Inönü–Wigner Contraction and D = 2 + 1 Supergravity

Eur. Phys. J. C (2017) 77:48 DOI 10.1140/epjc/s10052-017-4615-1

Inönü–Wigner contraction and D = 2 + 1 supergravity

P. K. Concha1,2,a, O. Fierro3,b, E. K. Rodríguez1,2,c 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia, Chile 3 Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Rivera 2850, Concepción, Chile

Received: 25 November 2016 / Accepted: 5 January 2017 / Published online: 25 January 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com

Abstract We present a generalization of the standard cannot always be obtained by rescaling the gauge fields and Inönü–Wigner contraction by rescaling not only the gener- considering some limit as in the (anti)commutation relations. ators of a Lie superalgebra but also the arbitrary constants In particular it is well known that, in the presence of the exotic appearing in the components of the invariant tensor. The Lagrangian, the Poincaré limit cannot be applied to a (p, q) procedure presented here allows one to obtain explicitly the AdS CS supergravity [7]. This difficulty can be overcome Chern–Simons supergravity action of a contracted superal- extending the osp (2, p) ⊗ osp (2, q) superalgebra by intro- gebra. In particular we show that the Poincaré limit can be ducing the automorphism generators so (p) and so (q) [9]. performed to a D = 2 + 1 (p, q) AdS Chern–Simons super- In such a case, the IW contraction can be applied and repro- gravity in presence of the exotic form. We also construct a duces the Poincaré limit leading to a new (p, q) Poincaré new three-dimensional (2, 0) Maxwell Chern–Simons super- supergravity which includes additional so (p)⊕so (q) gauge gravity theory as a particular limit of (2, 0) AdS–Lorentz fields. supergravity theory. The generalization for N = p +q grav- Here, we present a generalization of the IW contraction by itinos is also considered. considering not only the rescaling of the generators but also the constants of the non-vanishing components of an invariant tensor. The method introduced here ensures the construc- 1 Introduction tion of any CS action based on a contracted (super)algebra. In particular, we show that the Poincaré limit can be applied The three-dimensional (super)gravity theory represents an to a (p, q) AdS supergravity in the presence of the exotic interesting toy model in order to approach higher-dimensional Lagrangian without introducing extra fields as in Ref. [9]. (super)gravity theories, which are not only more difficult but Subsequently, we apply the method to different (p, q) AdS– also leads to tedious calculations. Additionally, the D = 2+1 Lorentz supergravities whose IW contraction leads to diverse model has the remarkable property to be written as a gauge (p, q) Maxwell supergravities. The possibility to turn the IW theory using the Chern–Simons (CS) formalism [1,2]. In par- contraction into an algebraic operation is not new and has ticular, the three-dimensional supersymmetric extension of already been presented in the context of asymptotic symme- General Relativity [3,4] can be obtained as a CS gravity the- tries and higher spin theories in Ref. [26]. Other interesting ory using (A)dS or Poincaré supergroup. A wide class of results using diverse flat limit contractions in supergravity N -extended Supergravities and further extensions have been can be found in Ref. [27]. studied in diverse contexts in, e.g., [5–25]. At the bosonic level, the Maxwell symmetries have lead The derivation of a supergravity action for a given superal- to interesting gravity theories allowing to recover General gebra is not, in general, a trivial task and its construction is not Relativity from Chern–Simons and Born–Infeld (BI) the- always ensured. On the other hand, several (super)algebras ories [28–31]. On the other hand, the AdS–Lorentz and can be obtained as an Inönü–Wigner (IW) contraction of its generalizations allow one to recover the Pure Lovelock a given (super)algebra [38,39]. Nevertheless, the Chern– [32–34] Lagrangian in a matter-free configuration from CS Simons action based on the IW contracted (super)algebra and BI theories [35,36]. At the supersymmetric level, the Maxwell superalgebra provides a pure supergravity action in a e-mail: [email protected] the MacDowell–Mansouri formalism [37]. More recently, a b e-mail: ofi[email protected] three-dimensional CS action based on the minimal Maxwell c e-mail: [email protected] 123 48 Page 2 of 18 Eur. Phys. J. C (2017) 77 :48 superalgebra has been presented in Ref. [22] using the expan- exotic Lagrangian, which has no Poincaré limit [7], is added sion procedure. Here, we show that the same result can be to the AdS CS supergravity. obtained using our alternative approach. Besides, we show The three-dimensional Chern–Simons action is given by ( , ) that the Maxwell limit can also be applied in a p q enlarged supergravity leading to a (p, q) Maxwell supergravity with ( + ) 2 I 2 1 = k Ad A + A3 , (1) an exotic Lagrangian. CS 3 The organization of the present work is as follows: in Sect. 2, we apply our approach to N = 1 and N = p+qAdS where A corresponds to the gauge connection one-form CS supergravities. In particular, we show that the Poincaré and ... denotes the invariant tensor. In the case of the limit can be applied to (p, q) AdS CS supergravity theories in osp (2|1) ⊗ sp (2) superalgebra, the connection one-form is the presence of the exotic Lagrangian. In Sect. 3, we discuss given by the Inönü–Wigner contraction of an expanded supergravity. In particular, we describe the general scheme. In Sect. 4,we 1 ab ˜ 1 a ˜ 1 α ˜ A = ω Jab + e Pa + √ ψ Qα, (2) apply our procedure to a N = 1 expanded CS supergravity. 2 l l In Sect. 5,wepresenttheCSformulationofthe(2, 0) and ( , ) ˜ ˜ ˜ p q Maxwell supergravities and discuss their relations to where Jab, Pa and Qα are the osp (2|1) ⊗ sp (2) generators. (2, 0) and (p, q) AdS–Lorentz supergravities, respectively. The gauge fields ea, ωab and ψ are the dreibein, the spin Section 6 concludes our work with some comments and pos- connection and the gravitino, respectively. Here, the length sible developments. scale l is introduced purposely in order to have dimensionless ˜ ˜ ˜ generators TA ={Jab, Pa, Qα} such that the connection one- A μ form A = Aμ TAdx must also be dimensionless. Since the a = a μ 2 Inönü–Wigner contraction and the invariant tensor dreibein e eμdx is related to the spacetime metric gμν a b through gμν = eμeν ηab, it must have dimensions of length. a/ The standard Inönü–Wigner contraction [38,39]ofaLie Then the “true” gauge field should√ be considered as e l. ψ/ (super)algebra g consists basically in properly rescaling the In the same way, we consider l as the supersymmetry ψ = ψ μ generators by a parameter σ and applying the limit σ →∞ gauge field since the gravitino μdx has dimensions 1/2 corresponding to a contracted (super)algebra. of (length) . ( | )⊗ ( ) Despite having the proper contracted (super)algebra fol- The (anti)-commutation relations for the osp 2 1 sp 2 lowing the IW scheme, the contracted invariant tensor cannot superalgebra are given by be trivially obtained. This is particularly regrettable since the ˜ ˜ ˜ ˜ ˜ ˜ Jab, Jcd = ηbc Jad − ηac Jbd − ηbd Jac + ηad Jbc, (3) invariant tensor is an essential ingredient in the construction of a Chern–Simons action. ˜ ˜ ˜ ˜ Jab, Pc = ηbc Pa − ηac Pb, (4) In this paper, we present a generalization of the standard ˜ ˜ ˜ Inönü–Wigner contraction considering the rescaling not only Pa, Pb = Jab, (5) of the generators but also of the constants appearing in the ˜ ˜ 1 ˜ ˜ ˜ 1 ˜ invariant tensor. The method introduced here allows one to Jab, Qα =− abQ , Pa, Qα =− a Q , 2 α 2 α obtain the non-vanishing components of the invariant tensor (6) of an IW contracted (super)algebra. Thus, the construction ˜ ˜ 1 ab ˜ a ˜ of any CS action based on an IW contracted (super)algebra is Qα, Qβ =− C Jab − 2 C Pa , αβ αβ ensured. In particular, we apply the method to different (p, q) 2 AdS–Lorentz superalgebras whose IW contraction leads to (7) ( , ) diverse p q Maxwell superalgebras. where C denotes the charge conjugation matrix, α repre- Let us first apply the approach to the AdS supergravity in = 1 , sents the Dirac matrices and ab 2 [ a b]. order to derive the Poincaré supergravity. The non-vanishing components of an invariant tensor for the osp (2|1) ⊗ sp (2) superalgebra are given by ( | ) ⊗ ( ) 2.1 Poincaré and osp 2 1 sp 2 supergravity ˜ ˜ Jab Pc = μ1 abc, (8) As in the bosonic level, the IW contraction of the AdS super- ˜ ˜ Jab Jcd = μ0 (ηadηbc − ηacηbd) , (9) algebra leads to the Poincaré one. Besides, the Poincaré CS ˜ ˜ supergravity action can be obtained considering a particular Pa Pb = μ0ηab, (10) limit after an appropriate rescaling of the fields of the super ˜ ˜ AdS CS action. Nevertheless, in the presence of torsion the Qα Qβ = 2 (μ1 − μ0) Cαβ , (11) 123 Eur. Phys. J. C (2017) 77 :48 Page 3 of 18 48 where μ0 and μ1 are arbitrary constants. Then, considering Let us note that the present approach allows one to trivially the invariant tensor (8)–(11) and the connection one-form in obtain the Poincar é limit from the osp (2|1) ⊗ sp (2) CS the general expression for the D = 3 CS action, we have action. One could suggest that the same result can be obtained considering l →∞,, nevertheless the presence of the exotic μ (2+1) 0 a b 2 a c b 2 a 2 ¯ I = k ω dω + ω ω ω + e Ta − ψ Lagrangian forbids such limit. Additionally, one can notice CS 2 b a 3 c b a l2 l that the gravitino does not contribute anymore to the exotic μ 1 ab c 1 a b c ¯ form. + abc R e + abce e e + 2ψ l 3l2 μ 1 ab c −d abcω e , (12) 2l 2.2 (p, q) Poincaré and osp (2|p) ⊗ osp (2|q) supergravity where Let us now consider the (p, q) AdS supergravity theories, ab = ωab + ωa ωcb, a = a + ωa b, R d c T de be which can be viewed as a direct sum of AdS superalgebras. 1 ab 1 a It is well known that the (p, q) Poincaré superalgebra can = dψ + ωab ψ + ea ψ. 4 2l be derived as a Inönü–Wigner contraction of the (p, q) AdS It is well known that the following rescaling of the gener- superalgebra. However, as was mentioned in Refs. [9,11,40], ators: the Poincaré limit at the level of the action requires enlarge- ment of the AdS superalgebra, considering a direct sum of ˜ → ¯ , ˜ → σ 2 ¯ , ˜ → σ ¯ Jab Jab Pa Pa Qα Qα the so (p) ⊕ so (q) algebra and the (p, q) AdS superalgebra. leads to the Poincaré superalgebra in the limit σ →∞.It Here, we show that our approach allows one to obtain the seems natural to construct a Poincaré CS supergravity action Poincaré limit without introducing additional gauge fields. combining the corresponding rescaling of the generators with In particular, the non-vanishing components of the invariant ( , ) the AdS invariant tensor given by Eqs. (8)–(11). However, tensor of the p q Poincaré superalgebra are obtained from such rescaling of the generators leads to a trivial invariant the AdS ones. tensor and then to a trivial CS action. In order to obtain the The supersymmetric extension of the AdS algebra con- N = + right Poincaré limit at the level of the action, a rescaling of the tains p q gravitinos, and it is spanned by the set of ˜ ˜ ˜ ij ˜ IJ ˜ i I arbitrary constants appearing in the invariant tensor should generators Jab, Pa, T , T , Qα, Qα which satisfy [9] also be considered. Indeed, a rescaling which preserves the curvatures structure is given by ˜ ˜ ˜ ˜ ˜ ˜ Jab, Jcd = ηbc Jad − ηac Jbd − ηbd Jac + ηad Jbc, (17) μ → μ ,μ→ σ 2μ . 0 0 1 1 T˜ ij, T˜ kl = δ jkT˜ il − δikT˜ jl − δ jlT˜ ik + δilT˜ jk, (18) Then, considering the rescaling of both the generators and T˜ IJ, T˜ KL = δ JKT˜ IL − δIKT˜ JL − δ JLT˜ IK + δILT˜ JK, (19) the constants, one can see that the limit σ →∞leads to ˜ , ˜ = η ˜ − η ˜ , ˜ , ˜ = ˜ , the non-vanishing components of the invariant tensor for the Jab Pc bc Pa ac Pb Pa Pb Jab (20) Poincaré superalgebra, ˜ ij, ˜ k = δ jk ˜ i − δik ˜ j , ˜ IJ, ˜ K = δ JK ˜ I − δIK ˜ J , T Qα Qα Qα T Qα Qα Qα ¯ ¯ Jab Pc P = μ1 abc, (13) (21) ¯ ¯ = μ (η η − η η ), ˜ ˜ i 1 ˜ i ˜ ˜ i 1 ˜ i Jab Jcd P 0 ad bc ac bd (14) Jab, Qα =− ab Q , Pa, Qα =− a Q , (22) 2 α 2 α ¯ ¯ = μ , Qα Qβ P 2 1Cαβ (15) ˜ ˜ I 1 ˜ I ˜ ˜ I 1 ˜ I Jab, Qα =− ab Q , Pa, Qα = a Q , (23) 2 α 2 α ¯ ¯ ¯ where Jab, Pa and Qα are the Poincar é generators. Con- ˜ i ˜ j 1 ij ab ˜ a ˜ ˜ ij Qα, Q =− δ C Jab − 2 C Pa + Cαβ T , sidering the Poincaré gauge connection one-form and the β 2 αβ αβ non-vanishing components of the Poincaré invariant tensor, (24) the CS action reduces to ˜ I ˜ J 1 IJ ab ˜ a ˜ ˜ IJ Qα, Qβ = δ C Jab + 2 C Pa − Cαβ T , 2 αβ αβ (2+1) μ0 2 I = k ωa dωb + ωa ωc ωb (25) CS 2 b a 3 c b a μ μ 1 ab c ¯ 1 ab c + abc R e + 2ψ − d abcω e , l 2l where i, j = 1,...,p and I, J = 1,...,q. Here, the T˜ ij and (16) T˜ IJ generators correspond to internal symmetry generators where the fermionic curvature is now given by and satisfy a so (p) and so (q) algebra, respectively. One can introduce the osp (2|p) × osp (2|q) connection 1 ab = dψ + ωab ψ. A 4 one-form given by 123 48 Page 4 of 18 Eur. Phys. J. C (2017) 77 :48

