Higher dimensional Chern-Simons supergravity

8 Pages·0.15 MB·English

by  Maximo Banados

Higher dimensional Chern-Simons supergravity
Ma´ximo Ban˜ados1, Ricardo Troncoso1,2 and Jorge Zanelli1,3
1Centro de Estudios Cient´ıficos de Santiago, Casilla 16443, Santiago, Chile
2Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Casilla 487-3 Santiago, Chile
3Departamento de F´ısica, Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile
∇g =0. (2)
A Chern-Simons action for supergravity in odd- A1...An+1
dimensional spacetimes is proposed. For all odd dimensions, [In the case of a supergroup, the invariant tensor should
thelocalsymmetrygroupisanontrivialsupersymmetricex- have the corresponding (anti-) symmetry properties.]
tension ofthePoincar´e group. In2+1dimensionsthegauge For a given n, the above condition may have no so-
group reduces to super-Poincar´e, while for D=5 it is super-
lutions. Indeed, (2) imposes strong restrictions on the
Poincar´e with a central charge. In general, the extension is
group. As we shall see below, for D > 3, one needs to
6 obtained by the addition of a 1-form field which transforms enlarge the bosonic sector of the theory in order to pro-
9 asanantisymmetricfifth-ranktensorunderLorentzrotations.
duce a supersymmetric extension of Chern-Simons grav-
9 Since the Lagrangian is a Chern-Simons density for the su-
ity that contains local Poincar´einvariance.
1 pergroup,thesupersymmetryalgebra closes off shellwithout
Itisdirecttoprovefrom(1)that,uptoatotalderiva-
theneed of auxiliary fields.
n tive, L is invariant under the gauge transformation
a PACS numbers: 04.50.+h, 04.65.+e, 11.10.K
J
δλA=∇λ, (3)
3
I. INTRODUCTION
where λ is an arbitrary zero-form parameter and ∇λ =
1
dλ+[A,λ] . If δ and δ are two trasformations, with
v λ η
In the search for an unified theory of all interactions parameters λ and η respectively, then
3
including gravity, higher dimensional models have be-
0
0 come standard in theoretical physics. Two important [δλ,δη]A=δ[λ,η]A. (4)
1 examples are D = 10 superstrings [1] and D = 11 su-
0 pergravity [2] which give rise, upon dimensional reduc- The three dimensional supergravity studied by
6 tion, to interesting effective models in four dimensions. Achu´carro and Townsend [4], as well as the five-
9 In the study of higher –and lower– dimensional mod- dimensional theory studied by Chamseddine [5], are
/
c els, Chern-Simons densities play an important role. Al- Chern-Simons theories in the sense described above.
q thoughChern-Simonsformsfirstappearedinthephysics Their supersymmetry transformations can be written in
- literatureinthecontextofanomalies,itisnowclearthat the form (3) and therefore the supersymmetry algebra
r
g theyhaveanintrinsicvalueasdynamicaltheoriesintheir closes off-shell without the need of auxiliary fields.
: own right. For example, pure gravity and extended su- Itgoeswithoutsayingthat,appartfromtheinvariance
v
i pergravityinthreedimensionsareChern-Simonstheories under (3), the Chern-Simons action is also invariant un-
X
for the groups SO(2,2) [3] and OSp(2|p)⊗OSp(2|q) [4], der diffeomorphisms. In 2+1 dimensions that symmetry
r respectively. Similarly, in five dimensions, supergravity is not independent from the local gauge group because,
a
can be written as a Chern-Simons action for the super- as a consequence of the equations of motion, the con-
groupSU(2,2|N)[5]. Chern-Simonsformsalsoprovidea nection is locally flat. This means that, given two con-
simpledescriptionforpuregravityinallodd-dimensional figurations that differ by a diffeomorphism, there always
spacetimes[6]. Theequationsofmotionofthesetheories exist a gauge transformation that deforms one into the
possess black hole solutions [7] generalizing those found other. In the canonicalformalism this is reflected by the
in 2+1 dimensions [8]. absence of new independent constraints associated with
The Chern-Simons Lagrangian is constructed as fol- diffeomorphisms.
lows. Let G a basis for the Lie algebra of a given (su- Indimensions greaterthan three,however,the Chern-
A
per)group G. Let A = AAG be the connection for G Simons equations of motion do not impose the flatness
A
and F = dA+A∧A = FAGA its curvature 2-form. The conditionand therefore the diffeomorphisminvarianceis
Chern-Simons Lagrangian, L, is a (2n+1)-form whose anindependentsymmetrygivingrisetoindependentcon-
exteriorderivative(definedin2n+2dimensions)satisfies: straints in the canonical formalism (see [9] for more de-
tails on this point). For our purposes here it is enough
dL=<F∧···∧F >, to observethat the gaugetrasformations(3) forma sub-
=gA1...An+1FA1∧...∧FAn+1 (1) group of the whole symmetry group.
