Gauge Supergravity in D= 2+ 2

JHEP10(2017)062 Springer for Sp(2). . In anal- fdual part a constant ψ July 25, 2017 ab 4) gauge in- | Γ : ω October 10, 2017 : September 10, 2017 September 17, 2017 : : Received Published Revised Accepted ently imposed, and the resulting superalgebra, close off-shell. The sitàdel Piemonte Orientale, 4) connection supermatrix, and are | 0) supergravity in 2 + 2 dimensions. 2 , , and Majorana gravitino Published for SISSA by https://doi.org/10.1007/JHEP10(2017)062 ab gauge ω + = (1 2), and the antiselfdual part of 2 | N = = 10 + 2 gauge supergravity, the action is given by Sp(2), including a Majorana-Weyl supercharge. Thus 4) curvature supermatrix two-form, and | for OSp(1 ⊕ D ψ 2) | , spin connection a 2) gauge invariant. V | . It is similar, but not identical to the MacDowell-Mansouri 5 γ . is the OSp(1 3 R 4) gauge supersymmetry survives. The gauge fields are the sel 1707.03411 | The Authors. c Supergravity Models, Superstring Vacua = 2 + 2 supergravity. The constant supermatrix breaks OSp(1

We present an action for chiral , ), where D [email protected] Γ 2 R and the Weyl projection of ab ω STr( via P. Giuria 1, Torino,Arnold-Regge Center, 10125 Italy via P. Giuria 1, Torino, 10125 ItalyE-mail: Dipartimento di Scienze eviale Innovazione T. Tecnologica, Univer Michel 11, Alessandria,INFN, 15121 Sezione Italy di Torino, The fields of the theory are organized into an OSp(1 Open Access Article funded by SCOAP Abstract: Leonardo Castellani given by the usual vierbein Gauge supergravity in D ogy withR a construction used for supermatrix containing selfduality condition on the spin“projected” connection action can be is consist OSp(1 Supersymmetry transformations, being part of a variance to a subalgebra OSp(1 ArXiv ePrint: Keywords: of action for half of the OSp(1 JHEP10(2017)062 2 3 3 4 4 5 5 6 7 8 1 2 4 7 ravity Sp(32) vities in ⊕ superalgebra 32) | Sp(1 gauge at has been explored mmetry “lives” on the fiber pears as the “square root” eonomic framework, see for symmetry are fused together dinate transformation along art of a superdiffeomorphism s part of a ] have shown how this approach ant, up to boundary terms. 7 – 4 – 1 – ]. The twelve dimensional action is written in terms of 16 = 1 (chiral) supergravity action ]. Recently it has been used to construct chiral gauge superg N = 2 + 2 supergravity Sp(2) transformation laws 15 ] and has allowed the construction of Chern-Simons supergra – matrices ⊕ D 11 5 supermatrix representation 12 – γ 2) 8 × | = 2 + 2, 4) connection and curvature 64) curvature supermatrix, but is invariant only under its O | | D ]). Recent advances in superintegration theory [ 3 = 2 + 2 = 10 + 2 dimensions [ – These two conceptually different ways of interpreting super In this paper we work in the gauge supersymmetry paradigm, th In Chern-Simons supergravities, on the other hand, supersy 1 3.6 Self-dual 3.1 Action 3.2 Invariances 3.3 OSp(1 3.4 The action3.5 in terms of Equations component of fields motion 2.1 The algebra 2.2 The 5 2.3 Connection and curvature D D in the