1 ab ˜ 1 a ˜ 1 ij ˜ 1 IJ ˜ 1 ¯ ˜ i One can note that the (p, q) Poincaré superalgebra can be A = ω Jab + e Pa + A Tij + A TIJ + √ ψi Q 2 l 2 2 l obtained from the (p, q) AdS one considering the following 1 ¯ ˜ I rescaling of the generators: + √ ψI Q . (26) l ˜ ¯ ˜ ij 2 ¯ ij ˜ 2 ¯ ˜ ¯ Jab → Jab, T → σ T , Pa → σ Pa, Qα → σ Qα The non-vanishing components of the invariant tensor for and the limit σ →∞: the (p, q) AdS superalgebra are given by J¯ , J¯ = η J¯ − η J¯ − η J¯ + η J¯ , (35) ˜ ˜ ab cd bc ad ac bd bd ac ad bc Jab Jcd = μ0 (ηadηbc − ηacηbd) , (27) ¯ ¯ ¯ ¯ Jab, Pc = ηbc Pa − ηac Pb, (36) J˜ P˜ = μ , (28) ¯ ¯ i 1 ¯ i ab c 1 abc Jab, Qα =− abQ , (37) 2 α ˜ ˜ = μ η , Pa Pb 0 ab (29) ¯ ¯ I 1 ¯ I Jab, Qα =− abQ , (38) 2 α ˜ i ˜ j = (μ − μ ) δij, Qα Qβ 2 1 0 Cαβ (30) ¯ i , ¯ j = δij a ¯ + ¯ ij, Qα Qβ C αβ Pa Cαβ T (39) ˜ I ˜ J = (μ + μ ) δIJ, Qα Qβ 2 1 0 Cαβ (31) ¯ I , ¯ J = δIJ a ¯ − ¯ IJ, Qα Qβ C αβ Pa Cαβ T (40) ˜ ij ˜ kl il kj ik lj T T = 2 (μ0 − μ1) δ δ − δ δ , (32) Here, T ij and T IJ behave as central charges and no longer ˜ IJ ˜ KL IL KJ IK LJ T T = 2 (μ0 + μ1) δ δ − δ δ , (33) satisfy a so (p) and a so (q) algebra, respectively. Indeed, when p or q is greater than 1, the (p, q) Poincaré superal- N where μ0 and μ1 are arbitrary constants. Considering the gebra corresponds to a central extension of the -extended connection one-form A and the non-vanishing components Poincaré superalgebra. of the invariant tensor in the three-dimensional CS general At the level of the action, we have to consider a rescaling expression (1), we obtain the osp (2|p) ⊗ osp (2|q) super- of the constants appearing in the invariant tensor. Indeed, a gravity action in three dimensions: rescaling which preserves the curvature structure is given by μ → μ ,μ→ σ 2μ . ( + ) μ 2 2 0 0 1 1 I 2 1 = k 0 ωa dωb + ωa ωc ωb + ea T CS b a c b a 2 a 2 3 l Then the limit σ →∞leads to the non-vanishing compo- μ 1 ab c 1 a b c nents of the invariant tensor for the Poincaré superalgebra, + abc R e + e e e l 3l2 ¯ ¯ = μ (η η − η η ) , 2 Jab Jcd P 0 ad bc ac bd (41) + (μ − μ ) AijdAji + Aik Akj A ji 0 1 J¯ P¯ = μ , (42) 3 ab cP 1 abc 2 ¯ i ¯ j = μ δij, + (μ + μ ) AIJdAJI + AIK AKJAJI Qα Qβ 2 1Cαβ (43) 0 1 3 P ¯ I ¯ J IJ 1 Qα Qβ = 2μ1Cαβ δ , (44) +2 (μ − μ ) ψ¯ i i + Aijψ j P 1 0 l ¯ ¯ ¯ i ¯ I where Jab, Pa, Qα and Qα correspond now to the Poincaré 1 ¯ I I IJ J +2 (μ + μ ) ψ + A ψ , (34) generators. As was noticed in Ref. [9], there are no compo- 1 0 l nents of the (p, q) Poincaré invariant tensor including the T¯ ij ¯ IJ , where and T generators. Indeed, considering the (p q) Poincaré connection one-form and the invariant tensor (41)–(44)inthe Rab = dωab + ωa ωcb, T a = dea + ωa ec, general expression for the CS action we ﬁnd c c 1 1 i = ψi + ω abψi + aψi , ( + ) μ0 2 d ab ea I 2 1 = k ωa dωb + ωa ωc ωb 4 2l CS 2 b a 3 c b a 1 1 I = ψ I + ω abψ I − aψ I . μ1 d ab ea + Rabec + 2ψ¯ i i + 2ψ¯ I I , (45) 4 2l l abc Let us note that an off-shell formulation for osp (2|p) × where osp (2|q) supergravity is not ensured when p + q > 1. i i 1 ab i Interestingly, diverse off-shell formulations for (p, q) AdS = dψ + ωab ψ , 4 supergravity when p + q ≤ 3 can be found in Refs. I I 1 ab I = dψ + ωab ψ . [41,42]. 4 123 Eur. Phys. J. C (2017) 77 :48 Page 5 of 18 48