It is the purpose of this paper to show that the above
whereg ≡<G ,...,G >iscompletelysym- schemecanbeextendedtosupergravityinallodddimen-
A1...An+1 A1 An+1
metric and satisfies the invariance condition sions,providedonechoosesthebosonicLagrangianinan
1
agprapnrgoiparniaftoerwthaey.boItsotnuircnssecotuotrtihsantotfoHriDlbe>rt’3s., RthaethLear-, δI =2Z ǫabcdRab∧Tcλd, (10)
thecorrectLagrangianisnon-linearinthecurvatureand
which is not zero for arbitrary λ.
yet, gives rise to first order equations for the tetrad and
An alternative approach, often followed in the super-
the spin connection.
gravity literature, is the so called 1.5 formalism. That
is to set Ta = 0 keeping only the tetrad transformation
in (7); the variation of wab is then calculated using the
II. GENERAL RELATIVITY AS A GAUGE
chain rule. This procedure brings the diffeomorphism
THEORY
invariance into the scene because if Ta = 0, the vari-
ation (7) for the tetrad is equal -up to a rotation- to
Inthissectionweshallreviewsomeaspectsoftheviel-
a Lie derivative with a parameter ξµ = eµλa. In sum,
bein –or gauge– formulation of general relativity. The a
the Hilbert action in 3+1 dimensions is invariant under
main point of this section is to display the differences
Lorentz rotations and diffeomorphisms, but not under
between gravity in odd and even dimensions.
the local translations (7), generated by P .
a
A completely different situationis observedin 2+1di-
mensions,inwhichcasetheHilbertactionislinearinthe
A. Poincar´e translations vs. diffeomorphisms
dreibein field,
Generalrelativityinfour andallevendimensionscan-
not be construed as a ‘truly’ localgauge theory [10]. On I2+1 =Z ǫabcRab∧ec. (11)
thecontrary,forodd-dimensionalspacetimes,gravitycan
be written as a Chern-Simons action for the Poincar´e It is straighforward to check, using the Bianchi iden-
group [4,3,6]. [If the cosmological constant is present, tity,thatthevariationof(11)under(7)givesaboundary
then the relevant group is the (anti)-de Sitter group.] termandhence,(7)isasymmetryofthe2+1theory. As
The Poincar´e group has generators P and J satisfy- Witten has pointed out, this simple fact has deep conse-
a ab
ing: quences[3]. Indeed, inthreedimensions,onecanreplace
thediffeomorphisminvariancebyalocalPoincar´einvari-
[Pa,Pb]=0, ance whose constraint algebra is a true Lie algebra.
[P ,J ]=η P −η P , Onemaywonderifthereexistsanactioninvariantun-
a bc ab c ac b
der (7) in higher dimensions. This generalization indeed
[J ,J ]=η J −η J +η J −η J . (5)
ab cd ac bd bc ad bd ac ad bc exist and is given by
We define the connection for this group
1 I2n+1 =Z ǫa1...a2n+1Ra1a2∧···∧Ra2n−1a2n∧ea2n+1 (12)
A=eaP + wabJ . (6)
a ab
2
which, clearly, exists only in odd dimensions. The fact
Letλbe anarbitraryparameterwithvaluesintheLie that (12) can be written for odd dimensional manifolds
algebra (λ = λaP + 1λabJ ). Under the infinitesimal onlyisassociatedtotheexistenceofChern-Simonsforms
a 2 ab
gaugetransformationδA=∇λ,ea andwab transformas in those dimensions.