group geometricex. approach [ (a.k.a. group manifold or rh since long ago [ of a gauge supergroup rather than on a (super) base space. It i in the OSp(1 of transformations leaving the Chern-Simons action invari interpolates between superspace and component actions. In most approaches toof supergravity, diffeomorphisms, local and supersymmetry hasGrassmann a directions. ap natural In this interpretation frameworkalgebra as supersymmetry in coor is superspace. p 1 Introduction 4 Conclusions A 3 The Contents 1 Introduction 2 OSp(1 odd dimensions [ JHEP10(2017)062 e 4 0) ing , (2.3) (2.1) (2.2) (2.4) al (1 = 10 + 2 4) connec- | 4) curvature D | deals with the = 2 + 2 chiral 3 D . Due to the presence ac e explicit expression of 5 superalgebra, entailing γ M is the OSp(1 sections for bd η n all even dimensions with = 2+2 are summarized in h the MacDowell-Mansouri ab R 1 is generated by a Majorana- lfduality condition. Section gauge . − cedure relies on the existence D M bd . αβ ) M 4) superalgebra, but only under a e spin connection transformation law, see | ab ]). They are candidate backgrounds ac = 3+1. The version we propose here η ), where 20 Cγ D – − ( 1 2 RΓ bc 18 recalls the definitions of OSp(1 ∧ 2 + M b a 2) signature, supersymmetry invariance being = 2 to find an action for P ad R , P 0), and part of a η ac – 2 – , , t η αβ + ) (1 STr( − a ad = 2 R a P M Cγ ab s ( bc bc 5 supermatrix representation. Section chiral − M η η ], and are related after dimensional reduction to integrabl × = = 0 (mod 8). = 2+2 dimensions have been considered by many authors in is a constant supermatrix involving 25 ] = ] = ] = t } b c – D β cd Γ − ¯ ,P ,P 21 Q s a , ,M ab α P ]. The actions were obtained in most cases by supersymmetriz [ ab ¯ ]. M Sp(2) that includes a Majorana-Weyl supercharge. Thus chir Q [ 28 { M – ⊕ 16 [ 2) 26 | ) satisying = 2 [ 4) connection and curvature | s, t 4) superalgebra is given by: ]. | D = 2 superstring [ 17 = 2+2 action, its invariances, field transformation laws, th , N 11 D , the action is not invariant under the full OSp(1 The paper is organized as follows. Section Here we apply it for the case Supergravity theories in It is “restored” in second order formalism, or by modifying th Γ 1 automatic off-shell closure without need of auxiliary fields differs because supersymmetry is of the “base space” type, as in usual supergravity in The OSp(1 2.1 The algebra chiral the past (for a veryfor partial the list of referencesmodels see in [ the self-dual Einstein-Palatini action with (2 of for ex. [ This section and the next one closely parallel the analogous tion and curvature, and their 5 the action in terms ofcontains component fields, some field conclusions. equations and Gammathe se appendix. matrix conventions in 2 OSp(1 Weyl supercharge and closesof off-shell. Majorana-Weyl The fermions, constructive and pro signatures can ( in principle be applied i subalgebra. Supersymmetry is part of this superalgebra: it supersymmetry survives. Thisaction, is for an which gauge important supersymmetry difference is wit completely broken subalgebra OSp(1 supermatrix two-form, and supergravity. This action is given by supergravity of ref. [ JHEP10(2017)062 are (2.6) (2.5) (2.7) (2.8) (2.9) (2.10) (2.11) (2.16) (2.15) (2.12) (2.13) (2.14) αβ C . ! ρ α 0 δ ). σ ρ σα 0 (vierbein), generate ) A.4 C a ab