The Poincaré CS action (45) can be directly obtained from [W0, W0] ⊂ W0, [W0, W2] ⊂ W2, (47) the (p, q) AdS one considering the rescaling of the constants [W0, W1] ⊂ W1, [W1, W2] ⊂ W1, (48) 2 (μ0 → μ0,μ1 → σ μ1), the rescaling of the fields: [W1, W1] ⊂ W0 ⊕ W2, [W2, W2] ⊂ W0, (49) ω → ω , ij → σ −2 ij, → σ −2 , ab ab A A ea ea where each subspace is generated by sets of generators ψi → σ −1ψi ,ψI → σ −1ψ I , = (i) = λ Wp X p i+p X p with and the limit σ →∞. Thus, the Poincaré limit can be applied without introducing extra fields and/or change of basis. Nev- i = 0, 2, 4 ...,n − p and p = 0, 1, 2 . ertheless, as in the N = 1 case, the gravitino does not contribute to the exotic form. Besides, no so (p) or so (q) Here X p are the generators of the original superalgebra g and λ gauge fields appear in the Lagrangian. In order to obtain more i+p is an element of a semigroup S satisfying some explicit interesting supergravity actions whose gravitino appears in multiplication law of the SM family [56]. the exotic term, it is necessary to consider our approach to The IW contraction of G is obtained considering the enlarged supersymmetries. rescaling of the expanded generators On the other hand, let us note that the term proportional (i) = σ i+p (i) X p X p to μ1 reproduces the action of Ref. [9] when F(A) = 0. Naturally, the same procedure can be applied to the direct sum and applying the limit σ →∞. of (p, q) AdS superalgebra and so (p)⊕so (q) algebra, which On the other hand, according to Theorem VII.2 of would lead to the most general action for (p, q) Poincaré Ref. [47], the invariant tensor for an S-expanded superal- superalgebra [9]. gebra G can be obtained from the original ones through i ( , ) ( , ) =˜α , T A j T B k G i K jk TATB g (50) 3 Inönü–Wigner contraction of an S-expanded = λ α˜ i supergravity with T(A, j) j TA.Here i are arbitrary constants and K jk is the 2-selector for the semigroup S defined as The development of the Lie (super)algebra expansion method 1, when i = i ( j, k) , has played an important role in deriving new (super)gravity K i = jk 0, otherwise, theories [43–55]. In particular, the semigroup expansion method (S-expansion) allows to find explicitly the non- with λi( j,k) = λ j λk. vanishing components of an invariant tensor for an expanded The IW contraction of the invariant tensor is obtained, (super)algebra in terms of the original one [47]. This feature considering the rescaling of the generators is particularly useful since the invariant tensor is a crucial j T( , ) = σ T( , ), ingredient in the construction of a Chern–Simons action. A j A j Nevertheless, the CS action for a contracted superalge- the rescaling of the constant α˜i , bra cannot be naively obtained by rescaling the generators α˜ → σ i α˜ appearing in the non-vanishing components of an invariant i i tensor and considering some limit. and applying the limit σ →∞. As in the previous section, we show that by applying the The approach considered here offers a close relation rescaling to both generators and constants of the invariant between expansion and contraction. Interestingly, as we shall tensor, the CS action for a contracted superalgebra can be see, our procedure allows us to obtain the contracted super- obtained. In particular, we present the general scheme in gravity in presence of expanded Pontryagin–Chern–Simons order to derive an IW contracted supergravity from an S- form. expanded one. In the following sections, we shall present known and new First, we shall consider the contraction of the following Maxwell supergravities considering the present IW approach subspace decomposition of the Lie S-expanded superalgebra to diverse S-expanded supergravities. G = S × g:

G = W0 ⊕ W1 ⊕ W2, (46) 4 Inönü–Wigner contraction and N = 1 supergravity where W0 corresponds to a subalgebra, W1 corresponds to the Here, we present a generalization of the Inönü–Wigner con- fermionic subspace, and W2 is generated by boost generators. traction combining the S-expansion method and the rescaling Such a decomposition satisﬁes of the invariant tensor and generators. In particular, we show 123 48 Page 6 of 18 Eur. Phys. J. C (2017) 77 :48