The key property of (12) is that it is linear in the
follows.
vielbein field rather than being linear in the curvature,
Translations: δea =Dλa, δwab =0, (7) as would be the case for the Hilbert action. This fact
Rotations: δea =λaeb, δwab =−Dλab, (8) makes (12) invariant under the transformation (7), up
b to a boundary term. Inspite of being non linear in the
where D is the covariantderivative in the connection w. curvature, this action yields first order differential equa-
tions for all the fields. This is not surprising as (12) is a
Itmightseempuzzlingthateventhoughthetetradand
particularcase of a Lovelockaction[11]. I describes
thespinconnectioncarryarepresentationofthePoincar´e 2n+1
a Chern-Simons theory of ISO(D − 1,1), obtained by
group,theHilbertactionconstructedpurelyoutofthose
contractionofSO(D−1,2),andpossessessolutionswith
fields, is not invariant under the Poincar´e group in four
conical singularities [7] analogous to those found in 2+1
dimensions. In fact, the action
dimensionswithoutcosmologicalconstant[12]. Itshould
be mentioned here that the action (12) has a propagat-
I =Z ǫabcdRab∧ec∧ed (9) ing torsion [13] and therefore, the 1.5 formalism is not
applicable in this case. The main goal of this paper is
is invariant under (8) but not under (7). The reason is to describe the supersymmetric extension of the action
simplythatunder(7)theactionchanges,modulobound- (12).
ary terms, by
2
B. Super-Poincar´e vs. supergravity to this case [4], there is an alternative route which leads
more directly to supergravity. The key feature of 2+1
Supergravity if often referred to as the square root of supergravity is that –unlike 3+1 supergravity–, the ac-
General Relativity [14] much in the same spirit as the tion is invariant under local Poincar´e transformations.
Dirac equation is the square root of the Klein-Gordon This means that in the 2+1 theory it is not necessaryto
equation. This is justified by the fact that the commu- expressthe righthandside of(14)interms ofdiffeomor-
tatoroftwosupersymmetrytransformationsgivesagen- phisms;the commutatoroftwosupersymmetrytransfor-
eral change of coordinates plus a rotation and another mationsgivesalocaltranslation,whichis asymmetryof
supersymmetry transformation. A concrete example is the action as well.
the usual N = 1 local supersymmetry transformations In other words, in 2+1 dimensions one can consider a
[15] truly first order formalism in which the spin connection
transforms independently of the vielbein and gravitino
1 just as dictated by group theory. In particular, under
δea = ε¯Γaψ ,
µ 2 µ supersymmetry, one can set
δψ =D ε. (13)
µ µ δwab =0 (16)
µ
Here ψ and ε are Majorana spinors; ε¯ is the Majorana
conjugate (ε¯) = C εα satisfying ε¯Γaη = −η¯Γaε, and which, together with (13), is a symmetry of the 2+1 ac-
D is the covαariantαdβerivative in the spin connection. It tion.
µ
The simplicity of the 2+1 theory can be explicitly ex-
is then straightforwardto prove that the commutator of
twotransformationswithparametersεandη actsonthe hibited. The action reads [4,16]
tetrad as
1 I =Z (ǫabcRab∧ec−ψ¯∧Dψ), (17)
[δ ,δ ]ea = D (ε¯Γaη). (14)
ε η µ 2 µ
where ψ is a two component Majorana spinor. [See the
The transformation(13)followsfromthe super-Poincar´e Appendix for a summary of conventions.]
algebrawhenoneconsiderstheveilbeinandthegravitino This action is invariant under Lorentz rotations,
asthecompensatingfieldsforlocaltranslations(P )and
a
supersymmetry transformations (Q), respectively. It is δωab =−Dλab, δea =λaea,
b
notsurprisingthereforethattherighthandsideof(14)is 1
alocaltranslation–actingonthetetrad–,withparameter δψ = λabΓabψ; (18)
4
λa = 1ε¯Γaη.
2
The trouble with supergravity in 3+1 dimensions is Poincar´etranslations,
that the action is not invariant under local translations.
Nevertheless, transformation (14) is still a symmetry of δwab =0, δea =Dλa, δψ =0; (19)
the action provided the transformation of the spin con-
nectionpreservesthetorsionequationTa = 1ψ¯Γa∧ψ(1.5 and supersymmetry transformations,
2
formalism). On the surface defined by the torsion equa-
1
tion, the right hand side of (14) can be rewritten as δwab =0, δea = ε¯Γaψ, δψ =Dε. (20)
2
Dµ(ε¯Γaη)=Lξeaµ+ξνwbaνebµ−ξνψνΓaψµ. (15) The fields ea, wab and ψ transform as components of
a connection for the super Poincar´e group and therefore
The first term in the right hand side of (15) represents a
the supersymmetry algebra implied by (18)-(20) is the
diffeomorphism with parameter ξµ =ε¯Γµη (Γµea =Γa),
µ super Poincar´eLie algebra. The invariance of the action
the secondtermis arotationwithparameterξµwa , and
bµ (17) under the super Poincar´e group should come as no
the third term is a supersymmetry transformation with
surprise because it is the Chern-Simons action for the
parameter −ξµψ . Equation (15) shows that, as far as
µ connection A = eaP + 1wabJ + Q¯ψ, whose genera-
the tetrad is concerned, the algebra of supersymmetry a 2 ab
tors are P ,J and Qα. Indeed, the action (17) can be
transformations closes. But the fact that we have used a ab
written as [4]
the torsion equation implies that the connection is no
longeranindependentvariable. Onthecontrary,itsvari- 2
ation is given in terms of δea and δψ, and differs from I = <A∧dA+ A∧A∧A>, (21)
Z 3
the form dictated by group theory. As a consequence,
the local supersymmetry algebra acting on the gravitino where the bracket <···> stands for a properly normal-
closes only on shell and auxiliary fields are required for ized trace on the algebra, with < J ,P >= ǫ and
ab c abc
its closure off shell. < Q ,Q >= −iC are the only non-vanishing traces.
α β αβ
Acompletelydifferentsituationisobservedinthecase [C is the charge conjugation matrix.]
αβ
of 2+1 gravity. Although the above discussion applies
3
III. FIVE DIMENSIONAL POINCARE´ (22) is also invariant under supersymmetry transforma-
SUPERGRAVITY tions,
δea =−i(ε¯Γaψ−ψ¯Γaε),
We turn now to the supersymmetric versionof the ac-
tion (12) in higher dimensions. To illustrate the ideas δwab =0,
we start with the five dimensional case which already δb=ε¯ψ−ψ¯ε,
contains the main ingredients. The general case will be
δψ =Dε. (25)
indicated in the next section.
The proof of the invariance of (22) under (25) is
straightforward. An important test of the consistency
A. The Action
of (25) is the fact that the commutator of two super-
symmetry transformations gives local translation (24).
The analog of the action (17) in five dimensions has Indeed, if δ and δ are supersymmetry transformations
ε η
three pieces; with parameters ε and η, we have,
I5/susy =IG+Ib+Iψ (22) [δε,δη]ea =−iD(ε¯Γaη−η¯Γaε),
[δ ,δ ]wab =0,
where I is the purely gravitationalterm, I is a second ε η
G b
bosonic term needed by supersymmetry, and Iψ is the [δε,δη]b=d(ε¯η−η¯ε),
fermionic term. The explicit formulas are: [δ ,δ ]ψ =0. (26)
ε η
1
IG = 2Z ǫabcdfRab∧Rcd∧ef, locTahlesusypmerm-Peotirniecsaor´eftgheenaecratitoonrs(2P2),arJeg,enQeαra,teQ¯db,yptlhues
a ab α
the Abelian generator K, responsible for the non-zero
Ib =Z Rab∧Rab∧b, transformation of b in (24). These generators form an
extension of the super-Poincar´ealgebrawhose only non-
Iψ =−Z Rab∧(ψ¯Γab∧Dψ+Dψ¯Γab∧ψ). (23) vanishing (anti-)commmutator is
Here the gravitino field is a Dirac spinor 1-form, and b {Qα,Q¯β}=−i(Γa)αβPa+δβαK, (27)
is a 1-form Lorentz pseudo-scalar. The form of I is dic-
b plus the Poincar´e algebra. The commutators of K, Qα,
tated by the transformation of I and I under super-
G ψ andQ¯ withtheLorentzgeneratorscanbe readofffrom
symmetry. [See section V for an alternative way to see α
their tensor character.
this through the integrability conditions of the classical
Note that K commutes with all the generators in the
equations of motion.]