γ V . is dual to the Majorana ( a = α γ αβ ψ ¯ ) a ors in ( Q ab βα ab V . C 2 1 ¯ β ψγ ! α Cγ a 2 1 ( Q + ψ Σ γ ψ 4 1 α a a − ab = ¯ Σ 0 γ Q b γ R + a α ψ . ¯ − ψV 5 supermatrices: ab a V + σ ¯ V Q ρ a 2 1

ω a ) 2 1 × ¯ a 1 4 ψγ V P a − ≡ a γ 1 2 − γ ( − ≡ a ab V , ψ + (spin connection) and γ Ω α R cb αβ Ω α b + ab ) ! ¯ 1 2 ω ab Q ∧ – 3 – ab ¯ a γ Q V 0 β c ab ω b β α a ab + a Ω α ) a ¯ Cγ ω ) ω ψ ω M ω γ ( 0 0 , ab ab a − 1 1 4 4 2 1 1 2 ab ) by factoring out the two spinor Majorana one-forms. γ γ γ − − ), one needs the identity ! ( ( ω 2 Ω ab − − −

a ab 0 d 1 2 ψ 1 2 2 1 A.4 ¯ R 2.4 ψ = 2 + 2 gamma matrices and charge conjugation = = ¯ 1 4 = d dψ dV dω ψ = a Ω D − ] = ] = σα R ≡ = ≡ Ω β β C

¯ ¯ a ρ β Q Q Σ = Σ = R ab 4) curvature two-form supermatrix is δ , , | R ≡ a R ab ,P + P Ω [ M ! [ σβ 0 C 4)-connection is given by 2) bosonic subalgebra, and the supercharge | ρ α , dual to the one-forms , ab δ a 0 0 γ 5 supermatrix representation P 1 2 ×

. Conventions on SO(3 α and 5 supermatrix representation: = ≈ ψ ab × ab M M We omit wedge products between forms. 2 given in the appendix. 2.2 The 5 The above superalgebra can be realized by the 5 gravitino the Sp(4) The corresponding OSp(1 where where simple matrix algebra yields: To verify the anticommutations ( deducible from the Fierz identity ( In the 5 We have also used the Fierz identity for 1-form Majorana spin 2.3 Connection andThe curvature 1-form OSp(1 JHEP10(2017)062 e a (3.2) (3.3) (3.7) (3.5) (3.6) (3.4) (3.9) (3.8) (3.11) ). These 5 = 0 (which γ ǫ 1) (1 + − 2 1 5 γ as: = ± ac R + M R P bd ǫ η + cd − = 0 and ( . ǫ γ ± a bd ! . R ε ǫ ! M abcd − ]) , where ǫ ǫ ) (3.1) ǫ 0 1) ac + 2 1 , = η a P − Γ γ [ α . − − R 5 a ¯ ǫ γ Q ε RRΓ ! ab ± δ ( bc 1 2 γ 0 RR + ab M ¯ ǫ ⇒ 1) 0 + 5 5 0 1 and STr( γ − γ = + α ad 4) curvature two-form 1 2 M | − a ab η ¯ STr( Q ] = 0 (3.10) γ – 4 –

γ 5 Ω Z ab αβ = β a + β + ) γ ǫ α ab 2 Z ε ( ≡ ) ¯ M ε Q ab ab ¯ ǫ ± 2 − + ad 1 4 , + ab γ Γ − ǫ γ

5 − ab M = Cγ (

γ Sp(2) subalgebra: ( Ω is the constant matrix: bc = M 2 S 1 1 2 2 η ≡ ] = ⊕ − Γ ǫ ǫ 1 + 5 supermatrices. δS ǫ , = 2) d | ] = ] = [ ] = Γ } × = [ + β ± + cd cd = 1 (chiral) supergravity action = + β ¯ + ab ¯ Q Q γ , Ω , ,M ,M N ǫ + ab + α δ ± − ab ab ¯ 4), generated by 4) gauge parameter: Q M | | [ M M 1) = 0). These restrictions on the gauge parameters determin { [ [ 4) gauge transformations: − | 5 γ ) varies as ( = 2 + 2, ǫ 3.1 D is the OSp(1 ǫ 3.2 Invariances Under the OSp(1 All boldface quantities are 5 where STr is the supertrace and where Computing the commutator yields implies also ¯ Thus the action is invariant under gauge variations with generators close on the OSp(1 the action ( with 3.1 Action The action is written in terms of the OSp(1 3 The subalgebra of OSp(1 JHEP10(2017)062 , ) − ab