On a Five-Dimensional Chern–Simons Ads Supergravity Without Gravitino

Physics Letters B 777 (2018) 275–280

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Physics Letters B

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On a ﬁve-dimensional Chern–Simons AdS supergravity without gravitino ∗ Y.M.P. Gomes , J.A. Helayel-Neto

Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr Xavier Sigaud 150, Urca, CEP 22290-180, Rio de Janeiro, Brazil a r t i c l e i n f o a b s t r a c t

Article history: Based on recent discussions on the so-called unconventional supersymmetry, we propose a 5D Chern– Received 29 November 2017 Simons AdS-N -SUGRA formulation without gravitino fields and show that a residual local SUSY is Accepted 14 December 2017 preserved. We explore the properties of CS theories to find a solution to the field equations in a 5D Available online 18 December 2017 manifold. With a Randall–Sundrum-type ansatz, we show that this particular dimensional reduction is Editor: M. Cveticˇ compatible with SUSY, and some classes of 4D solutions are then analyzed. © 2017 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 invariant traces, and this is the reason why supersymmetry breaks down. The action in four dimensions must be seen as an effec- An alternative method to build up a theory with SUSY is by tive description, due to, for instance, a quartic fermionic coupling implementing a gauge theory for a super-algebra that includes an that shows up and prevents the model from being renormalizable internal gauge group, G, along with a local SO(1, D − 1) algebra [1,3,4]. that has to be set up to connect these two symmetries through The paradigm that the procedure still keeps from standard SUSY fermionic supercharges [1–4]. In these references, the field mul- is that fermion and bosons can be combined into a unique non- tiplet is composed by a (non-)Abelian field, A, a spin-1/2 Dirac trivial representation of a supergroup [16–19]. The differences ap- fermion, ψ, the spin connection, ωab, the d-bein, ea, and additional pear already in the scenario where the SUSY works. In this pro- gauge fields which complete the degrees of freedom to accomplish posal, SUSY is an extension of the symmetries of the tangent space. [ 1 ⊕ 1 ] the supersymmetrization. These additional fields are dependent on Since Dirac fermions are in the ( 2 , 0) (0, 2 ) -representation of the structure of the group and the space–time which we intend Lorentz group, SUSY is implemented as an extension of the tangent to work in. The representations of the fields are not all the same. space symmetries. This approach allows us to implement SUSY in The Dirac spinor transforms under the fundamental representation, any manifold, by looking for the symmetries of the tangent bundle. while the gauge connection belongs to the adjoint representation Another difference is found in the field representations [1]. of G. In this framework, the metric is completely invariant under A Chern–Simons AdS5 supergravity is a gauge model based on the symmetries G, SO(1, D − 1) and supersymmetry. a SUSY extension of the AdS5 gravity. Based on the no-gravitini Due to the properties above, the model displays important dif- approach [1] and on the structure of SO(4, 2) group, we work with ferences in comparison with standard SUSYs. For example, there a field that is a 1-form gauge connection [5]: is no superpartners with degenerate masses, nor an equal number 1 ˆ â ˆ ab ˆ k ¯ r ˆ ¯ r ˆ ˆ of degrees of freedom of bosons and fermions. There is not even A = e Ja + ω Jab + A Tk + (ψ Q r + Q ψr ) + bK, (1) a spin-3/2 fermion, i.e., a gravitino, in the spectrum of the model 2 ˆ a [1,3,4]. where the hat stands for 5-dimensional forms; = eˆ γa, with It is remarkable that, in odd dimensions, the Chern–Simons a, 0, ..., 4; k = 1, ..., N 2 − 1 and r = 1, ..., N . This 1-form has val- (CS) form is quasi-invariant under the whole supergroup. On ues in the SU(2, 2|N ) super-algebra, whose bosonic sector is given other hand, for even dimensions, the symmetry breaks into G × by SU(2, 2) ⊗ SU(N ) ⊗ U (1), where SU(2, 2) SO(4, 2) [5]. − = ˆ SO(1, D 1). For example, for D 4, the super-group can have no The infinitesimal gauge transformation is given by δ Aˆ = d + [ ˆ } = a + 1 ab + k + ¯ r + ¯ r + K A, , with Ja 2 Jab Tk χ Q r Q χr b . In components, we have: * Corresponding author. E-mail addresses: [email protected] (Y.M.P. Gomes), [email protected] 1 â ˆ a ˆ ab ab ˆ ¯ r ˆ a ¯ r a ˆ (J.A. Helayel-Neto). δe = d + ω b + eb + (ψ γ χr + χ γ ψr ), (2a) 2 https://doi.org/10.1016/j.physletb.2017.12.037 0370-2693/© 2017 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. 276 Y.M.P. Gomes, J.A. Helayel-Neto / Physics Letters B 777 (2018) 275–280

1 ab ˆ ab ac b bc a ¯ r ˆ ab r ab ˆ 3. 5D topological action δωˆ = d + ωˆ + ωˆ + (ψ γ χr + χ¯ γ ψr ), c c 4 (2b) The topological action can be written as a Chern–Simons action in 5 dimensions [5]: ˆ k = ˆ k + k ˆ m − ¯ r k s ˆ + ¯ r ˆ k s δ A d f lm Al i(ψ (τ )r χs χ (τ )r ψs), (2c) 1 1 S5D = AFF − FAAA + AAAAA , (6) ˆ ˆ 2 10 δ(ψr ) = ∇χr (2d) where ... stands for the supertrace. The only non-vanishing su- ˆ = ˆ + ¯ r ˆ + ¯ r ˆ δb db i(ψ χr χ ψr ), (2e) pertraces are:

ˆ ˆ 1 1 ˆ 1 1 where ∇ χ = dχ +[i( − )b+ eˆ γ a + ωˆ γ ab]χ + Aˆ (τ k) sχ . 1 i j k ijk r r 4 N 2 a 4 ab r k r s Ja Jbc Jde =− εabcde , T T T =−f ˆ 1 The field-strength is given by Fˆ = dAˆ + [Aˆ , Aˆ }. In components, we 2 2 1 1 ˆ ˆ a 1 ˆ ab ˆ k ˆ¯ r ¯ r ˆ i j ij have F = F J + F J + F T + Q + Q + F K, where: K Jab Jcd =− ηab,cd , KT T =− δ a 2 ab k r r 4 N 1 1 1 ˆ a ˆ a a b ¯ r ˆ a ˆ ˆ a ¯ r ˆ a ˆ K =− KKK = + F = deˆ + ωˆ eˆ + ψ γ ψ = D ˆ eˆ + ψ γ ψ , (3a) Ja Jb ηab , b r ω r 4 N 2 42 i i ˆ ab ˆ ab a b 1 ¯ r ˆ ab ˆ ¯ α s =− α s ¯ α i s =− α i s F = R + eˆ eˆ + ψ γ ψr , (3b) Q r Jab Q β (ab)β δr , Q r T Q β δβ (τ )r 2 4 2 i 1 1 1 k ˆ k k l m r k s ¯ α s =− α s ¯ αK s =− + α s ˆ = ˆ + ˆ ˆ + ¯ ˆ ˆ Q r Ja Q β (a)β δr , Q r Q β ( )δβ δr F dA f lm A A ψ (τ )r ψs , (3c) 2 2 4 N ¯ Using these definitions, we can find the components of the ac- ˆ = ∇ˆ s ˆ ˆ r = ∇ˆ s ¯ s ˆ r ( )r (ψs), ( )r (ψ ), (3d) 5D tion S = SG + S SU(N ) + SU (1) + S f , where: ˆ ˆ ˆ ¯ r ˆ ˆ F = db + iψ ψr , (3e) 1 1 1 ˆ ab ˆ cd ê ˆ ab ˆc ˆd ê â ˆb ˆc ˆd ê SG =− abcde F F e − F e e e + e e e e e , (7) 2 2 10 ˆ ab = ˆ ˆ ab + ˆ ac ˆ b where R dω ω ωc . In the sequel, we shall specifically an- ˆ ˆ ˆ 1 ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆ alyze the SUSY transformations and see how the gravitino sector is S SU(N ) =− Tr AFF − AAAF + AAAAA + suppressed from the model. 2 10 i + ˆ · ¯ r ˆ ∇ˆ ∇ˆ s ˆ A ψ ( τ )r ψs , (8) 2. SUSY transformation 2 1 1 ˆ ˆ 2 ˆ 1 ˆ ab ˆ 1 ˆ a ˆ SU (1) = ( + )b(F ) + b − F Fab − F Fa + In the work of Ref. [3], to ensure that no gravitini appear in N 2 42 4 4 1 1 1 1 the spectrum in a 3D action, the authors show that the dreibein − ˆ i ˆ + + ¯ r ˆ ∇ˆ 2 s ˆ F Fi ( )ψ ( )r ψs , (9) remains invariant under gauge and supersymmetry transforma- N 2 4 N tions, but rotates as a vector under the Lorentz subgroup. To do = ¯ r ˆ Rˆ s ∇ˆ ˆ + this, we must look for the SUSY transformations. In the fermionic S f i ψ r ( ψ)s c.c.. (10) ˆ part, we have δ(ˆ ψ ) = ∇ χ , where χ is the local SUSY param- r r Here, eter. Any vector with spinor index can be split into irreducible 1 1 1 1 representations: 1 ⊗ 1/2 = 3/2 ⊕ 1/2of the Lorentz group. So, for ∇ˆ 2 s = ˆ ab + â ˆb + ˆ a + − ˆ ˆ s + ( )r (R e e )γab T γa i( )db δr ξ α = (P + P )aξ α = φα + α, where (P )b = δb − 1 γ γ b = 4 2 4 N a 3/2 1/2 b b a a 3/2 a a 5 a b − b α α + ˆ ˆ k + kk k ˆ k ˆ k k s δa (P1/2)a are the projectors, φa are the 3/2-component and a dA f A A (τ )r , (11) b is the 1/2-component. Therefore, we have (P 3/2) γbψ = 0, by def- ˆ k ˆ s ˆ 2 s k s a (∇τ ∇) = (∇ ) (τ ) , (12) inition. SUSY transformation yields: r r s and ˆ = â + â = ∇ˆ δ(ψr ) δe γaψr e γaδψr χr . (4) ˆ s 1 ˆ ab 1 ˆ a i 1 1 ˆ s ˆ i s R = − F γab − F γa + ( + )F δ + F (τi) . (13) r 4 2 2 4 N r r Applying the P3/2-projector to the equation above, we find that Rˆ s ⊃ ˆ ˆ s It should be noticed that, since r δr , the fermionic ⊃ ν∇ˆ = part of S f generates a Dirac-like action for the fermions (S f (P3/2)μ νχr 0 , (5) 5 ¯ r d xψ D/ψr ). Notice that the bosonic part is almost the same in comparison with the usual 5D AdS-SUGRA action [5–9]. The im- which implies that ∇ˆ χ = eâγ ρ , for an arbitrary spinor ρ. This r a r portant difference lies in the fermionic sector. condition guarantees that the symmetry transformations close off- shell without the need of introducing auxiliary fields [3]. Applying 3.1. Gauge transformation and the field equations P1/2-projector to the equation (4) we obtain that, under SUSY, δψ = ρ and δeâ = 0. The spinor ρ obeys the Killing equa- r r r The CS 5D action transforms under a gauge transformation as tion; the number of Killing spinors defines the number of un- δS5D = FFδA . We can see that due to this identity, one can