algebra and therefore, it is as a central charge in the su-
Since we are interested in the geometrical aspects
per Poincar´e algebra. This is, however, a peculiarity of
of this theory only, we have set all the coupling con-
five dimensions. For other odd dimensions, the genera-
stants equal to one. The Lorentz covariant derivative
tor K is a completely antisymmetric tensorof fifth rank,
D in the spinorial representation is given by Dψ =
dψ + 14wabΓab∧ψ. We work with Dirac spinors in or- wgehniecrhahtoars.a non-vanishingcommutatorwith the Lorentz
der to avoid the dimensional dependence of Majorana
spinors which only exist -on a Minkwoskian signature-
in dimensions 2,3,4 mod 8. All the main results of this
B. Chern-Simons formulation
paper carrythroughforMajoranaspinorswhen they ex-
ist. The Dirac conjugate is defined as ψ¯ = ψ†Γ and in
0
the action, ψ and ψ¯ are varied independently. [See the The fact that the symmetries of the action (22) close
Appendix.] without the need of any auxiliary fields strongly suggest
Besides localLorentzrotations,the aboveactionis in- that (22) may be written as a Chern-Simons action for
variant under the Abelian translations, the supergroup. In this section we prove that this is
indeed the case.
δea =Dλa, ConsideraconnectionAforthe superalgebrafoundin
δwab =0, the last section,
δb=dρ, 1
A=eaP + wabJ +bK+ψ¯ Qα−Q¯ ψα. (28)
δψ =0, (24) a 2 ab α α
where λa is a 0-form Lorentz vector and ρ is a 0-form The super-curvature F = dA + A∧A is then found by
direct application of (5) and (27)
Lorentzpseudo-scalar. Theinvarianceof(22)under(24)
follows directly from the Bianchi identity. The action
4
F =FAG Justasinthe fivedimensionalcase,the supersymmet-
A
1 ric action in dimension 2n+1 has three terms;
=T˜aP + RabJ +F˜K+Dψ¯ Qα−Q¯ Dψα. (29)
a ab α α
2
I =I +I +I (34)
2n+1/susy G b ψ
Here T˜a := Ta−iψ¯Γa∧ψ and F˜ := db+ψ¯∧ψ; Ta is the where the bosonic ‘geometric’ term I is given by,
torsion2-form, and Rab is the 2-form Lorentz curvature. G
We recall that a Chern-Simons Lagrangian in five di-
mensions, Lg, is defined by the relation IG =Z Rabc∧Rab∧ec. (35)
gABCFA∧FB∧FC =dLg, (30) Ib is a second bosonic term involving a fifth-rank 1-form
field, babcde,
where the trilinear form g ≡< G ,G ,G > is an
ABC A B C 1
DinivffaerrieannttctheoniscoersoofftthheeiLnivearailagnetbtreanwsoitrhggenergaivtoerdsiffGeAr-. Ib =−6Z Rabc∧Rde∧babcde. (36)
ABC
ent five dimensional Lagrangians L .
g (Note that this term vanishes in three dimensions and
To prove that (22) is a Chern-Simons action we need in five dimensions babcde is a Lorentz pseudo-scalar.) Fi-
to find aninvarianttensor such that L is equal, up to a
g nally, the fermionic part is
total derivative, to the Lagrangian in (22). This tensor
indeed exists and is given by, i
Iψ = 3Z Rabc∧(ψ¯Γabc∧Dψ+Dψ¯Γabc∧ψ). (37)
<J ,J ,P > =ǫ ,
ab cd e abcde
Each term in the action (34) is independently invariant
<J ,J ,K > =η η −η η
ab cd ac bd ad bc
under local Lorentz transformations. The complete ac-
<Qα,J ,Q¯ > =−2(Γ )α. (31)
ab β ab β tion is invariant under the Abelian translations,
Itisstraightforwardtoprovethat,uptoatotalderivative δea =Dλa, δwab =0, δbabcde =Dρabcde,
and an overall factor, the 5-form L associated to the
g δψ =0, (38)
above tensor is equal to the supersymmetric Lagrangian
in (22). and supersymmetry transformations,
¿From the above result it is now evident that the ac-
tion is invariant under supersymmetry transformations δea =−i(ε¯Γaψ−ψ¯Γaε), δwab =0,
up to a total derivative. All the symmetry transforma- δbabcde =−i(ε¯Γabcdeψ−h.c.), δψ =Dε, (39)
tions of the action can now be collected together in the
formδA=∇λwhere ∇is the covariantderivativeofthe where D represents the Lorentz covariant derivative.