ab − ε 3.3 M rana (3.16) (3.17) (3.21) (3.22) (3.14) (3.12) (3.20) (3.13) (3.23) (3.18) (3.19) (3.15) and ab + leaving the ω ψ , ab + lds do not mix, ε and . . transform with the s the “matter fields” a 5 γ + V 3.6 + 2 Sp(2) transformations. ψ , ψ + Γ , 1 + ⊕ ab ab Σ 2 α γ ω − ab ± ab 2) ¯ + | ω Q 1 γ ǫ 2), while the antiselfdual | + = + ǫ + ¯ − ψ . RR + α + Sp(2) parameters and connections. + ¯ cd cd Σ ¯ ab − Σ Q η η − a γ × cd cd Sp(2) subalgebra as described above, Σ ψ γ ab da + da − − η η a ab + γ ε ε + = 0, see section ab ⊕ ψ STr ε γ ¯ ǫ da + da − γ bc bc ab + + − 1 4 ε ε + ab , ab − ε 2) ω ω ab − ¯ ǫ Z − | γ Sp(2) bc bc + − 4 1 ω ε + ab − + + bc − 1 4 R R ≈ cd ε η + + c ǫ bc cd cd generate OSp(1 1 4 + + + + – 5 – ) = 4 2) η η η M R ab ǫ c , ) + cd cd + α + Sp(2) gauge fields γ db + db − ǫ η η ab ¯ V ε ε ab − Q a ab abcd + + ) γ ε ⊕ db + db − ǫ γ ǫ ac ac + − ω ε ε ab − RRΓ ab a a + is inert under supersymmetry, This will be important 2 1 ω ω 4 1 + ε γ 2) ac ac + − R R to the OSp(1 | a ab ∓ − − − 4 2 1 1 ab + + R R − ǫ V ε ω STr( − − − − ab ab + ab − + 2 1 ab + ε Z M − dε dε dǫ = = = = ( = 2 a Sp(2) transformation laws for the curvatures deduced from ( − + = = = ( = = − ab ab + − 1 2 ) is a topological term, we have: ⊕ Σ Σ a . − + δR ab ab = + − δ δ δR δR = 2) are respectively Weyl andψ anti-Weyl projections of the Majo | δV S δψ δψ ± δω δω RR ± ab − P Sp(2) transformation laws ψ M and Weyl projected = is the Weyl projected supersymmetry parameter, and ⊕ STr( + ǫ ab ± R and + ψ 2) M | P + Sp(2) covariant derivative of the gauge parameters, wherea ) invariant: ψ = ) we deduce the transformation laws on the fields ⊕ transform homogeneously. Note also that gauge and matter fie 3.1 + 2) ǫ 3.3 | − ψ The antiselfdual connection Finally, the OSp(1 Thus we see that the OSp(1 are the selfdual and antiselfdual SO(2 , a ab − OSp(1 where V action ( separating into a gauge and a matter multiplet under OSp(1 from ( generates Sp(2). 3.3 OSp(1 read: ω Restricting the gauge parameter for the consistency of the selfduality condition 3.4 The actionRecalling in that terms of component fields The selfdual and gravitino, i.e. Moreover JHEP10(2017)062 , , λV (3.30) (3.32) (3.26) (3.31) (3.25) → V fdual action, abcd ǫ ) ρ . d abcd + = 2+2, or also to V ǫ c P − D ψ ≡ ψV by rescaling cd in + ab γ λ − . ¯ ¯ ψγ ψ eq.) (3.34) . + 3.6 erning self-dual supergravity, ab originating from the last term − ψ gauge (chiral) supersymmetry ρ + total derivative (3.27) d a − R 2. After inserting the curvature ψ, ρ a Σ (3.24) V / 2 1 ψ ργ c ) ab abcd − ψ 5 ǫ and + γ V ψR P + ¯ γ ab b − a a ab b γ Σ ψ − (sum of Euler and Pontryagin forms), V ω R ¯ ab V a ψγ cd 4 1 b a 1 2 R γ ) + V ( =0 (Einstein eq.) (3.33) = 0 ( − 1 4 − abcd − ) of = (1 + 1 2 ǫ ¯ a ψ − − −R ) read: a − cd − R dψ ψ − ab − P = 0), see section d 3.26 − = = R a RRP – 6 – γ ≡ ψV R ψ 3.25 b . Contrary to the MDM case, here the Poincaré ab − + total derivative (3.28) ρ ab − V a + a λ 2 1 R 2 γ Tr( − ab R Σ R λ ab − b ¯ ργ a γ 2 , and ψ − R a Z ω ab ) and ( − + total derivative (3.29) − γ a − 2.3 