˜ ˜ the D = 3 CS action based on a N = 1 Maxwell superalge- erators {Zab, Za}, it also has more than one spinor genera- bra. tor. The explicit (anti)commutation relations can be found in A non-standard supersymmetrization of the Maxwell Appendix A for N = 1. algebra was introduced in Refs. [57,58] which can be The construction of a Chern–Simons action for the mini- obtained as an IW contraction of the standard AdS–Lorentz mal AdS–Lorentz superalgebra requires the gauge connection superalgebra1 [59,60]. Nevertheless, the non-standard one-form A: Maxwell supersymmetric action and its physical relevance 1 ab 1 ãb ˜ 1 ab 1 a A = ω Jab + k Zab + k Zab + e Pa remains poorly explored due to its unusual anticommuta- 2 2 2 l tion relations. Indeed, the P generators of the non-standard a 1 ã ˜ 1 α 1 α + h Za + √ ψ Qα + √ ξ α. (52) Maxwell superalgebra are not expressed as bilinear expres- l l l sions of the fermionic generators Q, Since we have considered a dimensionless connection one- 1 ab form, a factor l has to be introduced for the dreibein field and Qα, Qβ =− C Zab. 2 αβ the dreibein like field hã. The same argument applies for the This feature prevents construction of a supergravity action spinor fields. based on this peculiar supersymmetry. Despite this particu- Another crucial ingredient necessary to write down a CS larity, there is an alternative in order to construct a N = 1 supergravity action is the invariant tensor. Following Theo- supergravity action based on the Maxwell supersymmetries. rem VII.2 of Ref. [47], the non-vanishing components of an A particular Maxwell superalgebra, also called the min- invariant tensor for the super minimal AdS–Lorentz are given imal supersymmetrization of the Maxwell algebra [61–64] by differs from the non-standard Maxwell one since it possesses ˜ ˜ Jab JcdS =˜α0 Jab Jcd = α0 (ηadηbc − ηacηbd) , (53) an additional fermionic generator. Interestingly, a minimal Maxwell superalgebra can be derived as an Inönü–Wigner ˜ ˜ ˜ Jab Zcd =˜α2 Jab Jcd = α2 (ηadηbc − ηacηbd) , (54) contraction of a new minimal AdS–Lorentz superalgebra S introduced in Ref. [65]. Z˜ Z =˜α J˜ J˜ = α (η η − η η ) , (55) ab cd S 2 ab cd 2 ad bc ac bd Before studying the explicit IW contraction at the level of the invariant tensor, we first present the explicit construc- ˜ ˜ JabZcdS =˜α4 Jab Jcd = α4 (ηadηbc − ηacηbd) , (56) tion of a CS supergravity invariant under the minimal AdS– Z˜ Z˜ =˜α J˜ J˜ = α (η η − η η ) , (57) Lorentz superalgebra. To this purpose, we will apply the ab cd S 4 ab cd 4 ad bc ac bd S-expansion procedure analogously to the four-dimensional Z Z =˜α J˜ J˜ = α (η η − η η ) , (58) case [65]. ab cd S 4 ab cd 4 ad bc ac bd J P = Z P = Z˜ Z˜ =˜α J˜ P˜ = β , 4.1 Minimal AdS–Lorentz exotic supergravity ab c S ab c S ab c S 2 ab c 2 abc (59) Following the procedure of Ref. [65], a minimal AdS– ˜ ˜ ˜ ˜ ˜ JabZc = ZabZc = Zab Pc =˜α4 Jab Pc = β4 abc, Lorentz superalgebra can be derived as an S-expansion of S S S the osp (2|1) ⊗ sp (2) superalgebra. Indeed, considering (60) (4) S = {λ ,λ ,λ ,λ ,λ } as the abelian semigroup whose P P = Z˜ Z˜ =˜α P˜ P˜ = α η , (61) M 0 1 2 3 4 a b S a b S 4 a b 4 ab elements satisfy P Z˜ =˜α P˜ P˜ = α η , (62) a b S 2 a b 2 ab λα+β , if α + β ≤ 4, ˜ ˜ λ λ = Qα Qβ = αβ =˜α Qα Qβ = 2 (β − α ) Cαβ , (63) α β λ , α + β> , (51) S S 2 2 2 α+β−4 if 4 ˜ ˜ Qαβ S =˜α4 Qα Qβ = 2 (β4 − α4) Cαβ , (64) (4) and after extracting a resonant subalgebra of SM × { ˜ , ˜ , ˜ } ( | ) ⊗ ( ) (osp (2|1) ⊗ sp (2)), the minimal AdS–Lorentz superalge- where Jab Pa Qα generate the osp 2 1 sp 2 super- bra is obtained [65]. This algebra corresponds to a superalgebra (see Eqs. (8)–(11)) and where we have defined symmetric extension of the so (2, 2) ⊕ so (2, 1) algebra = ˜ ˜ α ≡˜α μ ,α≡˜α μ ,α≡˜α μ , {Jab, Pa, Zab} and is generated by {Jab, Pa, Zab, Za, Zab, 0 0 0 2 2 0 4 4 0 Qα,α}. This superalgebra, as in the Maxwell case, is quite β2 ≡˜α2μ1,β4 ≡˜α4μ1. different from the standard AdS–Lorentz superalgebra dis- cussed in Refs. [19,59]. In fact, besides extra bosonic gen- Here α˜0, α˜2, α˜4 are arbitrary constants as μ0 and μ1.The CS supergravity action can be written considering the con- 1 Also known as Poincaré semi-simple extended superalgebra. nection one-form (52) and the non-vanishing component of 123 Eur. Phys. J. C (2017) 77 :48 Page 7 of 18 48 the invariant (53)–(64) in the general three-dimensional CS 4.2 The Maxwell limit expression One is tempted to consider an appropriate rescaling of the ( + ) 2 2 1 = + 3 . fields at the level of the action (65) and apply some limit in ICS k Ad A A 3 order to derive the Maxwell supergravity action. However, Thus, we have modulo boundary terms although a minimal Maxwell superalgebra can be obtained as an IW contraction of the minimal AdS–Lorentz superalgebra, ( + ) α 2 I 2 1 = k 0 ωa dωb + ωa ωc ωb the IW contraction of the action (65) would reproduce a trivial CS 2 b a 3 c b a CS action. To reproduce a non-trivial CS supergravity action β 1 N = + 2 Rabec + eaebec based on the 1 Maxwell symmetries, it is necessary to l abc 2 3l extend the rescaling of the generators to the α and β constants 1 +K abec K˜ abh˜c + hãh˜be + 2ψ ¯ + 2ξ¯ appearing in the non-vanishing components of the invariant 2 l tensor. A rescaling which preserves the curvature structure 1 + α Ra k˜b + K a k˜b + K˜ a kb + ea H is given by 2 b a b a b a 2 a l 4 4 2 α4 → σ α4,β4 → σ β4,α2 → σ α2, 1 ã 2 ¯ 2 ¯ + h Ka − ψ − ξ 2 2 l l β2 → σ β2,α0 → α0. l β 1 + 4 Rabh˜c + hãh˜bh˜c + K˜ abec + K abh˜c Then, considering the rescaling of the constants and the gen- l abc 2 3l erators 1 + eaebh˜c + 2ξ ¯ + 2ψ¯ l2 ˜ 2 ˜ 4 2 Zab → σ Zab, Zab → σ Zab, Pa → σ Pa, Jab → Jab, 1 + α Ra kb + K a kb + K˜ a k˜b + ea K ˜ → σ 4 ˜ , → σ , → σ 3, 4 b a b a b a 2 a Za Za Qα Qα l 1 ã 2 ¯ 2 ¯ + h Ha − ξ − ψ , (65) and applying the limit σ →∞, we recover the N = 1 l2 l l Maxwell non-vanishing components of the invariant tensor, where Jab JcdM = α0 (ηadηbc − ηacηbd) , (66) ab = ab + a d + ã ˜d , ˜ ab = ãb + a ˜d + d ã , K Dk k d k b k d k b K Dk k d k b kb k d ˜ Jab Zcd = α2 (ηadηbc − ηacηbd) , (67) a = ã + a ˜b + ã b, a = a + a b + ã ˜b, M H Dh k bh k be K T k be k bh 1 1 1 Z˜ Z˜ = α (η η − η η ) , (68) = dψi + ω abψi + k abψi + k˜ abξi ab cd M 4 ad bc ac bd 4 ab 4 ab 4 ab Jab ZcdM = α4 (ηadηbc − ηacηbd) , (69) 1 a i 1 ˜ a i + ea ξ + ha ψ , 2l 2l 1 1 1 = β , i = dξi + ω abξi + k abξi + k˜ abψi Jab Pc M 2 abc (70) 4 ab 4 ab 4 ab ˜ = ˜ = β , 1 a i 1 ˜ a i Zab Pc Jab Zc 4 abc (71) + ea ψ + ha ξ . M M 2l 2l P P M = α η , (72) a b 4 ab TheCSaction(65) is locally gauge invariant under the min- Qα Qβ = 2 (β − α ) Cαβ , (73) M 2 2 imal AdS–Lorentz superalgebra and is split into five inde- Qαβ M = 2 (β4 − α4) Cαβ . (74) pendent pieces proportional to α0,α2,α4,β2 and β4..In particular, the term proportional to α corresponds to the 0 Here, the generators now satisfy the (anti)commutation rela- exotic form, while the α and α terms contain exotic like 2 4 tions of a minimal Maxwell superalgebra (see Appendix B Lagrangians plus fermionic terms. for p = 1, q = 0). Then using the Maxwell connection Let us note that the CS action (65) reproduces the three- one-form dimensional generalized cosmological constant term intro- ãb = ã = 1 ab 1 ãb ˜ 1 ab 1 a 1 ã ˜ duced in Refs. [56,66] when k h 0. A gen- A = ω Jab + k Zab + k Zab + e Pa + h Za eralized supersymmetric cosmological term has also been 2 2 2 l l 1 α 1 α introduced in a four-dimensional MacDowell–Mansouri like +√ ψ Qα + √ ξ α action constructed out of the curvature two-form, based on l l an AdS–Lorentz superalgebra [65,67]. Besides, the bosonic and the non-vanishing components of the invariant tensor, part corresponds to the AdS–Lorentz CS action presented in we derive the three-dimensional CS supergravity action for Refs. [68,69]. the N = 1 Maxwell superalgebra [22]: 123 48 Page 8 of 18 Eur. Phys. J. C (2017) 77 :48 α (2+1) = 0 ωa ωb + 2 ωa ωc ωb IM−CS k bd a c b a algebra. In order to apply the S-expansion procedure, let us M 2 3 β first consider a decomposition of the original superalgebra + 2 Rabec + 2ψ ¯ l abc osp (2|2) ⊗ sp (2) = V0 ⊕ V1 ⊕ V2, 2 + α Ra k˜b − ψ ¯ 2 b a l where V0 corresponds to a subalgebra generated by the β 4 ab ˜c ãb c ¯ ¯ ˜ + abc R h + Dωk e + 2ξ + 2ψ Lorentz generators Jab and by the internal symmetry genera- l ˜ ij tor T (with i = 1, 2), V1 corresponds to the fermionic sub- a b 1 a ã ˜b 2 ¯ 2 ¯ + α R k + e T + Dωk k − ξ − ψ , ˜ ( | )⊗ 4 b a l2 a b a l l space and V2 is generated by Pa. In particular, the osp 2 2 ( ) (75) sp 2 generators satisfy the following (anti)commutation relations: where ˜ , ˜ = η ˜ − η ˜ − η ˜ + η ˜ , Rab = dωab + ωa ωcb, T a = dea + ωa ec, Jab Jcd bc Jad ac Jbd bd Jac ad Jbc (76) c c 1 ab T˜ ij, T˜ kl = δ jkT˜ il − δikT˜ jl − δ jlT˜ ik + δilT˜ jk, (77) = dψ + ωab ψ, 4 J˜ , P˜ = η P˜ − η P˜ , (78) 1 ab 1 ˜ ab 1 a ab c bc a ac b = dξ + ωab ξ + kab ψ + ea ψ. 4 4 2l ˜ ˜ ˜ Pa, Pb = Jab, (79) Thus, we reproduce the CS action presented in Ref. [22] ˜ ij, ˜ k = δ jk ˜ i − δik ˜ j , using a different approach. Unlike the non-standard Maxwell, T Qα Qα Qα (80) the minimal Maxwell superalgebra allows one to properly ˜ ˜ i 1 ˜ i ˜ ˜ i 1 ˜ i Jab, Qα =− ab Q , Pa, Qα =− a Q , (81) write a three-dimensional CS supergravity action invariant 2 α 2 α under local Maxwell supersymmetry transformations (up to ˜ i ˜ j 1 ij ab ˜ a ˜ ˜ ij Qα, Q =− δ C Jab − 2 C Pa + Cαβ T . boundary terms). However, as was shown at the algebraic β 2 αβ αβ level, this requires the introduction of an additional Majo- (82) rana spinor field ξ. The presence of a second spinorial generator was already introduced in Refs. [70,71] in the context The subspace structure satisfies of D = 11 supergravity and superstring theory, respectively. [V , V ] ⊂ V , [V , V ] ⊂ V , [V , V ] ⊂ V , (83) Subsequently, the introduction of a second spinorial gener- 0 0 0 0 1 1 0 2 2 , ⊂ ⊕ , , ⊂ , , ⊂ ator in the Maxwell symmetries was proposed in Ref. [61]. [V1 V1] V0 V2 [V1 V2] V1 [V2 V2] V0. More recently, a family of Maxwell superalgebras was pre- (84) sented, which generalize the superalgebra introduced by D ( ) Let us now consider S 4 = {λ ,λ ,λ ,λ ,λ } as the rele- ’Auria, Fré and Green and contains the minimal Maxwell M 0 1 2 3 4 vant abelian semigroup whose elements satisfy the multipli- one [72–74]. cation law Let us note that the bosonic term reproduces the CS action presented in Refs. [69,75]. λα+β , if α + β ≤ 4, λαλβ = (85) λα+β−4, if α + β>4. 5 Inönü–Wigner contraction and N -extended (4) = ∪ ∪ Supergravity Let SM S0 S1 S2 be a subset decomposition with S = {λ ,λ ,λ } , Wenow present the explicit derivation of the three-dimensional 0 0 2 4 = {λ ,λ } , (p, q) Maxwell supergravity action using our approach. In S1 1 3 particular, we first show the explicit construction of the (2, 0) S2 = {λ2,λ4} , AdS–Lorentz supergravity using the semigroup expansion where S , S , and S satisfy the resonance condition [com- method. The (2, 0) Maxwell supergravity is then derived 0 1 2 pare with Eqs. (83)–(84)] from the (2, 0) AdS–Lorentz superalgebra applying the IW contraction to both generators and constants appearing in the S0 · S0 ⊂ S0, S1 · S1 ⊂ S0 ∩ S2, (86) invariant tensor. S0 · S1 ⊂ S1, S1 · S2 ⊂ S1, (87) · ⊂ , · ⊂ . 5.1 N = 2 AdS–Lorentz supergravity S0 S2 S2 S2 S2 S0 (88) Then according to Ref. [47], A non-trivial N = 2 AdS–Lorentz superalgebra can be obtained as an S-expansion of the osp (2|2) ⊗ sp (2) super- GR = W0 ⊕ W1 ⊕ W2, 123 Eur. Phys. J. C (2017) 77 :48 Page 9 of 18 48

