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hep-th/0312193 CBPF-NF048/03 UFES-DF-OP2003/5 Gauge Fixing of Chern-Simons N -Extended Supergravity 4 0 Wander G. Neya,b, Olivier Piguetc,1 and Wesley Spalenzaa,1 0 2 aCentro Brasileiro de Pesquisas Físicas - (CBPF) - Rio de Janeiro, RJ n bCentro Federal de Educaçaõ Tecnológica - (CEFET) - Campos dos Goytacazes, RJ a c Universidade Federal do Espírito Santo - (UFES) - Vito´ria, ES J 0 E-mails: [email protected], [email protected], [email protected] 2 3 Abstract v 3 We treat the N−extended supergravity in 2+1 space-time dimensions as a Yang-Mills 9 gaugefieldwithChern-Simonsactionassociated totheN-extendedPoincarésupergroup. We 1 2 fixthe gauge of this theory within the Batalin-Vilkovisky scheme. 1 3 0 1 Introduction / h t - p Since Einstein achieved his theory, gravity is treated as a particular geometry of space-time. The e metric is what determines this geometry which, in the first order formalism, is described by the h vierbein – or more precisely the dreibein in the context of the present paper – and a Lorentz : v connection. The dreibein components form a basis of the tangent vector space, equiped with a i X Lorentz structure. The metric is then derived from the dreibein. The Lorentz conection, considered as an independent field, turns out to be functional of the dreibein by virtue of the field r a equations. In this case the action is that of Einstein-Palatini. N-extended supergravity is a supersymmetric generalization of gravity with N supersymmety generators. We shall be interested in N-extended supergravity in 3-dimensional space-time, a general presentation of which having including conformal N-extended supergravity,been given in [1]. Witten [2]describedgravityin 2+1dimensions asa gaugetheory,writing the gravityfields as a Yang-Mills field with a Chern-Simons action and showing the equivalence of the Chern-Simons and Einstein-Palatini actions. The gauge group of this theory is that of Poincaré, which may be extended to the de Sitter or anti-de Sitter groups corresponding to a positive or negative, cosmologicalconstant, respectively. Theequivalenceof3-dimensionalgravitywithaChern-simonstheorycanbegeneralizedtothe case of N-extended supergravity, taking the Poincaré, de Sitter or anti-de Sitter supergroup as gauge group [3, 4]. All one needs is an invariant quadratic form for these supergroups from which one can construct the Chern-Simons action, the latter being then interpreted as the N-extended supergravity action. The author of [4] did it for the anti-de Sitter supergroup considered as the product of two ortho-simplectic supergroups: AdS(p,q)=OSP+(p,2;R) OSP−(q,2;R) (where × 1SupportedinpartbytheConselhoNacionaldeDesenvolvimento Cient´ıficoeTecnolo´gico CNPq–Brazil. 