Chern-Simons Gravity: From 2+1
to 2n+1Dimensions
Jorge Zanelli 1;2
1
Centrode EstudiosCientos(CECS),Casilla1469, Valdivia,Chile
2
DepartamentodeF
isia,Universidadde Santiagode Chile,Casilla307,Santiago2,Chile
Reeived7January,2000
These leturesprovideanelementary introdutionto ChernSimons Gravityand Supergravity in
d=2n+1dimensions.
I Introdution
The presentsituation of High Energy Physisis both
exiting andparadoxial. Ononesidewehavea
mon-umentaltheoretialmahinerybuiltonthemost
beau-tiful ideas {strings, membranes, dualities, M theory,
AdS-CFT orrespondene{, with hardly any
experi-mentally testable preditions for the near future. On
the other hand,there is a growing orpus of puzzling
observationsomingmostly from astrophysis,
inlud-ing gammaraybursts, missing mass, mirowave
inho-mogeneities, indiations of an aelerated expansion,
whih defy present standard wisdom about the
uni-verse, aquired through observations of our more
im-mediate neighborhood. The new observations are for
themostpartsurprising,andoneouldexpetbynow
that the surprises will ontinue. In an eort to ope
with thenew data,a numberof exotiproposalshave
beenputforward. Thus,inreenttimesnotionssuhas
quintesseneandothernonminimal extensionsof
Gen-eral Relativity(GR) inludinganonvanishing
osmo-logial onstant, and extradimensions atmarosopi
saleshavegainedaeptanein respetablejournals.
Inthissituation,areasonableattitudefora
theoret-ialphysiistwould betoritially examineall
onsis-tent alternativesto ourtime-honored foundations. In
this vein, here we disuss a lass of gravity theories
whihsharetheessentialgeometrifoundationofGR:
a)Generalovariane
b)Seondordereldequationsforthemetri
At the same time, other features of GR are
re-laxed:
) Spaetime is allowed to have any number of
dimensions
d) Spaetime is not neessarily asymptotially
at
e) The ation is extremized under independent
variationsofthemetriandaÆneonnetion
Condition()isabsolutelyneessaryintheontext
ofstringtheoryand(d)is mostnaturalin thelightof
thereentlydisoveredAdS-CFTorrespondene.
Requirement(e), onthe other hand,allows for
in-dependent propagation ofthe metri and aÆne
stru-turesof spaetime. This is satisfatory in viewof the
fatthatthemetriandaÆnefeaturesofspaetimeare
geometriallyindependent: onehastodowiththe
mea-surementofdistaneswhiletheotherrelatestoparallel
transport. Anaturalwaytoimplementondition()is
tousetherstorderformalism,wheretheindependent
eldsarethevielbeinone-forme a
=e a
dx
;andthespin
onnetion! ab
=! ab
dx
.
The familyof gravity theoriesobtained with these
postulateswhere studiedby Lovelokin theearly 70's
[1℄. The ve dimensional ase had already been
dis-ussedby Lanzos in the late 30's [2℄. Morereently,
Zwiebah [3℄ and Zumino [4℄ showed the
Lanzos-Lovelok (LL) theories to be appropriate to desribe
the eetive low-energy gravity theory found in the
weakouplinglimitofstringtheory.
Intheir simplest version,theLL theorieshavethe
sameelds,symmetriesandloaldegreesoffreedomas
ordinarygravity. Theationis apolynomialofdegree
[d=2℄inurvature 1
, whihanalsobewritten interms
oftheRiemann urvatureR ab
=d! ab
+! a
!
b
andthe
vielbeinas 2
I
G =
Z
[d=2℄
X
p=0
p L
(p)
; (1)
where
p
arearbitraryonstants,andL (p)
isgivenby
L (p)
=
a1ad R
a
1 a
2
R a
2p 1 a
2p
e a
2p+1
e a
d
: (2)
What makes theLL theories so speial is the fat
that they omply with the requirement b) above. An
arbitraryLagrangiandensityonstrutedwiththe
met-riand urvature tensors, onthe ontrary,wouldgive
rise to fourth order eld equations for the metri, in
general. A powerful reason to hoose the
Einstein-Hilbert(EH)ationin fourdimensionsisthatthe
Ein-stein equations are seond order. This feature of the
EHationstemsfromthefatthattheEHLagrangian
intwodimensionsisthedensityofatopologial
invari-ant: theEulerharateristiofthemanifold. Similarly,
theLLtheoriesind-dimensionsarelinearombinations
(witharbitraryoeÆients)oftheEulerdensitiesofall
dimensionsbelowd. Thus,GeneralRelativityisa
par-tiularaseofLLtheory.