supergroup and λ is a zero-form Lie algebra-valued vec- The proof of the invariance of (34) under supersym-
tor in the adjoint representation. From this formula it is metry transformations is straighforward. One starts by
alsoevidentthatthe algebraofsupersymmetrytransfor- varying the fermionic part. Up to a boundary term one
mations closes as dictated by group theory, easily obtains
i
[δλ,δη]A=δ[λ,η]A. (32) δIψ = 12Z Rabc∧Rde∧(ψ¯{Γabc,Γde}ǫ−c.c.). (40)
The Chern-Simons action (22) can be obtained from the
Using the formula (A4) of the Appendix, we find that
action found in [5] by an appropriateWigner-Inonucon-
(40) has a term proportional to a 5-rank Dirac matrix
traction. The closure of the supersymmetry algebra,
Γabcde plus a term proportional to a Dirac matrix Γa.
however,wasnotmentionedin[5]. Inthenextsectionwe
It is direct to see that the first term is cancelled by the
provethat the aboveschemeis notexclusiveofthe three
variation of I while the second term is cancelled by the
and five dimensional theories but it can be extended to b
variation of I .
any odd-dimensional spacetime. G
As inthe lowerdimensionalcases(D =3,5),the com-
mutator of two supersymmetry transformations gives a
localAbeliantranslation(38). Thus,thesupersymmetric
IV. THE GENERAL CASE
extension of the Poincar´e algebra that leaves the action
invariant has generators G = [P ,J ,K ,Qα,Q¯ ].
A a ab abcde α
Inthissectionweshowhowtheresultsofthe previous
The only non-vanishing (anti-)commutator is
sectionsaregeneralizedtoanyodd-dimensionalmanifold.
Inordertosimplifythenotationweintroducethesymbol {Qα,Q¯ }=−i(Γa)αP −i(Γabcde)αK , (41)
β β a β abcde
R defined by,
abc
plus the Poincar´ealgebra. The commutatorsofthe Q,Q¯
Rabc :=ǫabca1···aD−3Ra1a2∧···∧RaD−2aD−3. (33) and K with the Poincar´e generators can be read from
their tensorial character.
5
The action I is also a Chern-Simons action. VI. COMMENTS AND PROSPECTS
2n+1/susy
The connection now is
We have shown in this note that the successful meth-
1
A=eaPa+ wabJab+babcdeKabcde+ψ¯αQα−Q¯αψα, ods used in three dimensions to construct supersymet-
2
ric extensions of general relativity can be generalized to
(42) any odd-dimensional spacetime. We have restricted our-
selves, however, to Poincar´e supergravity. The full anti-
andtheLagrangianisdefinedby<F∧···∧F >=dL2n+1 de Sitter extension remains an open problem. [In five
where the invariant (n+1) multilinear form < ··· > is
dimensions, a Chern-Simons action for anti-de Sitter su-
defined by
pergravityhasbeenknownforsometime[5]. Thataction
reduces to the action consideredhere after a proper con-
<J ···J P > =ǫ ,
a1a2 aD−2aD−1 aD a1···aD traction is performed.]
1
There are good reasons to seek a full anti-de Sitter
<J ···J K > =− ǫ η ,
a1a2 fg abcde 12 a1···aD−3abc [fg][de] Chern-Simons formulation of supergravity. First, the
<QJa1a2···JaD−4aD−3Q¯ > =2inΓa1···aD−3 (43) bosonic Lagrangian in the Poincar´e case does not con-
tain the Hilbert term thus making the contact with four
(the remaining brackets are zero). This completes the dimensional theories rather obscure [6]. Secondly, the
construction of the (2n+1)-dimensional Chern-Simons Poincar´etheoryinodddimensionsdoesnotpossessblack
action for supergravity. hole solutions while the anti-de Sitter theory does [7].