ω b , ρ a − + cd ± ψ V 4 1 cb ρ V = 4 a − 2.16 R ab a 4¯ abcd ω ψV γ − S ε Σ cd V a c dR c a − − a ab ¯ γ ab dρ ψ ǫ V ω ργ 4 2 1 1 2¯ ab abcd − −R − ∓ − ε R d ab = = = = V c dω a a − ab ± V ρ ). The Poincarélimit however exists in the “projected” sel R ≡ R a ab − ψR ρ a ab R R 3.25 R ¯ ψγ Z is set to zero (and therefore 0 is singular, due to the term = , and dividing the action by ab − S and Σ as defined in section → λψ ω λ √ R As in the MDM case, we can introduce a length parameter This action is similar to a MacDowell-Mansouri (MDM) action -type actions previously considered in the literature conc → 2 ψ and used the identities definitions the action takes the form but with the important difference thatclosing it off-shell. is invariant under a limit 3.5 Equations ofThe motion variational equations for the action ( and the Bianchi identities in the action ( consequences of the definitions ( R up to boundary terms. Carrying out the supertrace leads to: We have dropped the topological term with with where JHEP10(2017)062 3 ), ab − 4). the | R Sp(2) 3.36 (3.39) (3.38) (3.36) (3.37) ). As a × = 3 + 1: 2) . To settle | D 3.36 − ψ ], one can fur- 20 in the last term. – . Sp(2) of OSp(1 2) transformations | ab − − 18 ⊕ . ψ R 2) | quation of motion ( and d in [ ouri action in + ], it cannot be obtained as a = 0 breaks OSp(1 er hand, we know that in the on: eq.) (3.35) part of ψ . under the transformations of gauge supersymmetry. 4) connection, but the action 20 h the field equations and the , | ab − + – a ab − , absent in the MDM case. This ω ψ = 0 V a 18 R cd η R − cd db ± supersymmetry is completely broken. ab − ψ ω ) in the action by discarding the ω 4) curvatures = 0. The torsion equation a | R γ cd = 0 ( =0 (torsioneq.) ac ± ab = 0, i.e. under OSp(1 + 3.37 ε ω gauge 1 2 ab + ), and the ab − − abcd abcd ε R . As in refs. [ ε ε + – 7 – ab ± − − = 3.25 ab 3.3 ψ ψ as a function of ), and could provide a mass term for dω ω ab 1 cd cd ab + − γ γ = R , with , due to the second term in the torsion equation (after ω and λ − 2 1 1 ab ab ± − . Indeed the condition has no dynamics and becomes an auxiliary field. This − Σ 3.3 + ≡ R λ R 2 1 3.5 − ψ + ab − ψ + ω a and V abcd + ε ab 2). + Σ d | = 2+2 supergravity action, made out of the fields contained in ω a V γ equation acquires the term c D 4 = 2 + 2 supergravity = 0, as in the MDM case, because of the second term in ( in the first term of ( R − ψ a D ab = 0, so that given in section R R − = 0 allows to express − ρ ab − , ψ ω + , ψ 4) connection. It is invariant only under a subalgebra OSp(1 | a To make contact with the selfdual supergravities considere Second order formalism is retrieved by solving the torsion e The reason is that supersymmetry survives but not in the form of ,V 3 ab + this issue one needs toMDM solve model the the torsion gravitino equation. is On massless the even oth if rescaling this term is multiplied by does not imply transformation rules of sections requirement is consistent with Bianchi identities, and wit OSp(1 This closely resembles what happens for the Mac Dowell-Mans to its first factor OSp(1 there too the supergavity fields are organized in a OSp(1 term contains a mass parameter We have presented a field equation from the projected action. 