broken supersymmetries, i.e., supersymmetries respected by the readily find the field equations in terms of component fields, and background [3]. For instance, if ρr = 0, we have χr = constant (co- variantly constant), and we obtain a global SUSY. For a general they are given by: solution, a Hamiltonian analysis must be carried out to extract the 1 1 i ¯ â →− ˆ bc ˆ de − ˆ ˆ − ˆ r ˆ = exact solution for the SUSY parameter [4]. δe εabcde F F Fb F γa r 0 , (14) 2 4 2 Y.M.P. Gomes, J.A. Helayel-Neto / Physics Letters B 777 (2018) 275–280 277

1 1 i ¯ ˆ ab ˆ cd ˆ e ˆ ˆ ˆ r ˆ Since the 5 D gamma-matrices can be split as γ a = (γ I , γ ), we δω →− εabcde F F − Fab F − γab r = 0 , (15) 5 2 4 2 then have: ˆ 1 ab 1 a 1 i 1 1 2 δb →− Fˆ Fˆ − Fˆ Fˆ − Fˆ Fˆ + ( − )(Fˆ ) + ab a i 2 2 ˆ = I + 4 + I + 4 4 4 N N 4 γ eI γ5e γ (eI )χ γ5eχ dχ . (26) 1 1 1 ¯ − − ˆ r ˆ = ( ) r 0 , (16) As we can see, the equation F = 0satisfies the field equations 2 4 N for the topological action. Therefore, we can analyze this solution ˆ i ikj ˆ j ˆ k 1 ˆ ˆ i ˆ¯ r i s ˆ δ A → f F F + Fi F + (τ ) s = 0 , (17) in terms of the reduced components (see Appendix). N 2 r s ˆ NOTE – Chamseddine gauge-fixing: In the dimensional reduc- R s = 0 . (18) r tion of the 5D Chamseddine action to a 4D action for gravity [10], F = 4 = I = I = IJ = It can be checked that 0is a solution to the field equation. it can be shown that we can fix e eχ b ωχ 0, due to Let us analyze this solution. In components, we have: the condition ∂χ f = 0, for any field f . However, in our case, this is not possible anymore, due to the supersymmetric character of ˆ a = → ˆ a = ˆ â =−¯ r ˆ a ˆ F 0 T Dωˆ e ψ γ ψr (19a) the transformation. If we wish to preserve SUSY, we should not ˆ ab ˆ ab a b 1 ¯ r ˆ ab ˆ use the Chamseddine gauge-fixing. The central question is: which F = 0 → R + eˆ eˆ =− ψ γ ψr (19b) 2 is the gauge-fixing that maintains SUSY and switches off the spuri- ˆ k = → ˆ ˆ k + k ˆ l ˆ m =−¯ r ˆ k s ˆ ous degrees of freedom? Using the Killing equation, we check that F 0 dA f lm A A ψ (τ )r ψs (19c) â = 4 = I = δSUSY e 0. Therefore, we can fix, in principle, e eχ 0, but ˆ ˆ r Fˆ = 0 → db =−iψ¯ ˆ ˆ ψ . (19d) IJ r we still cannot fix bI and ωχ . It is useful to pay attention to some special structures. For Since the fünfbein does not transform under SUSY, the analy- â ˆ ˆ a ˆ sis of the residual transformation of e gives us some clues about instance, if we take the 3-form S = eâ T , we have that S = ˆ ˆ the ansatz we may assume for the fünfbein. One of the possible −ψ¯ r ˆ ˆ ˆ ψ = i db, where represents the Hodge dual in the r ansätze is shown in the next Section. 5D Manifold. However, by using the Cartan identities, we have ˆ d(eˆ Tˆ a) = Tˆ a Tˆ − eˆ eˆ Rˆ ab. By virtue of this identity, we find the a a a b 5. Randall–Sundrum dimensional reduction equation that follows:

ˆ ˆ ˆ ˆ a ˆ ˆ ˆ ˆ ab A Randall–Sundrum-like ansatz is proposed with the assump- id db = T Ta − eaeb R . (20) tion that the geometry of 5D space–time has the following struc- ˆ On the other hand, by defining the co-derivative d† = d, ture [12–14]: ˆ ˆ ˆ ˆ we may introduce a Laplacian operator, 2ˆ = d†d + dd†, and, us- − ˆ† ˆ ˆ ˆ a ˆ ˆ ˆ ˆ ab 2 = 2σ (χ) μ ν + 2 2 ing a gauge condition d b = 0, we have 2b = (T Ta − eaeb R ). ds5D e gμν(x)dx dx G(χ) dχ . (27) ˆ Therefore, the b-field has a dynamics which respects the equation We can translate (27) in terms of the following fünfbein: above. In this sense, we can interpret the topological sector as the ˆ − source of the U (1) field, b. Going further, if the fünfbein is in- hI (x)e σ (χ) 0 eâ = μ , (28) vertible, we can define the following operation on some n-form, α 0 G(χ) μ μ ˆ ˆ a = ˆ ˆ a μ2 μn−1 = ˆ ˆ (Ea V ) Ea Vμμ μ ...μ − dx ...dx V , where Ea is the in- 1 2 n 1 where σ is called conformal function. A special choice and appli- verse of the fünfbein, i.e., (Eˆ eˆb) = δb. This operation can nor- a a cation of this ansatz in standard AdS SUGRA can be viewed in the mally be extended to forms with any number of Lorentz indices. ˆ ˆ ˆ a ˆ ab paper by Garavuso and Toppan [15]. The 4D metric can be written So, we can define the 1-form T = (E T ) = 10 (eˆ eˆ R ). There- J a a b as g (x) = η hI h . This ansatz fixes e4 = eI = 0. This can, in fore, the equation (20) can be rewritten as μν IJ μ ν χ principle, be a problem. But, due to the no-gravitini condition, this ˆ ˆ a ˆ 1 ˆ choice is viable. Going further, the inverse of the fünfbein is given 2ˆ b = (T Ta) − T . (21) 10 by:

We here omit the fermionic structure of the torsion for simplic- −1 μ σ (χ) ˆ (h (x)) e 0 ity. This shows us that the torsion is the only source for the b-field Eˆ α = I , (29) a 0 1 and it is a propagating excitation in five-dimensional space–time. G(χ) − where we assume that hI has an inverse, i.e., hI (h 1) Jμ = IJ. 4. Dimensional reduction μ μ η This opens up the opportunity to define the inverse of the viel- I = −σ I μ = I μ We have that the index a = 0, ..., 4 = I, 4; where the index I bein (in 4D). We have that e e hμdx eμdx , which implies I = μ = σ −1 μ refers to the SO(1, 3) Minkowski group. So, the fields can be split eμeν I gμν . Therefore, we may define E I e (h ) I so that I μ = I into two pieces [10,11]. eμ E J δ J . We can define a similar operation “ ” in 4D, and we μ I = I = I ab IJ ˆ I a I 4 can rewrite the identity above as E J eμ (E J e ) δ J . We shall ˆ ={ˆ ,λb } , eˆ ={eˆ , eˆ } . (22) ω ω use this operation from now on, and the 4D character is implicit Besides that, we are also interested in considering the action in the forms without “ ˆ ”. Now, we can look at the field equations in a 4-dimensional version; so, we must split the coordinates as F = 0. This ansatz gives us that the torsion part yields the follow- xα = (xμ, χ) and the 1-forms can be written as follows below: ing equations: ˆ IJ = IJ + IJ ; ˆ I = I + I I + I J = I =−¯ r I ω ω ωχ dχ b b bχ dχ (23) de ω J e T ψ γ ψr , (30a) ˆ I = I + I ; ˆ4 = 4 + 4 − I + IJ =− ¯ r I e e eχ dχ e e eχ dχ (24) σ e ωχ e J Gψ γ γ5ψr , (30b) ˆ = + ; ˆ k = k + k I =−¯ r I b b bχ dχ A A Aχ dχ . (25) λb eI ψ γ5γ ψreI , (30c) 278 Y.M.P. Gomes, J.A. Helayel-Neto / Physics Letters B 777 (2018) 275–280