In principle, a Chern-Simons anti-de Sitter supergrav-
ity can be constructed from the knowledge of the as-
V. INTEGRABILITY OF THE EQUATIONS OF sociated supergroup and an invariant tensor only (find-
MOTION. ing the invariant tensor, however, may prove to be a
non-trivial task). In five dimensions, the relevant super-
A remarkable feature of supersymmetric theories in groupisSU(2,2|1)[17]whileintheimportantexampleof
generalandsupergravityinparticularisthefactthatthe eleven dimensions the supergroup is OSp(32|1) [18]. As
integrability conditions for the fermionic field equations the spacetime dimension increases, one faces a growing
arethe bosonic equations. The study ofthe integrability multiplicity of choices for the invariant tensor. To illus-
conditions of the fermionic equation in our model sheds trate this issue, consider the problem of classifying all
some light on the role of the bosonic field babcde. the invariants that can be constructed out the Lorentz
In the notation introduced in Sec. IV, the fermionic curvature in a given dimension [19]. In four dimensions
field equations are we only have,
RabcΓabc∧Dψ =0, (44) ǫabcdRab∧Rcd, Rab∧Rab, (48)
while in eight dimensions we have,
and similarly for the Dirac conjugate spinor. Taking the
covariantderivative of(44)we find the integrabilitycon- ǫabcdefghRab∧Rcd∧Ref∧Rgh, Rab∧Rab∧Rcd∧Rcd,
dition,
Rab∧Rbc∧Rcd∧Rda. (49)
Rabc∧RdeΓabcΓde∧ψ =0. (45) Of course,all the above scalarsdefine Chern-Simons La-
grangians in dimension five and seven respectively. A
This equation should be satisfied for any ψ. Using ele-
similar proliferation of scalars appears in supergravity.
mentary properties of the Dirac matrices we obtain the
A good candidate for the right theory could be a linear
following equations for the bosonic fields,
combinationofallpossibleinvariantssuchthat,underan
appropriated Wigner-Inonu contraction, reduces to the
Rabc∧RabΓc =0, (46) Poincar´etheories studied here.
Rabc∧RdeΓabcde =0. (47) The particular case of eleven dimensions seems to
be particulary suited to admit an anti-de Sitter Chern-
Eq. (46) is the equation of motion for the vielbein field,
Simons formulation. As shown in [18], the super anti-
whileEq. (47)istheequationofmotionfortheb−field.
de Sitter group is OSp(32|1). A natural basis for the
Had we not included b, supersymmetry would not have
Lie algebra of Sp(32) is given by the Dirac matrices
beenachievedandthe integrabilityconditionswouldnot
Γ ,Γ ,Γ ,andthisbasisiseasilyextendedtoexpand
a ab abcde
havebeensatisfied. [ThisdoesnotruleoutaLagrangian
the superalgebra of OSp(32|1). Thus, the supergroup
withoutthe bfield. However,the fermionic equationsfor
OSp(32|1) naturally accommodates the field content of
such a theory would impose additional equations on the
thePoincar´eChern-Simonssupergravityconsideredhere.
bosonic fields.]
One could expect, therefore, that a Chern-Simons La-
grangianfor the supergroup OSp(32|1) in eleven dimen-
sions should reduce to the supersymmetric action (34)
6
uponcontraction. For example,is easyto check thatthe We define Γa1···ap as the totally antisymmetric product
superalgebraobtainedinSec. (IV) canbe obtainedfrom of gamma matrices,
OSp(32|1) by a Wigner-Inonu contraction:
1
Γa1···ap = sgn(σ)Γa ···Γa , (A2)
Ga→λ−1Pa, p!Xσ σ(1) σ(p)
G →J ,
ab ab
Two useful formulas implicitly used in the text are:
G →λ−1K ,
abcde abcde
Qα→λ−12Qα. (50) Γ =−(−i)n+1ǫ Γabc (A3)
a1···aD−3 3! a1···aD−3abc
At the level of the Lagrangian, however, the problem is
more complicated. Due to the ambiguity in the choice
1
of the invariant tensors and the large number of terms {Γabc,Γde}=Γabcde−[η[ab][de]Γc+perm(abc)], (A4)
2
in the super anti-de Sitter Chern-Simons action, it is a
non-trivial problem to find an expression such that, un- where η[ab][de] =ηadηbe−ηaeηbd.
der contraction, reduces to the action considered in this
paper [20].
