Gauge Fixing of Chern-Simons N-Extended Supergravity

0.15 MB·English

by  Wander G. Ney

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 F´isicas - (CBPF) - Rio de Janeiro, RJ

n bCentro Federal de Educac¸a˜o Tecnol´ogica - (CEFET) - Campos dos Goytacazes, RJ
a c Universidade Federal do Esp´irito 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-extendedPoincar´esupergroup. 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, con-
sidered 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 su-
persymmetric 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 Poincar´e, 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 Poincar´e, 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-Poincar´elimit 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-Poincar´e limit of null cosmo-
logical 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 Poincar´e 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(Poincar´eindex),α=1,2(spinindex)andI =1,...,N (rigidSO(N)index). The
tangent space metric is Minkowskiof signatureis ( ++), the Levi-Civita tensor for the Poincar´e
−
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) genera-
tors 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 Poincar´e super-
group 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˜ =d˜A˜+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 associ-
ated 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 .
T T L L S S
This is a gauge fixed action for the 2+1 dimension N extended supergravity.
−
7
3.1 Conclusion
In this work we have achieved the gauge fixing in the Batalin-Vilkovisky scheme, of the 2+1
dimensionalN-extendedsupergravityconsideredasatopologicalfieldtheoryoftheChern-Simons
type, the gaugegroupbeing the N-extendedPoincar´esupergroup. We haveused the formalismof
extended fields in order to express the BRST algebra, the Slavnov-Taylor identity and the action
in a compact and manageable way.
It wouldbe interesting to considerthe constructionin terms ofa super-BF topologicaltheory,
with the Lorentz group as a gauge group. This approach would have the advantage of making
possible a generalization to higher dimensions, the super B-field being then however subjected to
appropriate constraints [6], as in the non-supersymmetric case(see e.g. [11]).
References
[1] S.J. Gates, JR and H. Nishino, Int. J. Mod. Phys. A8 (1993) 3371;
[2] E. Witten, Nucl Phys. B311 (1988) 46;
[3] A. Achu´carro, P. K. Townsend, Phys. Lett. B180; (1986) 89;
A. Achu´carro, P. K. Townsend, Phys. Lett. B229 (1989) 383;
[4] E. Witten, Nucl. Phys. B323 (1989) 113;
[5] F. Gieres, J. Grimstrup, T. Pisar, M. Sweda, JHEP 0006 (2000) 018 (hep-th/0002167);
[6] K. Ezawa, Prog.Theor. Phys. 95 (1996) 863 ( hep-th/9511047)
[7] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 69 (1977) 309;
I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 102 (1981) 27;
I.A. Batalin and G.A. Vilkovisky, Phys. Rev. D 28 (1983) 2567;
[8] J. C. Wallet, Phys. Lett. B 235 (1990) 71; H. Ikemori, Class. Quant. Grav. 10 (1993) 233
(hep-th/9206061) M. Adub, J.P. Ader and L. Cappiello Nuov. Cim. 105 A (1992) 1507
[9] Wander G. Ney, Master Tesis (in portuguese);
[10] O. Piguet and S.P. Sorella, Algebraic Renormalization, Springer-Verlag (1995);
[11] L.Freidel,K.KrasnovandR.Puzio,Adv.Theor.Math.Phys3(1999)1289(hep-th/9901069)
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