I =− ¯ r I λbχ eI Gψ γ ψreI , (30d) is a source to the vector gauge field, b, with effective charge − q = 3σ 1 . The pseudo-scalar bilinear acts as a source to the com- = 2 = 2 where σ ∂χ σ , and we have used γ5 1. Note that the first ponent . Besides that, an unusual source, quartic in the fermionic equation gives us that the 4D torsion is algebraically solved in fields, also appears in the equation of motion of both compo- terms of a fermionic bilinear. In general, the torsion tensor can nents. be written as:

I 1 I I 1 I L I 6. Conclusions T = (δ tK − δ t J ) + s + q , (31) JK 3 J K 6 JKL JK We have presented here a 5D Chern–Simons AdS-super-gravity where the q-tensor are in general discarded. Therefore, from the model without a gravitino. The formulation proposed in [1] gener- first identity of the previous set of equations, we see that we may ates effective models where the spin-1 fermionic field is replaced write a 3-form S such that S = e T I =−ψ¯ r ψ and a 1-form I r by a composition of a Dirac fermion and the d-bein. Exploring a T = (e T I ) =−4ψ¯ r ψ , both components of the 2-form torsion, I r natural solution for topological actions, F = 0, we find non-trivial T I = (e e T I ) =−ψ¯ r I ψ , i.e.: JK K J γ J γ γK r solutions for the fields. Specially, we find that the torsion plays an important role in terms of fermionic condensates. Analyzing the t =−ψ¯ r γ ψ ; s =−ψ¯ r γ γ ψ ; qI = 0 . (32) I I r I I 5 r JK gauge transformations, we have shown that the Randall–Sundrum Following this line of arguments, from the second equation, we dimensional reduction respects the gauge transformation with the ¯ r have that σ (χ) = 4G(χ)ψ γ5ψr . In other words, we have a di- no-gravitini assumption. The 4-dimensional equations give us a un- rect relation between the conformal function, σ , the χ -component usual Ricci scalar dependence on the fermionic bilinears. A study ¯ r of the fünfbein, G(χ), and the chiral scalar bilinear, ψ γ5ψr . In of the fermionic behavior in the 4-dimensional brane is a next step ¯ r the case ψ γ5ψr = 0, the conformal function will be some ar- in our endeavor, so that we may go deeper into the properties of IJ ¯ r IJ our effective model we have discussed in the present contribu- bitrary constant. We can directly see that ωχ =−Gψ γ γ5ψr , I =−¯ r I I =− ¯ r I = I tion. λb ψ γ5γ ψr and λbχ Gψ γ ψr Gt . The only component of the spin connection that does not come out as a fermionic bilinear is the 4-dimensional spin connection, ωIJ. Acknowledgements Using the equations of motion and the global conformal sym- a a −1 metry, eˆ → eˆ , ψr → ψr , present in the connection, we can Thanks are due to Prof. S.A. Dias for a critical reading and help- express the Ricci scalar by means of the following equation: ful suggestions on our manuscript. This work was funded by the Brazilian National Council for Scientific and Technological Develop- 8 R(ω˜ ) =− − (1 + λ2)2(ψ¯ r γ IJγ ψ )(ψ¯ sγ γ ψ ) + ment (CNPq). 