1 p+q = N is the total number of supersymmetry generators). They considered various limiting cases, in particular the super-Poincarélimit of vanishing cosmologicalconstant. OurpurposeistoimplementaconvenientgaugefixingofthisN-extendedsuper-Chern-Simons theory, taking into account the existence of a local vector supersymmetry generally associated to diffeomorphisminvarianceasusualinsuchtopologicaltheories[5]. Becauseofthelatterinvariance thecompletegaugealgebraclosesonlyon-shell. WeshallusethereforetheBatalin-Vilkovisyversion ofthe BRSTgaugefixingscheme[7]. Inordertoobtainthe gaugefixedactioninaconcisewaywe shall use a formalism of extended fields, i.e. superpositions of forms of all possible degrees [8, 10]. We choose to work in the present paper directly in the super-Poincaré limit of null cosmological constant. The construction for super-de Sitter with nonvanishing cosmological constant is similar [9] and will not be explicited out here. The plan of the paper is the following. Section 2 reviews the construction of 3-dimensional N-extended supergravityas a Chern-Simons theory fololowing [4] and then analyses all the gauge invariances, putting them together in a BRST operator and writing the corresponding Slavnov- Taylor identity. The gauge fixing is performed in Section 3. The paper ends with a conclusion section. 2 N Extended Supergravity − In order to construct an N-extended supergravity theory a la Chern-Simons in 3 dimensioanl space-time, with zero cosmological constant, we choose the N-extended Poincaré supergroup as a gauge group, whose Lie superalgebra is2: [Ja,Jb]=ǫabcJ , [Pa,Pb]=0, [Ja,Pb]=ǫabcP , c c 1 (2.1) [Ja,QI]= (γa)βQI, [Pa,QI]=0, QI,QJ =δIJ(γ ) Pa . α −2 α β α h α βi a αβ Pa, Ja and QI are the generators of space-time translations, Lorentz transformations and super- α symmetry transformations, respectively, the QI’s being Majorana spinors. The indices take the valuesa=0,1,2(Poincaréindex),α=1,2(spinindex)andI =1,...,N (rigidSO(N)index). The tangent space metric is Minkowskiof signatureis ( ++), the Levi-Civita tensor for the Poincaré − indices ǫabc = ǫ is defined by ǫ123 = 1 and the spin Levi-Civita tensor ǫ = ǫαβ by ǫ12 = 1. abc αβ The latter is used in orderto lower and rise the spin indices as u =µβǫ , uα =ǫαβu . We also α βα β have ǫαγǫ = δα. The Dirac matrices (γaβ) are chosen real: γ0 = iσ , γ1 =σ , γ2 =σ , the γβ − β α − y z x σ’s being the Pauli matrices. In this representation the Majorana spinors have real components: u∗ =u and u¯α (u+γ0)α = εαβu = uα. α α ≡ β Let us collect in one array X the basis elements of the superalgebra: A X = P , J , QI ; a,b=0,1,2; α=1,2; I =1, ,N . { A} { a b α ··· } In order to construct a Chern-Simons action, we need a nondegenerate invariant quadratic form Φ ,Φ =g ΦAΦB , with Φ =ΦA X , (2.2) h 1 2i ab 1 2 1,2 1,2 A invariant under the adjoint action of the superalgebra δ Φ =[Φ ,X ] . A 1,2 1,2 A suchaquadraticformmaybederivedfromthefollowingquadraticCasimiroperatorofthealgebra (2.1): 1 C =CABX X =PaJ QIαQI. (2.3) A B a− 4 α 2The“gradedbracket”(, )isananticommutator ifbothentriesareodd,andacommutator, otherwise. 2 Namely: 0 (δ ) 0 1 ab (gAB)= 2(−1)[XA] CA−B1 = (δab) 0 0 , (2.4) (cid:0) (cid:1) 0 0 (2ǫαβ)   where [X ]=0,1 if X is an even, odd generator,respectively. A A Writing the Lie algebra valued Yang-Mills connection as A=AAX =eaP +ωaJ +ψIαQI , (2.5) A a a α whereea,ωa andψαI arethe dreibein, spinconnectionandgravitino1-forms,respectively,wecan now write the Chern-Simons as 1 S = A,dA+A2 , CS 2 h i R (2.6) 1 1 = ea dω + ǫ ωbωc +ψIdψIα+ ωaψIαψIβγ . − (cid:18) a 2 abc (cid:19) α 2 aαβ R the symbol , denoting the invariant quadratic form (2.2). This action is obviously invariant h i underrigidSO(N)transformations,underwhichψ transformsasavectorandtheremainingfields as scalars. We may turn this invariance into a local SO(N) invariance substituting the derivative dψI throughthe covariantderivativeD ψI =dψI+AIJψJ where AIJ is a non dynamicalSO(N) O O O connection. ThefieldsAIJ thenplaytheroleofsourcesoftheconservedNoethercurrentsjIJµ=εµνρψI ψJα O αν ρ of the SO(N) symmetry. We may observe that the connection A cannot be made a dynamical O field for the reasonthat a quadratic Casimir operator such as (2.3) containing the SO(N) generators is not invertible, hence does not lead to the invariant quadratic form necessary for writing a kinetic action. This is in contrast with what happens in the super-anti-de Sitter case, where the relevant quadratic Casimir operator is indeed invertible [4, 9]. The firsttermin(2.6) is the Einstein-Palatiniactionandthe othersarethe kinetic termofthe gravitino and its interaction with the spin conection. The equations of motion derived from the action (2.6) read3 δS 1 ∗ =dω + ǫ ωbωc =0 , (Curvature), δea a 2 abc δS 1 ∗ δωa =dea+ǫabcebωc+ 2ψIαψIβγaαβ =0 , (Torsion), (2.7) δS 1 ∗ =2 dψIα ωaγα ψIβ =0 . (Rarita Schwinger). δψI (cid:18) − 2 aβ (cid:19) α We see that the curvature is zero. The torsionequation gives us the spin connection as a function of the dreibein and the gravitino. The Rarita Schwinger equation expresses the vanishing of the gravitino covariant derivative with respect to the gauge group SO(1,2). The local symmetries of the action are expressed as its invariance under the BRST transformations sA= dC [A,C], sC = C2, s2 =0 , (2.8) − − − where the Faddeev-Popov ghost C is written in the adjoint representation of the Poincaré supergroup as: C =CAX =caP +caJ +cIαQI , A T a L a S α 3Thesymbol=∗ meansequaluptoanequationofmotion(”onshell”). 3 the fields ca, ca, cIα are the ghosts associated to space-time translations, Lorentz and supersym- T L S metrytransformations,respectively. Theyare0-formsofghostnumber1bydefinition. Theghosts ca, ca are odd and cIα is even. In components, the BRST transformations read T L S sea = dca +ǫa ωccb +ǫa eccb γa ψIαcIβ , sωa = dca +ǫa ωccb , − T bc T bc L− αβ S − L bc L 1 1 sψIα = dcIα+ γ αωacIβ + γ αψIβca , (2.9) − S 2 aβ S 2 aβ L 1 1 1 sca =ǫa cccb γa cIαcIβ , sca = ǫa cccb , scIα = γ αcacIβ . T bc T L− 2 αβ S S L 2 bc L L S 2 aβ L S The BRST invariance can be expressed through a Slavnov-Taylor identity. We introduce the Batalin-Vilkovisky anti-fields A∗ and C∗ associated to A and C: A∗ =ω∗aP +e∗aJ +ψ∗αIQI , C∗ =c∗aP +c∗aJ +c∗IαQI , (2.10) a a α L a T a S α and an action coupling them to the BRST transformations of A and C: ∗ ∗ S = ( A ,sA + C ,sC ) ext Z h i h i (2.11) = e∗se+ω∗sω+ψ∗Isψ∗αI +c∗sc +c∗sc +c∗IαscI . Z α L L T T S Sα (cid:0) (cid:1) The exterior field A∗ is a 2-form of ghost number 1 and C∗ a 3-form of ghost number 2. The GrassmannparitiesofthefieldsA,C,A∗ andC∗ ar−edeterminedthroughtheirtotaldegre−e: ghost numberplusformdegree. Thefieldiseven(commuting)orodd(anticommuting)ifitstotaldegree is even or odd respectively. The total action S =S +S (2.12) CS ext obeys the Slavnov-Tayloridentity δS δS (S)= =0. (2.13) S Z δϕ∗ δϕ ϕ=XA,C We can work in a more compact way defining an extended field [8] A˜ written as the sum of forms of all