(4) is a resonant subalgebra of SM × osp (2|2) ⊗ sp (2) where addition to the invariant tensor, the gauge connection one- form: ˜ ˜ ij W0 = (S0 × V0) = {λ0,λ2,λ4} × Jab, T , = 1 ωab + 1 ãb ˜ + 1 ab + 1 a + 1 ã ˜ ˜ i A Jab k Zab k Zab e Pa h Za W1 = (S1 × V1) = {λ1,λ3} × Qα , 2 2 2 l l 1 ij 1 ˜ ij ˜ 1 ij 1 ¯ i 1 ¯ i ˜ + A Tij + B Yij + B Yij + √ ψi Q + √ ξi . W2 = (S2 × V2) = {λ2,λ4} × Pa . 2 2 2 l l (94) The expanded superalgebra corresponds to a (2, 0) AdS– Lorentz superalgebra and is generated by the set of gener- The associated curvature two-form F = dA+ AA is given ˜ ˜ ij ˜ ij ij i i ators {Jab, Pa, Zab, Za, Zab, T , Y , Y , Qα,α}, which by are related to the original ones through = 1 ab + 1 ˜ ab ˜ + 1 ab + 1 a + 1 ˜ a ˜ ˜ ˜ i ˜ i F R Jab F Zab F Zab F Pa H Za Jab = λ0 Jab, Pa = λ2 Pa, Qα = λ1 Qα, 2 2 2 l l ˜ ˜ ˜ ˜ i ˜ i 1 ij 1 ˜ ij ˜ 1 ij 1 ¯ i 1 ¯ i Zab = λ2 Jab, Za = λ4 Pa,α = λ3 Qα, + F Tij + G Yij + G Yij + √ i Q + √ i , 2 2 2 l l = λ ˜ , ij = λ ˜ ij, ˜ ij = λ ˜ ij, Zab 4 Jab T 0T Y 2T (95) ij ˜ ij Y = λ4T . where The explicit (anti)commutation relations can be derived using the multiplication law of the semigroup and the original ab = ωab + ωa ωcb, superalgebra (the N -extended AdS–Lorentz superalgebra R d c can be found in Appendix A). a = a + ωa b + a b + ã ˜b − 1 ψ¯ i aψi − 1 ξ¯i aξi , F de be k be k bh In order to construct the explicit supergravity action let 2 2 ˜ a = ã + ωa ˜b + ã b + a ˜b − ψ¯ i aξi , us first derive the invariant tensor for the (2, 0) AdS–Lorentz H dh bh k be k bh superalgebra. According to Theorem VII.2 of Ref. [47], it ˜ ab = ãb + ωa ˜cb − ωb ˜ca + a ˜cb − b ˜ca + 2 a ˜b F dk ck ck k ck k ck e h is possible to show that the non-vanishing components of l2 the invariant tensor for the N = 2 AdS–Lorentz are, besides + 1 ψ¯ i abψi + 1 ξ¯i abξi , those given by Eqs. (53)–(62), 2l 2l ab = ab + ωa cb − ωb ca + ã ˜cb + a cb F dk ck ck k ck k ck i j i j ˜ i ˜ j ij 1 1 1 Qα Qβ = αβ =˜α2 Qα Qβ = 2 (β2 − α2) Cαβ δ , + eaeb + hãh˜b + ξ¯i abψi , l2 l2 l (89) i j ˜ i ˜ j ij ij ij ik kj Qαβ =˜α4 Qα Qβ = 2 (β4 − α4) Cαβ δ , (90) F = dA + A A , ˜ ij ˜ ij ik ˜ kj ˜ ik kj ij kl ˜ ij ˜ kl il kj ik lj G = d B + A B + B A T T =˜α0 T T = 2 (α0 − β0) δ δ − δ δ , + B˜ ikBkj + BikB˜ kj + ψ¯ i ψ j + ξ¯i ξ j , (91) Gij = dBij + AikBkj + Bik Akj + B˜ ikB˜ kj + BikBkj + 2ψ¯ i ξ j , T ijY˜ kl = Y˜ ijY kl =˜α T˜ ijT˜ kl 2 1 1 1 i = dψi + ω abψi + k abψi + k˜ abξi il kj ik lj 4 ab 4 ab 4 ab = 2 (α2 − β2) δ δ − δ δ , (92) 1 1 + e aξi + h˜ aψi ij kl ˜ ij ˜ kl ij kl ˜ ij ˜ kl a a T Y = Y Y = Y Y =˜α4 T T 2l 2l + Aijψ j + Bijψ j + B˜ ijξ j , il kj ik lj = 2 (α4 − β4) δ δ − δ δ , (93) 1 1 1 i = dξi + ω abξi + k abξi + k˜ abψi 4 ab 4 ab 4 ab { ˜ , ˜ , ˜ ij, ˜ i } ( | ) ⊗ ( ) 1 a i 1 ˜ a i where Jab Pa T Qα generate the osp 2 2 sp 2 + ea ψ + ha ξ superalgebra and where we have defined 2l 2l + Aijξ j + Bijξ j + B˜ ijψ j .

α0 ≡˜α0μ0,α2 ≡˜α2μ0,α4 ≡˜α4μ0, Then considering the connection one-form (94) and the non- β ≡˜α μ ,β ≡˜α μ ,β ≡˜α μ . 0 0 1 2 2 1 4 4 1 vanishing components of the invariant tensor ((53)–(62) and (89)–(93)) in the general three-dimensional CS expression, Here α˜0, α˜2, α˜4 are arbitrary constants as well as μ0 and we ﬁnd the (2, 0) AdS–Lorentz CS action supergravity up to μ1. To construct the CS supergravity action we require, in a surface term: 123 48 Page 10 of 18 Eur. Phys. J. C (2017) 77 :48