2 5 r IJ 5 s ¯ r 4 2 ¯ r ¯ s I Appendix −10 ψ ψr + (ψ γI ψr )(ψ γ ψs) + 3 2 The representation of the generators is given in terms of (4 + ¯ r ¯ s I 2 ¯ r 2 + (ψ γI γ5ψr )(ψ γ γ5ψs) − 4 (ψ γ5ψr ) , (33) N × + N 24 ) (4 ) supermatrices [6,7]:

IJ = ˜ IJ + IJ where the spin connection is written as ω ω K , with 1 ( )α 0 (γ )α 0 = 2 γab β = a β K IJ being the 1-form the contortion, related with the torsion by Jab , Ja , 00 00 I = ˜ I + I J = I J = J = the relation T D(ω)e K J e K J e and K I (E K IJ) tI . Eq. (33) is the main result of our paper. Independently from the 00 00 T = , Q α = , possible dependence of the -coordinate on the fermionic ﬁeld, k k s s − r α χ 0 (τ )r δs δβ 0 we may state that our result can be interpreted as an effective 0 δr δα i δα 0 cosmological constant where the fermionic matter is distributed. ¯ s = s β K = 4 β Q α , 1 s . (37) This can also can affect the internal structure of stars, specially the 00 0 N δr most dense ones. The torsion components, with the correct dimension, are writ- From that, the following algebra can be written: ten as follows: [ J ab, J cd]=ηad J bc − ηac J bd + ηbc J ad − ηbd J ac , 1 1 t =− ψ¯ r γ ψ ; s =− ψ¯ r γ γ ψ ; qI = 0 (34) I I r I I 5 r JK [ J a, J b]=s2 J ab , [ J a, J bc]=ηab J c − ηac J b , ˆ a s a a ¯ s s ¯ s a As we have seen in Section 3, the gauge ﬁeld b acquires dynam- [ J , Q s]=− γ Q s , [ J , Q ]= Q γ , 2 2 ics in the 5-dimensional space. Now, after dimensional reduction, ˆ = = ab 1 ab ab ¯ s 1 ¯ s ab we can look its components, b (b, bχ ), and we have that: [ J , Q s]=− γ Q s , [ J , Q ]= Q γ , 2 2

3σ − 1 2 = ¯ r + 1 1 ¯ s 1 1 ¯ s b ( )(ψ ψr ) [K, Q s]=−i( − )Q s , [K, Q ]=i( − )Q , 2 4 N 4 N νρλ +2G (ψ¯ r γ γ γ ψ )(ψ¯ r γ ψ )dxμ , (35) [ k ]= k r [ k ¯ s]=− k s ¯ r μ ν 5 ρ r λ r T , Q s (τ )s Q r , T , Q (τ )r Q 1 i 1 2 =− ψ¯ r ψ − μνρλ[(ψ¯ r I ψ )(ψ¯ r ψ ) + ¯ r r a r ab r k r γ5 r γμγ γν r γργI γλ r {Q s, Q }=− δ γ Ja − δ γ Jab + iδ K + (τ ) Tk . (38) 2 2 s 4 s s s +(ψ¯ r γ γ γ ψ )(ψ¯ r γ γ γ ψ )] , (36) μ 5 ν r ρ 5 λ r All the other relations vanish. In the dimensionally reduced sce- where 2 here means 2ˆ in the framework of the RS dimensional nario, we have that the covariant derivative can be written as μ ¯ r μ ˆ reduction. As one can notice, the vector current, j = ψ γ ψr , ∇=(∇, ∇χ dχ), where: Y.M.P. Gomes, J.A. Helayel-Neto / Physics Letters B 777 (2018) 275–280 279 1 1 1 1 1 I I I r 4 I 4 s I 4 IJ F = T + b e + ¯ + e + e (45a) ∇ = d + i( − )b + eI γ + e γ5 + ωIJγ + λ 4 ψ ( γ5 )γ ( γ5 )ψr , r 4 N 2 2 4 I = I + IJ − I + Fχ ∂χ e ωχ e J D(ω)eχ λ I s k s + b γ γ δ + A (τ ) , (39) J I 5 r k r +ψ¯ r ( + γ e4)γ I (γ e + γ e4 )ψ , (45b) 2 5 J χ 5 χ r s 1 1 1 I 1 4 4 = 4 + I + ¯ r + 4 + 4 (∇χ ) = ∂χ + i( − )bχ + (eI )χ γ + e γ5 + F de λb eI ψ ( γ5e )γ5( γ5e )ψr , (45c) r 4 N 2 2 χ 4 = 4 + I − 4 − I + 1 1 Fχ ∂χ e λbχ eI deχ λbI eχ + IJ + I s + k s (ωIJ)χ γ λ(bI )χ γ5γ δr (Ak)χ (τ )r . (40) + ¯ r + 4 J + 4 4 2 ψ ( γ5e )γ5(γ J eχ γ5eχ )ψr . (45d) In terms of the definitions of the dimensional reduction found in Finally, for Fˆ ab, we find: (23), (24) and (25), we may rewrite the gauge transformations in â terms of the component fields. For e , we have: F IJ = R IJ + eI e J + λ2bI b J + 1 1 ¯ r 4 IJ 4 I I IJ I ¯ r 4 I + ψ ( + γ5e )γ ( + γ5e )ψr , (46a) δe = d + ω J + λb 4 + (ψ ( + γ5e )γ χr + 2 2 IJ = IJ + I LJ − IJ + 2 I J + I J + 1 r I 4 Fχ dωχ ω L ωχ ∂χ ω λ bχ b eχ e + χ¯ γ ( + γ5e )ψr ), (41a) 2 1 + ¯ r + 4 IJ K + 4 ψ ( γ5e )γ (γK eχ γ5eχ )ψr , (46b) 4 4 I 1 ¯ r 4 2 δe = d + λb I + ψ ( + γ5e )γ5χr + 2 I4 I I 4 ¯ r 4 I 4 F = λD(ω)b + e e + ψ ( + γ5e )γ γ5( + γ5e )ψr , 1 ¯ r 4 + χ γ5( + γ5e )ψr , (41b) (46c) 2 1 I4 = I − I − IJ + I 4 − I 4 + I = I + IJ + I + ¯ r J + 4 I + Fχ λ(D(ω)bχ ∂χ b ωχ b J ) eχ e e eχ δeχ ∂χ ωχ J λbχ 4 ψ (γ J eχ γ5eχ )γ χr 2 + ¯ r + 4 I J + 4 ψ ( γ5e )γ γ5(γ J eχ γ5eχ )ψr . (46d) 1 ¯ r I J 4 + χ γ (γ J eχ + γ5e )ψr , (41c) 2 χ For the gauge fields, we find: 1 4 4 I ¯ r I 4 k = k + k l m + δe = ∂χ + λb I4 + ψ (γI e + γ5e )γ5χr + F dA f A A χ χ 2 χ χ lm +ψ¯ r ( + γ e4)(τ k) s( + γ e4)ψ , (47a) 1 r I 4 5 r 5 s + χ¯ γ5(γI e + γ5e )ψr , (41d) 2 χ χ k = k + k l m − k + Fχ ∂χ A f lm Aχ A dAχ where = eI γ . For ωˆ ab, it follows that: + ¯ r + 4 k s J + 4 I ψ ( γ5e )(τ )r (γ J eχ γ5eχ )ψs , (47b) ] 1 = + ¯ r + 4 + 4 IJ = IJ + [IK J + ¯ r + 4 IJ + F db iψ ( γ5e )( γ5e )ψr , (47c) δω d ω K ψ ( γ5e )γ χr 4 = − + ¯ r + 4 J + 4 Fχ dbχ ∂χ b iψ ( γ5e )(γ J eχ γ5eχ )ψr (47d) 1 r IJ 4 [I J] + χ¯ γ ( + γ5e )ψr + λb , (42a) 4 4 By adapting the RS ansatz, we find the solutions which we have presented in Section 5. I 1 4I K I 1 ¯ r 4 I δb = d + b + ψ ( + γ5e )γ γ5χr + K 2 λ λ References 1 r I 4 + χ¯ γ γ5( + γ5e )ψr , (42b) 2λ [1] Pedro D. Alvarez, Pablo Pais, Jorge Zanelli, Unconventional supersymmetry and ] 1 its breaking, Phys. Lett. B 735 (2014) 314–321. IJ = IJ + [IK J + ¯ r J + 4 IJ + δωχ ∂χ ωχ K ψ (γ J eχ γ5eχ )γ χr [2] Pedro D. Alvarez, Mauricio Valenzuela, Jorge Zanelli, Supersymmetry of a dif- 4 ferent kind, J. High Energy Phys. 1204 (2012) 058, https://doi.org/10.1007/ 1 ] JHEP04(2012)058. + ¯ r IJ J + 4 + [I J χ γ (γ J eχ γ5eχ )ψr λbχ , (42c) [3] Pedro D. Alvarez, et al., Supersymmetric 3D model for gravity with SU (2) 4 4 gauge symmetry, mass generation and effective cosmological constant, Class. I 1 4I K I 1 ¯ r J 4 I Quantum Gravity 32 (17) (2015) 175014. δb = ∂χ + b + ψ (γ J eχ + γ5e )γ γ5χr + χ λ χ K 2λ χ [4] Alfredo Guevara, Pablo Pais, Jorge Zanelli, Dynamical contents of unconven- 1 tional supersymmetry, J. High Energy Phys. 2016 (8) (2016) 85. + ¯ r I J + 4 [5] Gaston Giribet, et al., Black hole solutions in Chern–Simons AdS supergravity, χ γ γ5(γ J eχ γ5eχ )ψr . (42d) 2λ J. High Energy Phys. 2014 (8) (2014) 83. 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