Finally we mention that the -Poincar´e- supersymmet- Acknowledgments
ric Chern-Simonsactionsfound inthis paper arenotthe
only possibilities. In dimensions greater than five, the
TheauthorsaregratefultoM.Contreras,C.Mart´ınez
fermionic Lagrangian accepts other Poincar´e-invariant
and, especially, C. Teitelboim for helpful discussions.
terms that give rise to other supergravities. For exam-
This work was supported in part by grants 1930910and
ple, in eleven dimensions one can add to the fermionic
2950047 from FONDECYT (Chile), 27-953/ZI-DICYT
lagrangianthe term,
(University of Santiago) and grant No. PG-023-95 of
Departamento de Postgrado y Post´ıtulo (University of
[Rab∧Rab]2∧ψ¯∧Dψ (51) Chile). M.B. was partially supported by a grant from
Fundaci´on Andes. R.T. thanks CONICYT for financial
This term, however, requires extra bosonic fields to re-
support. Institutional support to CECS from a group
spect supersymmetry. This is easily seeing by studying
of Chilean private companies (COPEC, CGEI, Empre-
the integrabilityconditions generatedby (51). One finds
sasCMPC,ENERSIS,MINERALAESCONDIDA,IBM
the equation over the bosonic fields,
and XEROX-Chile) is also acknowledged.
[Rab∧Rab]2∧Rcd =0. (52)
Thus, consistency requires an extra bosonic term in the
Lagrangian of the form [Rcd∧Rcd]2∧Rab∧cab which in-
volves the 1-form c . Varying this term with respect
ab
to c gives (52). Thus, the integrability condition is
ab
satisfied and the action is supersymmetric. A complete [1] Foracompletetreatmentsee,M.B.Green,J.H.Schwarz
classification of all possible fermionic Lagrangians for a and E. Witten, Superstrings, Cambridge University
givendimensionandtheircorrespondingsupersymmetry Press, 1987.
[2] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B76,
algebrasis beyondthe scope ofthis work. We wouldlike
409 (1978).
to point out, however, that the method outlined here
[3] E. Witten,Nucl. Phys. B 311, 46 (1988).
seems to provide a simple way to generate extensions of
[4] A. Achu´carro and P.K. Townsend, Phys. Lett. B180, 89
thesuper-Poincar´ealgebrainvolvingextrabosonicfields.
(1986).
[5] A.H. Chamseddine, Nucl. Phys. B 346, 213 (1990).
[6] A.H. Chamseddine, Phys. Lett. B233, 291 (1989).
APPENDIX A: GAMMA MATRICES
[7] M. Ban˜ados, C. Teitelboim and J.Zanelli, Phys. Rev.
D49, 795 (1994). See also, JJ Giambiagi Festschriff, H.
The Clifford algebra in D = 2n+1 dimensions with Falomir, R. Gamboa, P. Leal and F. Shaposnik eds.
Minkowskian signature can be generated by a set of (World Scientific, Singapore, 1991).
2n×2n matrices; the unit I and D matrices Γa, satisfy- [8] M.Ban˜ados,C.TeitelboimandJ.Zanelli,Phys.Rev.Lett
ing {Γa,Γb}=2ηabI, where a,b,...,= 1,2,...,2n+1 and 69, 1849 (1992).
ηab=diag(−,+,···,+). In this signature, Γ† =Γ Γ Γ . [9] M.Ban˜ados,L.J.GarayandM.Henneaux,Phys.Rev.D,
a 0 a 0
It is alwayspossible to finda representationofΓa ma- in press.
trices in which [10] T. Regge, Phys. Rep. 137, 31 (1986).
[11] D. Lovelock, J. Math. Phys. 12, 498 (1971).
Γ:=Γ1Γ2···ΓD =(−i)n+1I. (A1) [12] S. Deser, R. Jackiw and G. ’t Hooft, Ann. Phys. 152,
220 (1984).
7
[13] M. Ban˜ados, L.J. Garay and M. Henneaux, in prepara-
tion.
[14] C.Teitelboim,Phys. Rev. Lett.38,1106(1977);C.Teit-
elboim and R.Tabensky,Phys. Lett. 69 B, 240 (1977).
[15] ForareviewseeP.vanNieuwenhuizen,Phys. Rep.68,4
(1981).
[16] ForearlierworkonthreedimensionalsupergravityseeN.
MarcusandJ.H.Schwarz,Nucl.Phys.B228,145(1983);
S.Deser and J.H. Kay,Phys. Lett. B120, 97 (1983).
[17] W. Nahm,Nucl. Phys. B135, 149 (1978).
[18] J. W. van Holten and A. Van Proeyen, J. Phys. A15,
3763 (1982).
[19] A. Mardones and J. Zanelli, Class. & Quan. Grav. 8,
1545 (1991).
[20] M. Ban˜ados, R. Troncoso and J. Zanelli, in preparation.
8