. [47] E. Weimar-Woods, Contractions, generalized Inönü–Wigner contractions and [72] P. Concha, E. Rodríguez, Generalized pure Lovelock gravity, Phys. Lett. B 774 deformations of finite-dimensional Lie algebras, Rev. Mod. Phys. 12 (2000) (2017) 616, arXiv:1708 .08827 [hep -th]. 1505. [73] O. Fierro, F. Izaurieta, P. Salgado, O. Valdivia, (2 + 1)-dimensional supergravity [48] D.V. Soroka, V.A. Soroka, Semi-simple extension of the (super) Poincaré algebra, invariant under the AdS–Lorentz superalgebra, arXiv:1401.3697 [hep -th]. Adv. High Energy Phys. 2009 (2009) 234147, arXiv:hep-th/0605251. K.[74] P. Concha, E.K. Rodríguez, P. Salgado, Generalized supersymmetric cosmolog- [49] J. Gomis, K. Kamimura, J. Lukierski, Deformations of Maxwell algebra and their ical term in N = 1supergravity, J. High Energy Phys. 08 (2015) 009, arXiv: dynamical realizations, J. High Energy Phys. 0908 (2009) 039, arXiv:0906.4464 1504 .01898 [hep -th]. [hep -th]. K.[75] P. Concha, M.C. Ipinza, L. Ravera, E.K. Rodríguez, On the supersymmetric [50] P. Salgado, S. Salgado, so(D − 1, 1) ⊕ so(D − 1, 2) algebras and gravity, Phys. extension of Gauss–Bonnet like gravity, J. High Energy Phys. 09 (2016) 007, Lett. B 728 (2014) 5–10. arXiv:1607.00373 [hep -th]. [51] J. Díaz, O. Fierro, F. Izaurieta, N. Merino, E. Rodriguez, P. Salgado, O. Valdivia, [76] A. Baunadi, L. Ravera, Generalized AdS–Lorentz deformed supergravity on a A generalized action for (2 + 1)-dimensional Chern–Simons gravity, J. Phys. A, manifold with boundary, arXiv:1803.08738 [hep-th]. Math. Theor. 45 (2012) 255207. [52] S. Bonanos, J. Gomis, K. Kamimura, J. Lukierski, Maxwell superalgebra and su- [77] J. Lukierski, Generalized Wigner–Inönü contractions and Maxwell (super)alge- perparticle in constant gauge background, Phys. Rev. Lett. 104 (2010) 090401, bras, Proc. Steklov Inst. Math. 272 (1) (2011) 183, arXiv:1007.3405 [hep -th]. arXiv:0911.5072 [hep -th].