( + ) α 2 I 2 1 = k 0 ωa dωb + ωa ωc ωb As in the previous case, the CS supergravity action for the CS 2 b a 3 c b a (2, 0) Maxwell supergroup can be derived combining the IW 2 + (α − β ) AijdAji + Aik Akj A ji contraction with the S-expanded invariant tensor. Indeed, it 0 0 3 α β is necessary to extend the rescaling of the generators to the 2 ab c 1 a b c + abc R e + e e e β l 3l2 and constants appearing in the non-vanishing components ( , ) 1 of the invariant tensor of the 2 0 AdS–Lorentz superalge- + K abec + K˜ abh˜c + hã h˜be l2 bra. A rescaling which preserves the curvature structure is ˜ ij ji jk ki given by + (α2 − β2) B dA + A A 4 2 α4 → σ α4,α2 → σ α2,α0 → α0, + B˜ ij dBji + A jk Bki + B jk Aki + B˜ jk B˜ ki + B jk Bki 4 2 β4 → σ β4,β2 → σ β2,β0 → β0. + Aij + Bij d B˜ ji + A jk B˜ ki + B˜ jk Aki 2 2 Then, considering the rescaling of both constants and gener- + B˜ jk Bki + B jk B˜ ki − ψ¯ i i − ξ¯i i l l ators, and applying the limit σ →∞, we obtain the (2, 0) 1 1 Maxwell non-vanishing components of the invariant tensor, + α Ra k˜b + K a k˜b + K˜ a kb + ea H + hã K 2 b a b a b a l2 a l2 a β 4 ab ˜c 1 ã ˜b ˜c ˜ ab c ab ˜c 1 a b ˜c = α (η η − η η ) , + abc R h + h h h + K e + K h + e e h Jab Jcd M 0 ad bc ac bd (97) l 3l2 l2 ˜ = α (η η − η η ) , + (α − β ) ˜ ij ˜ ji + jk ˜ ki Jab Zcd 2 ad bc ac bd (98) 4 4 B d B A B M ˜ ˜ ˜ jk ki ˜ jk ki jk ˜ ki ij ji jk ki Z Z = J Z = α (η η − η η ) , (99) + B A + B B + B B + B dA + A A ab cd M ab cd 4 ad bc ac bd + ij + ij ji + jk ki + jk ki J P M = β , (100) A B dB A B B A ab c 2 abc 2 2 ˜ = ˜ = β , + B˜ jk B˜ ki + B jk Bki − ξ¯i i − ψ¯ i i Zab Pc Jab Zc 4 abc (101) l l M M 1 1 Pa PbM = α4ηab, (102) + α Ra kb + K a kb + K˜ a k˜b + ea K + hã H , (96) 4 b a b a b a l2 a l2 a i i where and are the fermionic components of the cur- i j ij Qα Qβ = 2 (β2 − α2) Cαβ δ , (103) vature two-form and M i j ij Q = 2 (β − α ) Cαβ δ , (104) α β M 4 4 ab = ab + a d + ã ˜d , ˜ ab = ãb + a ˜d + d ã , K Dk k d k b k d k b K Dk k d k b kb k d T ijT kl = 2 (α − β ) δilδkj − δikδlj , (105) a = ã + a ˜b + ã b, a = a + a b + ã ˜b, M 0 0 H Dh k bh k be K T k be k bh ij ˜ kl il kj ik lj T Y = 2 (α2 − β2) δ δ − δ δ , (106) As in the N = 1 case, the (2, 0) Maxwell supergravity cannot M ij kl ˜ ij ˜ kl il kj ik lj be trivially obtained considering the rescaling of the fields in T Y = Y Y = 2 (α4 − β4) δ δ −δ δ , (96) and applying some limit. M M (107) N = 5.2 2 Maxwell supergravity where α0,α2,α4,β2 and β4 are arbitrary constants and the generators now satisfy the (2, 0) Maxwell (anti)commutation As the osp (2|2)⊗sp (2) superalgebra has its (2, 0) Poincaré relations. In order to write down a CS action we require the limit, the (2, 0) AdS–Lorentz superalgebra possesses its gauge connection one-form given by proper IW contracted superalgebra. Indeed, after rescaling the generators 1 ab 1 ãb ˜ 1 ab 1 a 1 ã ˜ A = ω J + k Z + k Z + e Pa + h Za 2 ab 2 ab 2 ab l l ˜ → σ 2 ˜ , → σ 4 , → σ 2 , → , 1 ij 1 ˜ ij ˜ 1 ij 1 ¯ i 1 ¯ i Zab Zab Zab Zab Pa Pa Jab Jab + A Tij + B Yij + B Yij + √ ψi Q + √ ξi . 2 2 2 l l ˜ → σ 4 ˜ , i → σ i ,i → σ 3i , ij → σ 4 ij, Za Za Qα Qα Y Y (108) Y˜ ij → σ 2Y˜ ij, T ij → T ij, Considering the non-vanishing components of the invariant N = and applying the limit σ →∞, we obtain the N = 2 tensor, the CS action for the 2 Maxwell superalgebra reduces to Maxwell superalgebra whose (anti)commutation relations can be found in Refs. [72,73] (see Appendix B for p = 2, ( + ) α 2 I 2 1 = k 0 ωa dωb + ωa ωc ωb q = 0). M−CS 2 b a 3 c b a 123 Eur. Phys. J. C (2017) 77 :48 Page 11 of 18 48 i i i i ij ji 2 ik kj ji where the and parameters are related to the Q and + (α0 − β0) A dA + A A A 3 generators, respectively. β We remark that the generalized cosmological constant + 2 Rabec + α Ra k˜b l abc 2 b a term appearing in the (2, 0) AdS–Lorentz supergravity model is no longer present after the IW contraction. This is analo- + (α − β ) B˜ ij dAji + A jk Aki 2 2 gous to the Poincaré limit from the AdS one. However, unlike 2 the Poincaré supergravity theory, the internal symmetry fields + Aij d B˜ ji + A jkB˜ ki + B˜ jk Aki − ψ¯ i i l appear explicitly in the exotic Lagrangian. Additionally, the β spinorial fields contribute to the exotic like part. 4 ab ˜c ãb c + abc R h + Dωk e l 5.3 (p, q) AdS–Lorentz supergravity and the Maxwell limit ˜ ij ˜ ji jk ˜ ki ˜ jk ki + (α4 − β4) B d B + A B + B A In this section we present the three-dimensional N = p + q ij ji jk ki + B dA + A A extended AdS–Lorentz Supergravity and its Maxwell limit applying the IW contraction not only at the generators level + Aij dBji + A jkBki + B jk Aki + B˜ jkB˜ ki but also to the constants appearing in the invariant tensor. To this purpose we expand the three-dimensional (p, q) exotic − 2ξ¯i i − 2ψ¯ i i supergravity theory [11], in order to obtain the local AdS– l l Lorentz supersymmetric extension. In particular, we gener- a b ã ˜b 1 a + α4 R k + Dωk k + e Ta , (109) alize the Poincaré limit showed in Sect. 2 to the Maxwell b a b a l2 limit. where The (p, q) AdS–Lorentz superalgebra can be obtained as i i 1 ab i ij j an S-expansion of the osp (2|p) ⊗ osp (2|q) superalgebra. = dψ + ωab ψ + A ψ , (4) 4 Indeed, considering SM = {λ0,λ1,λ2,λ3,λ4} as the rele- i i 1 ab i 1 ˜ ab i vant semigroup whose elements satisfy = dξ + ωab ξ + kab ψ 4 4 λα+β , if α + β ≤ 4 1 a i ij j ˜ ij j λαλβ = + ea ψ + A ξ + B ψ . λ , α + β> 2l α+β−4 if 4 This CS supergravity action is invariant up to boundary terms and considering the resonant condition (see N = 2 case), N = ˜ under the local gauge transformations of the 2 Maxwell we obtain a new superlagebra generated by {Jab, Pa, Zab, ˜ ij ˜ ij ij IJ ˜ IJ IJ i i I J supergroup. In particular, under the supersymmetric transfor- Za, Zab, T , Y , Y , T , Y , Y , Qα,α, Qα,α } mations, the fields transform as whose generators satisfy the (p, q) AdS–Lorentz superalgebra. In particular, besides satisfying the (anti)commutation ab a i a i δω = 0,δe =¯ ψ , (110) relations appearing in Appendix A,theI -index generators 1 satisfy δkãb =− ¯i abψi , (111) l 1 1 IJ, KL = δ JK IL − δIK JL − δ JL IK + δIL JK, δkab =− ¯ i abψi − ¯i abξ i , (112) T T T T T T l l (120) T IJ, Y˜ KL = δ JKY˜ IL − δIKY˜ JL − δ JLY˜ IK + δILY˜ JK, δhã =¯i aψi +¯ i aξ i , (113) (121) ij δ A = 0, (114) T IJ, Y KL = δ JKY IL − δIKY JL − δ JLY IK + δILY JK, 2 δB˜ ij =− ψ¯ [i j], (115) (122) l ˜ IJ, ˜ KL = δ JK IL − δIK JL − δ JL IK + δIL JK, 2 Y Y Y Y Y Y δBij =− ψ¯ [i j], (116) l (123) i i 1 ab i ij j Y˜ IJ, Y KL = δ JKY˜ IL − δIKY˜ JL − δ JLY˜ IK + δILY˜ JK, δψ = d + ω ab + A , (117) 4 (124) i i 1 ab i 1 a i 1 ãb i δξ = d + ω ab + e a + k ab IJ KL JK IL IK JL JL IK IL JK 4 2l 4 Y , Y = δ Y − δ Y − δ Y + δ Y , (118) (125) + Aij j + B˜ ij j , (119) 123 48 Page 12 of 18 Eur. Phys. J. C (2017) 77 :48 ij kl =˜α ˜ ij ˜ kl = (α − β ) δilδkj − δikδlj , I 1 I I 1 I T T 0 T T 2 0 0 Jab, Qα =− abQ , Pa, Qα = a , 2 α 2 α (141) (126) IJ KL ˜ IJ ˜ KL IL KJ IK LJ T T =˜α0 T T = 2 (α0 + β0) δ δ − δ δ , ˜ I 1 I ˜ I 1 I Z , Qα =− , Za, Qα = a Q , ab 2 ab α 2 α (142) (127) ij ˜ kl ˜ ij kl ˜ ij ˜ kl T Y = Y Y =˜α2 T T I 1 I I 1 I Z , Qα =− Q , P ,α = Q , ab ab α a a α il kj ik lj 2 2 = 2 (α2 − β2) δ δ − δ δ , (143) (128) IJ ˜ KL = ˜ IJ KL =˜α ˜ IJ ˜ KL I 1 I ˜ I 1 I T Y Y Y 2 T T J ,α =− , Z ,α = , ab ab α a a α 2 2 IL KJ IK LJ (129) = 2 (α2 + β2) δ δ − δ δ , (144) ˜ I 1 I I 1 I ij kl = ˜ ij ˜ kl = ij kl =˜α ˜ ij ˜ kl Zab,α =− abQ , Zab,α =− ab , T Y Y Y Y Y 4 T T 2 α 2 α (130) = α − β δilδkj − δikδlj , 2 ( 4 4) (145) IJ, K = (δ JK I − δIK J ), ˜ IJ, K T Qα Qα Qα Y Qα IJ KL ˜ IJ ˜ KL IJ KL ˜ IJ ˜ KL T Y = Y Y = Y Y =˜α4 T T = (δ JK I − δIK J ), α α (131) IL KJ IK LJ = 2 (α4 + β4) δ δ − δ δ , (146) Y IJ, Q K = (δ JKQ I − δIKQ J ), T IJ,K ˜ ˜ ˜ ij ˜ IJ ˜ i ˜ I α α α α where Jab, Pa, T , T , Qα, Qα correspond to the orig- = (δ JK I − δIK J ), (132) inal osp (2|p) ⊗ osp (2|q) generators and where we have α α ˜ IJ K JK I IK J IJ K defined Y ,α = (δ Qα − δ Qα ), Y ,α α ≡˜α μ ,α≡˜α μ ,α≡˜α μ , = (δ JK I − δIK J ), (133) 0 0 0 2 2 0 4 4 0 α α β ≡˜α μ ,β≡˜α μ ,β≡˜α μ . I J 1 IJ ab ˜ a ˜ IJ 0 0 1 2 2 1 4 4 1 Qα, Qβ = δ C Zab + 2 C Pa − Cαβ Y , 2 αβ αβ (134) α˜ , α˜ , α˜ μ μ Here 0 2 4 are arbitrary constants as 0 and 1. Let us I J 1 IJ ab a ˜ IJ now consider the gauge connection one-form of this extended Qα,β = δ C Zab + 2 C Za − Cαβ Y , 2 αβ αβ superalgebra: (135) 1 ab 1 ãb ˜ 1 ab 1 a 1 ã ˜ A = ω Jab + k Zab + k Zab + e Pa + h Za I J 1 IJ ab ˜ a ˜ IJ 2 2 2 l l α,β = δ C Zab + 2 C Pa − Cαβ Y . αβ αβ 2 1 ij 1 IJ 1 ˜ ij ˜ + A Tij + A TIJ + B Yij (136) 2 2 2 1 ˜ IJ ˜ 1 ij 1 IJ + B YIJ + B Yij + B YIJ Here, the T ij, Y˜ ij, Y ij, T IJ, Y˜ IJ and Y IJ generators corre- 2 2 2 spond to internal symmetry generators with i = 1,...,p 1 ¯ i 1 ¯ I 1 ¯ i 1 ¯ I + √ ψi Q + √ ψI Q + √ ξi + √ ξI . and I = 1,...,q. l l l l Using Theorem VII.2 of Ref. [47], it is possible to show (147) that the non-vanishing components of the invariant tensor for The curvature two-form F = dA+ AA is given by the N = p+qAdS–Lorentz are, besides those given by Eqs. (53)–(62), 1 ab 1 ˜ ab ˜ 1 ab 1 a 1 ˜ a ˜ F = R Jab + F Zab + F Zab + F Pa + H Za i j i j 2 2 2 l l Qα Q = α β β 1 ij 1 IJ 1 ˜ ij ˜ + F Tij + F TIJ + G Yij ˜ i ˜ j ij 2 2 2 =˜α2 Qα Qβ = 2 (β2 − α2) Cαβ δ , (137) 1 ˜ IJ ˜ 1 ij 1 IJ + G YIJ + G Yij + G YIJ I J I J 2 2 2 Qα Qβ = αβ 1 ¯ i 1 ¯ I 1 ¯ i 1 ¯ I + √ i Q + √ I Q + √ i + √ I , (148) ˜ I ˜ J IJ l l l l =˜α2 Qα Qβ = 2 (β2 + α2) Cαβ δ , (138) i j ˜ i ˜ j ij where Qαβ =˜α4 Qα Qβ = 2 (β4 − α4) Cαβ δ , (139) a = a + a b + ã ˜b − 1ψ¯ i aψi I J ˜ I ˜ J IJ F Dωe k be k bh Qαβ =˜α4 Qα Qβ = 2 (β4 + α4) Cαβ δ , (140) 2 − 1ξ¯i aξ i − 1ψ¯ I aψ I − 1ξ¯ I aξ I , 2 2 2 123 Eur. Phys. J. C (2017) 77 :48 Page 13 of 18 48 ˜ a = ã + ã b + a ˜b − ψ¯ i aξ i − ψ¯ I aξ I , 1 1 H Dωh k be k bh + 2 (β + α ) ψ¯ I I + ξ¯ I I 2 2 l l ˜ ab = ãb + a ˜cb − b ˜ca F Dωk k ck k ck a ˜b a ˜b ˜ a b + α2 R k + K k + K k 2 a ˜b 1 ¯ i ab i 1 ¯ I ab I b a b a b a + e h + ψ ψ − ψ ψ l2 2l 2l 1 a 1 ã + e Ha + h Ka 1 ¯i ab i 1 ¯ I ab I l2 l2 + ξ ξ − ξ ξ , 2l 2l β 1 + 4 Rabh˜c + hãh˜bh˜c ab ab ã ˜cb a cb 1 a b 1 ã ˜b abc 2 F = Dωk + k k + k k + e e + h h l 3l c c 2 2 l l 1 + K˜ abec + K abh˜c + eaebh˜c 1 ¯i ab i 1 ¯ I ab I l2 + ξ ψ − ξ ψ , l l ˜ ij ji ˜ + (α4 − β4) B F B + ij ji ( ) + ij + ij ji ( ) F IJ = dAIJ + AIK AKJ, B F A A B F B ˜ IJ = ˜ IJ + IK ˜ KJ + ˜ IK KJ + ˜ IK KJ ˜ IJ JI ˜ G d B A B B A B B + (α4 + β4) B F B + B IKB˜ KJ − ψ¯ I ψ J − ξ¯ I ξ J , + B IJF JI (A) + AIJ + B IJ F JI (B) IJ = IJ + IK KJ + IK KJ + ˜ IK ˜ KJ G dB A B B A B B IK KJ ¯ I J 1 1 + B B − 2ψ ξ , + 2 (β − α ) ξ¯i i + ψ¯ i i 4 4 l l I I 1 ab I = Dωψ + k ψ 1 1 ab + 2 (β + α ) ξ¯ I I + ψ¯ I I 4 4 4 l l + 1 ˜ abξ I − 1 aξ I − 1 ˜ aψ I kab ea ha a b a b ˜ a ˜b 1 a 1 ã l l + α4 R k + K k + K k + e Ka + h Ha , 4 2 2 b a b a b a l2 l2 + IJψ J + IJψ J + ˜ IJξ J , A B B (149) I I 1 ab I 1 ˜ ab I = Dωξ + kab ξ + kab ψ 4 4 where i , I , i and I are the fermionic components of 1 a I 1 ˜ a I the curvature two-form and − ea ψ − ha ξ 2l 2l IJ J IJ J ˜ IJ J ij ( ) = ij + ik kj, IJ ( ) = IJ + IK KJ, + A ξ + B ξ + B ψ , F A dA A A F A dA A A Fij B˜ = d B˜ ij + Aik B˜ kj + B˜ ik Akj + B˜ ik Bkj + Bik B˜ kj, Rab, Fij, G˜ ij, Gij, i i N = and and are defined as in the F IJ B˜ = d B˜ IJ + AIKB˜ KJ + B˜ IK AKJ + B˜ IKB KJ + B IKB˜ KJ, 2 case (see Eq. (95)). ij ij ik kj ik kj ˜ ik ˜ kj ik kj Considering the connection one-form (147) and the non- F (B) = dB + A B + B A + B B + B B , IJ IJ IK KJ IK KJ ˜ IK ˜ KJ IK KJ vanishing components of the invariant tensor ((53)–(62) and F (B) = dB + A B + B A + B B + B B , ab = ab + a d + ã ˜d , ˜ ab = ãb + a ˜d + d ã , (137)–(146)) in the general three-dimensional CS expres- K Dk k d k b k d k b K Dk k d k b kb k d N = + a = ã + a ˜b + ã b, a = a + a b + ã ˜b. sion, we find the CS action of p qAdS–Lorentz H Dh k bh k be K T k be k bh supergravity up to a surface term: ( , ) ( , ) ( + ) α 2 As the 2 0 AdS–Lorentz superalgebra has its 2 0 I 2 1 = k 0 ωa dωb + ωa ωc ωb CS 2 b a 3 c b a Maxwell limit, the IW contraction of the (p, q) AdS–Lorentz + (α − β ) ij ji ( ) + (α + β ) IJ JI ( ) superalgebra leads to the (p, q) Maxwell superalgebra. 0 0 A F A 0 0 A F A β Indeed, by rescaling the AdS–Lorentz generators as 2 ab c 1 a b c + abc R e + e e e l 3l2 ˜ 2 ˜ 4 2 1 Zab → σ Zab, Zab → σ Zab, Pa → σ Pa, Jab → Jab, + K abec + K˜ abh˜c + hãh˜be l2 ˜ 4 ˜ i i I I i 3 i Za → σ Za, Qα → σ Qα, Qα → σ Qα, → σ , ˜ ij ji + (α2 − β2) B F (A) I → σ 3 I , ij → σ 4 ij, IJ → σ 4 IJ, Y Y Y Y + B˜ ijF ji (B) + Aij + Bij F ji B˜ Y˜ ij → σ 2Y˜ ij, Y˜ IJ → σ 2Y˜ IJ, T ij → T ij, T IJ → T IJ, ˜ IJ JI + (α2 + β2) B F (A) and applying the limit σ →∞, we obtain the N = p + + B˜ IJF JI (B) + AIJ + B IJ F JI B˜ q Maxwell superalgebra whose explicit (anti)commutation relations can be found in Appendix B. Extending the rescal- 1 ¯ i i 1 ¯i i + 2 (β2 − α2) ψ + ξ α β l l ing of the generators to the and constants appearing in 123 48 Page 14 of 18 Eur. Phys. J. C (2017) 77 :48

1 1 the invariant tensor of the (p, q) AdS–Lorentz superalgebra + 2 (β − α ) ψ¯ i i + 2 (β + α ) ψ¯ I I 2 2 l 2 2 l as β a ˜b 4 ab ˜c ãb c + α2 R k + abc R h + Dωk e 4 2 b a l α4 → σ α4,α2 → σ α2,α0 → α0, + (α − β ) ˜ ij ji ˜ + ij ji ( ) + ij ji ( ) β → σ 4β ,β→ σ 2α ,β→ β , 4 4 B F B B F A A F B 4 4 2 1 0 0 (150) ˜ IJ JI ˜ IJ JI IJ JI + (α4 + β4) B F B + B F (A) + A F (B) and considering the limit σ →∞, we obtain the non- ( , ) 1 1 vanishing components of the invariant tensor for the p q + 2 (β − α ) ξ¯i i + ψ¯ i i 4 4 l l Maxwell superalgebra: 1 1 + 2 (β + α ) ξ¯ I I + ψ¯ I I 4 4 l l Jab JcdM = α0 (ηadηbc − ηacηbd) , (151) a b ã ˜b 1 a + α R k + Dωk k + e T , ˜ 4 b a b a 2 a (164) Jab Zcd = α2 (ηadηbc − ηacηbd) , (152) l M ˜ ˜ where i , I , i and I are the fermionic components of Zab Zcd = Jab Zcd = α4 (ηadηbc − ηacηbd) , (153) M the (p, q) Maxwell curvature two-form, given by Jab PcM = β2 abc, Pa PbM = α4ηab, (154) i i 1 ab i = dψ + ωab ψ Z˜ P = J Z˜ = β , (155) 4 ab c M ab c M 4 abc +Aijψ j , I = ψ I + 1ω abψ I + IJψ J , i j ij I J d ab A Q Q = 2 (β − α ) Cαβ δ , Q Q 4 α β M 2 2 α β M 1 1 1 IJ i = ξ i + ω abξ i + ˜ abψi + aψi = 2 (β + α ) Cαβ δ , (156) d ab kab ea 2 2 4 4 2l i j ij I J ij j ˜ ij j Q = 2 (β − α ) Cαβ δ , Q +A ξ + B ψ , α β M 4 4 α β M IJ I I 1 ab I 1 ˜ ab I 1 a I = 2 (β + α ) Cαβ δ , (157) = dξ + ωab ξ + kab ψ − ea ψ 4 4 4 4 2l +AIJξ J + B˜ IJψ J , T ijT kl = 2 (α − β ) δilδkj − δikδlj , (158) M 0 0 and IJ KL IL KJ IK LJ T T = 2 (α0 + β0) δ δ − δ δ , (159) M ij ( ) = ij + ik kj, IJ ( ) = IJ + IK KJ, F A dA A A F A dA A A T ijY˜ kl = 2 (α − β ) δilδkj − δikδlj , 2 2 (160) ij ˜ ˜ ij ik ˜ kj ˜ ik kj M F B = d B + A B + B A , IJ ˜ KL IL KJ IK LJ T Y = 2 (α2 + β2) δ δ − δ δ , (161) M F IJ B˜ = d B˜ IJ + AIKB˜ KJ + B˜ IK AKJ, ij kl ˜ ij ˜ kl il kj ik lj T Y = Y Y = 2 (α4 − β4) δ δ − δ δ , M M Fij (B) = dBij + Aik Bkj + Bik Akj + B˜ ikB˜ kj, (162) F IJ (B) = dBIJ + AIKB KJ + B IK AKJ + B˜ IKB˜ KJ. T IJY KL = Y˜ IJY˜ KL = 2 (α + β ) δILδKJ − δIKδLJ . M M 4 4 One can note that the (p, q) Maxwell supergravity action can (163) be obtained directly from the AdS–Lorentz one considering Let us note that the generators appearing in the invariant ten- the rescaling of the constant (given by Eq. (150)) and the sor ...M satisfy the (p, q) Maxwell superalgebra. Addi- gauge fields tionally, the rescaling of the constants considered here pre- ˜ −2 ˜ −4 ωab → ωab, kab → σ kab, kab → σ kab, serves the curvature structure. e → σ −2e , h˜ → σ −4h˜ , Then the three-dimensional (p, q) Maxwell supergravity a a a a i −1 i I −1 I CS action can be derived considering the (p, q) Maxwell ψ → σ ψ ,ψ → σ ψ , connection one-form (analogous to Eq. (147)) and the non- ξ i → σ −3ξ i ,ξI → σ −3ξ I , Aij → Aij, vanishing components of the invariant tensor (151)–(163): AIJ → AIJ, B˜ ij → σ −2 B˜ ij, B˜ IJ → σ −2 B˜ IJ, ij −4 ij IJ −4 IJ B → σ B , B → σ B , α β (2+1) 0 a b 2 a c b 2 ab c I = k ω dω + ω ω ω + abc R e M−CS 2 b a 3 c b a l and then the limit σ →∞. Thus the procedure presented + (α − β ) AijF ji (A) + (α + β ) AIJF JI (A) here ensures the explicit Maxwell limit considering the 0 0 0 0 ˜ ij ji ij ji ˜ rescaling not only of the fields but also of the constants + (α2 − β2) B F (A) + A F B appearing in the invariant tensor. Naturally, the Poincaré limit ˜ IJ JI IJ JI ˜ + (α2 + β2) B F (A) + A F B approach presented in Ref. [9] could be applied here, but it 123 Eur. Phys. J. C (2017) 77 :48 Page 15 of 18 48 would require the introduction of additional gauge fields. and reproduction in any medium, provided you give appropriate credit This would lead not only to new Maxwell supergravity the- to the original author(s) and the source, provide a link to the Creative ories but also to new AdS–Lorentz ones. Commons license, and indicate if changes were made. Funded by SCOAP3.