4 Conclusions which for ther impose consequence, the component of The resulting actionω (the “projected” action) is invariant whose gauge fields are 3.6 Self-dual We can impose the selfduality condition on the spin connecti we can implement the selfduality condition ( with These equations admit the vacuum solution OSp(1 Recalling that JHEP10(2017)062 ⊕ 2) | (A.3) (A.4) (A.1) forms ]. , ommons . SPIRE , IN [ = 1 2 5 eories , γ 4 (1981) 398 nd breaks the OSp(1 γ 3 ] = 0 (A.2) algebra of the action. 5 γ redited. 2 ge Zanelli. 136 γ eas in the present paper also ]. 1 a Andrianopoli, Paolo Aschieri, C, γ γ [ ab = ψγ . 5 SPIRE ab 5 IN γ ¯ C, ψγ ][ cd and take the trace on spinor indices). 2 1 A Review of the Group Manifold Approach − γ Annals Phys. , γ cd , = − ]. γ Supergravity Actions with Integral Forms abcd 1) satisfies a T ε − or 2 1 C , – 8 – ψγ 2). c 1 | a − γ SPIRE − Supergravity and superstrings: A Geometric ¯ IN , = ψγ [ ,C 1 ab +1 1 4 − γ , arXiv:1409.0192 [ C = ), which permits any use, distribution and reproduction in a ¯ ψ Cγ = (+1 ψ − (1992) 1583 ab = T (2014) 419 a are symmetric. A 7 γ matrices CC-BY 4.0 ab , η The following Fierz identity holds for Majorana spinor one- γ This article is distributed under the terms of the Creative C Group geometric methods in supergravity and superstring th ab Cγ η B 889 , volumes 1–3, World Scientific, Singapore (1991). = 2 and } a b ): = 2 + 2 , γ Cγ = 2 + 2 charge conjugation matrix a γ ψC D and Its Application to Conformal Supergravity perspective Int. J. Mod. Phys. Nucl. Phys. D { A selfdual condition can be imposed on the spin connection, a = 0) supersymmetry survives, being part of the invariance sub , ¯ [1] L. Castellani, R. D’Auria and P. Fré, [2] L. Castellani, P. Fréand P. van Nieuwenhuizen, [3] L. Castellani, [4] L. Castellani, R. Catenacci and P.A. Grassi, ψ Open Access. Fierz identity. Attribution License ( (to prove it, just multiply both sides by References any medium, provided the original author(s) and source are c ( Duality relation. (1 itself is invariant only under the Lorentz subalgebra, wher so that Sp(2) invariance to the first factor OSp(1 Clifford algebra. It is a pleasure toBianca acknowledge useful Cerchiai, discussions Riccardo with D’ Laur Auria, Mario Trigiante and Jor A Acknowledgments The JHEP10(2017)062 , ]. , , ]. ]. , ]. SPIRE ]. IN [ All Odd ]. SPIRE , Spinning SPIRE SPIRE IN itational , supergravity [ IN , IN SPIRE s ][ = 2 IN ][ SPIRE ]. ][ N = 12 IN (1991) 469 ]. , ][ , D , dition (February 2008) -dimensions (1991) 83 SPIRE IN SPIRE [ B 361 ]. IN strings 50 Years of Yang-Mills (2 + 2) , ]. B 367 hep-th/0408137 , in = 2 ]. SPIRE arXiv:1503.07886 hep-th/9207042 ]. [ [ N IN (1987) 85 SPIRE ][ hep-th/9203081 [ IN SPIRE ]. Nucl. Phys. ]. ]. ][ , SPIRE IN invariant action for (1977) 1376] [ ]. IN Nucl. 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