possible degrees (in our case, degrees 0 to 3): A˜=C+A+A∗+C∗ , (2.14) the total degree of A˜ being equal 1. We also define an extended exterior derivative d˜= b+d, where b is a BRST type operator such that: b2 = [b,d] = 0 and therefore d˜2 = 0. The extended zero curvature condition is F˜ =dÃ˜+A˜2 =0. (2.15) This condition gives us the b transformation of the extended field bA˜= dA˜ A˜2 , (2.16) − − andhenceofits componentsC,A, C∗ andA∗. The operatorbcanbe interpretedasthe linearized Slavnov-Tayloroperator associated with an action S(ϕ,ϕ∗): δS δ δS δ b= = + , (2.17) SS Z δϕ∗ δϕ δϕδϕ∗ ϕ=XA,C provided the action S is a solution of the equations δS δS = ϕ=bϕ , = ϕ∗ =bϕ∗ (ϕ=C, A) . (2.18) δϕ∗ SS δϕ SS 4 The general solution of the latter equations is the action (2.12). This argument shows that one could have proceeded in a reversed way as well, namely beginning with the construction of a nilpotemntoperatorbactingonthefieldsandantifields,andthenderivingthe actionasasolution of (2.18). This is the procedure we shallfollow inthe next sub section. Note that the fulfilment of (2.18) automatically ensures [9] the validity of the Slavnov-Tayloridentity (2.13). 2.1 Diffeomorphism and Vector SUSY Asatopologicalone,ourtheorymustbeinvariantunderthediffeormorfismsorgeneralcoordinates transformations. As we want to include them in the BRST operator, we treat the infinitesimal diffeormorfism parameter as a ghost vector ξ = ξµ∂ , the components ξ being odd. The diffeor- µ µ morfism transformation ∗ ∗ δ ϕ= ϕ , δ ϕ = ϕ , (2.19) diff ξ diff ξ L L where is the Lie derivative associated with the vector field ξ, may thus be added to the BRST ξ L operator. We do it in the extended field formalism, still including the local vector supersymmetry transformation which uses to accompany diffeomorphism invariance [5]. The ghost of the latter is an even vector field v = vµ∂ of ghost number 2 which happens to contribute to the BRST µ transformation of the diffeomorphism ghost ξ. In order to get this more general BRST operator, we define, as explained in the end of last subsection, a nilpotent b-operator which reads, for the extended field and the new ghosts: bA˜= dA˜ A˜2+ A˜ i A˜ , bξ =ξ2+v , bv =[ξ,v] . (2.20) ξ v − − L − For each form degree this gives bC = C2+ C i A , bA= dC [A,C]+ A i A∗ , ξ v ξ v − L − − − L − bA∗ = dA A2 [A∗,C]+ A∗ i C∗ , bC∗ = dA∗ [A∗,A] [C∗,C]+ C∗ , ξ v ξ − − − L − − − − L bξ =ξ2+v , bv=[ξ,v] (2.21) or, in components: 1 bca =ǫa cccb γa cIαcIβ + ca i ea, T bc T L− 2 αβ S S Lξ T − v 1 bca = ǫa cccb + ca i ωa, L 2 bc L L Lξ L− v 1 bcISα = 2γaαβcaLcISβ +LξcISα−ivψIα, (2.22) bea = dca +ǫa eccb +ǫa ωccb γa ψIαcIβ + ea i ω∗a, bωa =−dcTa +ǫbac ωcLcb +bc ωaT −i eα∗aβ, S Lξ − v − L bc L Lξ − v 1 1 bψIα = dcIα+ γα ωacIβ + γα ψIβca + ψIα i ψ∗Iα, − S 2 aβ S 2 aβ L Lξ − v 5 and, for the anti-fields: 1 be∗a = dea+ǫa ecωb γa ψIαψIβ − bc − 2 αβ +ǫa e∗ccb +ǫa ω∗ccb γa ψ∗IαcIβ + e∗a i c∗a, bc L bc T − αβ S Lξ − v T 1 bω∗a = dωa+ ǫa ωcωb+ǫa ω∗ccb + ω∗a i c∗a, − 2 bc bc L Lξ − v L 1 1 bψ∗Iα = dψIα+ γα ωaψIβ + γα ω∗acIβ − 2 aβ 2 aβ S 1 +2γaβαψ∗IβcaL+Lξψ∗Iα−ivc∗SIα, (2.23) bc∗a = de∗a+ǫa e∗cωb+ǫa ω∗ceb γa ψ∗IαψIβ T − bc bc − αβ +ǫa c∗ccb +ǫa c∗ccb γa c∗IαcIβ + c∗a, bc∗a = dbcω∗Ta+Lǫa ωbc∗cLωbT+−ǫaαcβ∗cScb +S c∗Laξ, T L − bc bc L L Lξ L 1 1 1 bc∗Iα = dψ∗Iα+ γ αω∗aψIβ + γ αψ∗Iβωa+ γ αc∗acIβ S − 2 aβ 2 aβ 2 aβ L S 1 + γ αψ∗Iβca + c∗Iα. 2 aβ L Lξ S Now the operator b can be interpreted as the following linearized Slavnov-Tayloroperator associated with an action S(ϕ,ϕ∗,v,ξ): δS δ δS δ δ = + + bu , (2.24) SS Z δϕ∗ δϕ δϕδϕ∗ δu ϕ=XA,C uX=v,ξ with the transformations bu explicitly4 given in (2.21). Indeed, the equations (2.18) are solved by the action 1 2 S(ϕ,ϕ∗,ξ,v)= A˜,dA˜+ A˜2 A˜+i A˜ . (2.25) ξ v −2Z h 3 −L i The integral of an extended form is defined as the integral of its 3-form terms. This action yields 1 2 S(ϕ,ϕ∗,ξ,v)= A, dA+ A2 + A∗, dC [A,C]+ A ξ Z (cid:18) −2h 3 i h − − L i (2.26) 1 + C∗, C2+ C i A A∗, i A∗ ξ v v h − L − i− 2h i(cid:19) 3 Gauge Fixing The gauge invariant action obtained in the preceding subsection has still to be gauge fixed. We shall use the Batalin-Vilkovisky scheme [7, 10]. The total action will then have the form: S =S +S +S . (3.1) CS ext gf In order to determine the gauge fixing part S , we choose the Landau gauge condition, which gf necessitates the introduction of a nondynamical background metric g 5. In differential form µν notation, the gauge condition reads d A=√g Aµd3x , µ ∗ ∇ where isthecovariantderivativewithrespecttothebackgroundmetric. Itwillbeimplemented µ ∇ through a Lautrup-Nakanishia Lagrangemultiplier B and its associatedFaddeev-Popovantighost C¯, both being 0-forms and Lie algebra valued: C¯ =C¯AX =c¯aP +c¯aJ +c¯αIQI , B =BAX =BaP +BaJ +BαIQI , (3.2) A T a L a S α A T a L a S α 4Thismeansthatweconsidertheghostξ andv asexternal fields. 5Thedynamicalmetricisrepresentedbythedreibeinea 6 their BRST transformations being defined by sC¯ =B , sB =0 . (3.3) The total gauge fixed action S(ϕ,ϕ∗,ξ,v,B,C¯) obeying the Slavnov-Tayloridentity δS δS δS δS δS (S)= +(ξ2+v) +[ξ,v] +B =0 , S Z (cid:18)δϕ∗ δϕ δξ δv δC¯(cid:19) Xϕ is given, according to Batalin and Vilkovisky, by S(ϕ,ϕ∗,B,C¯)=S(ϕ,ϕˆ∗)+ Bd A , (3.4) Z ∗ where the antifields ϕ∗ in (2.26) have been replaced by δΨ ∗ ∗ ϕˆ =ϕ + , δϕ the ”Batalin-Vilkovisky fermion” Ψ being a local functional of ghost number 1, chosen here as − Ψ=Ψ(ϕ,C¯)= dC¯ A= A dC¯. (3.5) Z ∗ Z ∗ We thus have Aˆ∗ =A∗+ dC¯ , Cˆ∗ =C∗ , (3.6) ∗ and the total gauge fixed action reads 1 2 S(ϕ,ϕ∗,ξ,v)= A, dA+ A2 + Aˆ∗, dC [A,C]+ A ξ Z (cid:18) −2h 3 i h − − L i (3.7) 1 + C∗, C2+ C i A Aˆ∗, i Aˆ∗ . ξ v v h − L − i− 2h i(cid:19) In terms of the components fields ea, etc., we have 1 1 S = eadω + ǫ eaωbωc+ψIdψIα ψIωaψIβγ α −Z (cid:18) a 2 abc α − 2 α aβ + eˆ∗a dc +ǫ eccb +ǫ ωccb γ ψIαcIβ + e R (cid:16)− Ta abc L abc T − aαβ S Lξ a(cid:17) +ωˆ∗a dc +ǫ ωccb + ω +ψˆ∗I dcIα+ 1cIβωaγ α+ 1γaαc ψIβ + ψIα − La abc L Lξ a α (cid:16)− S 2 S aβ 2 β La Lξ (cid:17) (cid:0) (cid:1) +cˆ∗a ǫ cccb 1γ cIαcIβ + c i e +cˆ∗a 1ǫ cccb + c i ω T (cid:16) abc T L− 2 aαβ S S Lξ Ta− v a(cid:17) L (cid:0)2 abc L L Lξ La− v a(cid:1) +cˆ∗I 1γα cacIβ + cIα i ψIα Sα(cid:16)2 aβ L S Lξ S − v (cid:17) 1ωˆ∗ai eˆ∗ 1eˆ∗ai ωˆ∗ 1ψˆ∗Ii ψˆ∗αI +Bad ω +Bad e +BI d ψαI , −2 v a− 2 v a− 2 α v T ∗ a L ∗ a Sα ∗ (cid:19) (3.8) where 1 dB = ǫ ρ∂ B , (3.9) ∗ µν 2√g µν ρ and eˆ∗a =e∗a dc¯a , ωˆ∗a =ω∗a dc¯a , ψˆ∗Iα =ψ∗Iα dc¯Iα −∗ T −∗ L −∗ S (3.10) cˆ∗a =c∗a , cˆ∗a =c∗a , cˆ∗Iα =c∗Iα . 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