6 Conclusion A Appendix

We have presented a generalization of the standard Inönü– The N -extended AdS–Lorentz superalgebra is generated by Wigner contraction combining the rescaling of the generators the set of generators and the constants appearing in the invariant tensor of a Lie ˜ ˜ ij ˜ ij ij i i superalgebra. The procedure considered here allows one not Jab, Pa, Zab, Za, Zab, T , Y , Y , Qα,α only to obtain the invariant tensor of a contracted superalge- = ,...,N ; α = ,..., bra but also to construct the contracted supergravity action. (with i 1 1 4) whose generators satisfy In particular, we have shown that the Poincaré limit can the (anti)-commutation relations: be applied to a (p, q) AdS supergravity in the presence of the , = η − η − η + η , exotic Lagrangian without introducing an so (p) ⊕ so (q) [Jab Jcd] bc Jad ac Jbd bd Jac ad Jbc (165) extension. Naturally, the internal symmetries generators of [Zab, Zcd] = ηbc Zad − ηac Zbd − ηbd Zac + ηad Zbc, the AdS superalgebra behave as central charges after the con- (166) traction and do not contribute explicitly in the construction [Jab, Zcd] = ηbc Zad − ηac Zbd − ηbd Zac + ηad Zbc, of the Poincaré action. Additionally, no gravitino fields con- (167) tribute to the exotic form. It is important to point out that ˜ ˜ ˜ ˜ ˜ the standard IW contraction does not allow one to apply the Jab, Zcd = ηbc Zad − ηac Zbd − ηbd Zac + ηad Zbc, Poincaré limit to the (p, q) AdS supergravity in the presence (168) of the Pontryagin–Chern–Simons form. ˜ , ˜ = η − η − η + η , We have also applied our generalized IW scheme to Zab Zcd bc Zad ac Zbd bd Zac ad Zbc expanded superalgebras. We have constructed a new class of (169) D = 2 + 1 (p, q) Maxwell supergravity theories as a partic- Z˜ , Z = η Z˜ − η Z˜ − η Z˜ + η Z˜ , ular limit of an AdS–Lorentz supergravity model. The results ab cd bc ad ac bd bd ac ad bc presented here show an explicit relation between contraction (170) and expansion. Besides, we have shown that the fermionic and the internal symmetries fields contribute to the exotic CS form. Nevertheless, we have clarified that a supergravity [Jab, Pc] = ηbc Pa − ηac Pb, [Zab, Pc] = ηbc Pa − ηac Pb, with Maxwell supersymmetry requires the introduction of an (171) additional spinorial field ξ. ˜ ˜ ˜ ˜ ˜ ˜ The procedure considered here could be useful in higher- Zab, Pc = ηbc Za − ηac Zb, Jab, Zc = ηbc Za − ηac Zb, dimensional supergravity models in order to derive non- (172) trivial Chern–Simons supergravity theories (work in progress). ˜ , ˜ = η − η , , ˜ = η ˜ − η ˜ , It would be interesting to explore the expansion and contrac- Zab Zc bc Pa ac Pb Zab Zc bc Za ac Zb tion method in the context of infinite-dimensional algebras (173) and hypergravity. , = , ˜ , = ˜ , ˜ , ˜ = , It would also be interesting to extend our study of the [Pa Pb] Zab Za Pb Zab Za Zb Zab Maxwell supergravities to black hole solutions. It is of par- (174) ticular interest to study black hole solutions with torsion for their thermodynamical properties [76–78]. In particular, one could explore the possibility of finding Maxwell “exotic” T ij, T kl = δ jkT il − δikT jl − δ jlT ik + δilT jk, (175) BTZ type black holes. T ij, Y kl = δ jkY il − δikY jl − δ jlY ik + δilY jk, (176) Acknowledgements This work was supported by the Newton-Picarte CONICYT Grant No. DPI20140053. (P.K.C. and E.K.R.) and by the T ij, Y˜ kl = δ jkY˜ il − δikY˜ jl − δ jlY˜ ik + δilY˜ jk, (177) Conicyt-PAI Grant No. 79150061 (P.K.C.). Y˜ ij, Y˜ kl = δ jkY il − δikY jl − δ jlY ik + δilY jk, (178) Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ˜ ij kl jk ˜ il ik ˜ jl jl ˜ ik il ˜ jk ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, Y , Y = δ Y − δ Y − δ Y + δ Y , (179) 123 48 Page 16 of 18 Eur. Phys. J. C (2017) 77 :48 Y ij, Y kl = δ jkY il − δikY jl − δ jlY ik + δilY jk, (180) [Jab, Jcd] = ηbc Jad − ηac Jbd − ηbd Jac + ηad Jbc, (192) [J , Z ] = η Z − η Z − η Z + η Z , (193) ab cd bc ad ac bd bd ac ad bc i 1 i i 1 i Jab, Qα =− abQ , Pa, Qα =− a , , ˜ = η ˜ − η ˜ − η ˜ + η ˜ , α α Jab Zcd bc Zad ac Zbd bd Zac ad Zbc (194) 2 2 (181) ˜ ˜ Zab, Zcd = ηbc Zad − ηac Zbd − ηbd Zac + ηad Zbc, (195) ˜ , i =−1 i , ˜ , i =−1 i , Zab Qα ab Za Qα a Q [J , P ] = η P − η P , [P , P ] = Z , (196) 2 α 2 α ab c bc a ac b a b ab (182) ˜ , = η ˜ − η ˜ , , ˜ = η ˜ − η ˜ , Zab Pc bc Za ac Zb Jab Zc bc Za ac Zb i 1 i i 1 i Zab, Qα =− abQ , Pa,α =− a Q , (197) 2 α 2 α (183) i 1 i ˜ i 1 i ij kl jk il ik jl jl ik il jk Jab,α =− ab , Za,α =− a , T , T = δ T − δ T − δ T + δ T , (198) 2 α 2 α (184) ij, kl = δ jk il − δik jl − δ jl ik + δil jk, T Y Y Y Y Y (199) ˜ i 1 i i 1 i Zab,α =− abQ , Zab,α =− ab , ij ˜ kl jk ˜ il ik ˜ jl jl ˜ ik il ˜ jk 2 α 2 α T , Y = δ Y − δ Y − δ Y + δ Y , (200) (185) Y˜ ij, Y˜ kl = δ jkY il − δikY jl − δ jlY ik + δilY jk, ij k jk i ik j ˜ ij k (201) T , Qα = (δ Qα − δ Qα), Y , Qα jk i ik j = (δ α − δ α), (186) IJ KL JK IL IK JL JL IK IL JK ij k jk i ik j ij k T , T = δ T − δ T − δ T + δ T , Y , Qα = (δ Qα − δ Qα), T ,α (202) jk i ik j = (δ α − δ α), (187) T IJ, Y˜ KL = δ JKY˜ IL − δIKY˜ JL − δ JLY˜ IK + δILY˜ JK, ˜ ij k jk i ik j ij k Y ,α = (δ Qα − δ Qα), Y ,α (203) jk i ik j IJ KL JK IL IK JL JL IK IL JK = (δ α − δ α), (188) T , Y = δ Y − δ Y − δ Y + δ Y , (204) i , j =−1 δij ab ˜ − a + ˜ ij, Qα Qβ C Zab 2 C αβ Pa Cαβ Y ˜ IJ ˜ KL JK IL IK JL JL IK IL JK 2 αβ Y , Y = δ Y − δ Y − δ Y + δ Y , (189) (205) i j 1 ij ab a ˜ ij Qα, =− δ C Zab − 2 C Za + Cαβ Y , β 2 αβ αβ (190) i 1 i i 1 i Jab, Qα =− ab Q , Pa, Qα =− a , α α i j 1 ij ab ˜ a ˜ ij 2 2 α, =− δ C Zab − 2 C Pa + Cαβ Y . β αβ αβ (206) 2 (191) ˜ i 1 i i 1 i Zab, Qα =− ab , Jab,α =− ab , 2 α 2 α (207) Let us note that the N = 1 case (whose internal symme- ij ij ˜ ij ij k jk i ik j ˜ ij k jk i ik j tries generators T , Y and Y are absent) reproduces the T , Qα = (δ Qα − δ Qα), Y , Qα = (δ α − δ α), minimal AdS–Lorentz superalgebra introduced in Ref. [65]. (208) ij k jk i ik j T ,α = (δ α − δ α), (209) I 1 I I 1 I B Appendix Jab, Qα =− ab Q , Pa, Qα = a , (210) α α 2 2 ˜ I 1 I I 1 I The (p, q) Maxwell superalgebra is generated by the follow- Zab, Qα =− ab , Jab,α =− ab , 2 α 2 α ing set: (211) IJ K JK I IK J ˜ IJ K ˜ ˜ ij IJ ˜ ij ˜ IJ ij IJ T , Qα = (δ Qα − δ Qα ), Y , Qα Jab, Pa, Zab, Za, Zab, T , T , Y , Y , Y , Y , = (δ JK I − δIK J ), i I i I α α (212) Qα, Qα,α,α IJ K JK I IK J T ,α = (δ α − δ α ), (213) with i = 1,...,p and I = 1,...,q.The(p, q) Maxwell generators satisfy the following (anti)commutation relations: 123 Eur. Phys. J. C (2017) 77 :48 Page 17 of 18 48 i , j =−1 δij ab ˜ − a + ˜ ij, Qα Qβ C Zab 2 C αβ Pa Cαβ Y 17. D. Butter, S.M. Kuzenko, J. Novak, G. Tartaglino-Mazzucchelli, 2 αβ Conformal supegravity in three dimensions: off-shell actions. JHEP (214) 1310, 073 (2013). arXiv:1306.1205 [hep-th] 18. M. Nishimura, Y. 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