A. Drawbaks of the Generi LL Ation
In spiteof their nie features, theLL theories
suf-fer from anoriginal sin: theyare endowedwith a
ol-letion of indeterminate dimensionful parameters
p ,
p=1;:::;
d
2
. This has twopuzzlingonsequenes
al-readyatthelassiallevel:
(i) The theorieshavealargenumberof physial
parameterswhihshouldbeexperimentally
deter-mined. This would makegravityless interesting
as a fundamental theory, beause it would have
morenaturalonstants,likeG
Newton
andthe
os-mologialonstant,tobeadjusted.
(ii) The eld equations admit solutions whih
have indeterminate spaelike dependene and
timelikeevolution. Thisresultsfromthefatthat
the eld equations are polynomials of degree p
in the derivativesof the metri (g). Thus, the
valuesoftheveloityanjump betweendierent
rootsoftheequationarbitrarilyandstill
extrem-ize theation[5℄,[6℄.
Aonsequeneof(ii) is thefat that theLegendre
transformationfromtheLagrangiantotheHamiltonian
annot be performed in general,making theanonial
quantizationprogramill-dened. Anotherproblem
re-lated with the seond issue is the existene of several
vaua with dierent topologies. This ould be an
in-terestingnovelfeatureofthesetheories,wereitnotfor
the fat that the perturbation expansions around the
dierentvauagiverisetoompletelydierenttheories
andtypiallyontainghosts[9℄.
There is one more reason to nd the presene of
thelarge numberofoeÆients in theLagrangian
un-desirable. The bare
p
's are dimensionful onstants
andthereforeouldreeivequantumorretionsbeyond
ontrol,makingthepossibilityofonstrutinga
quan-tum theory ofgravity evenmoreremote thanin
stan-dard GR. This would be so unless the valuesof these
onstantsare protetedby somefundamental
symme-try,likethezeromassforthephoton,ortheequal
num-ber of quarks and leptons, whih are \proteted" by
gaugeinvariane.
Thus,itwouldbeinterestingtonda\natural"way
toxthe's. Moreover,iftheriterionthatxesthese
oeÆients is basedon some symmetrypriniple, and
possiblyprotetthemfromrenormalizationwitha
rea-sonablesymmetrypriniple. Inthissenario,onends
twospeialfamilies,whihstandoutamongallLL
the-ories. TheyorrespondtoaBorn-Infeld-liketheoryin
evendimensions,andtheChern-Simonstheoryforthe
anti-deSitter(AdS)gaugegroupforoddD.Insetion
III wereviewthese theoriesin general. Insetion IV
the CS ase and their supersymmetri extensions are
analyzed in greater depth. The next setion is just a
ursoryreviewofnonabelianCStheorieswhihanbe
skippedbytheexpertsandbythoseeagertogettothe
juiy stu.
II Chern-Simons Theory in 3
Dimensions
3
Chern-Simons(CS)theoryhasaurioushistory. It
wasdisoveredin the ontextof anomalies in the70's
and used as a rather exoti toy model for gauge
sys-temsin 2+1dimensionseversine[10℄. Itwasonlyby
themid80'sthatitwasrealizedthatordinaryEinstein
gravityin2+1dimensionsisanaturalexampleofaCS
system,espeiallythroughtheworkofWitten[11℄. As
itturnsout,CSsystemsaremoreonspiuousthanit
might seem at rst sight. General Relativity in 2+1
dimensions(with orwithoutosmologial onstant)is
2
Wedgeprodutbetweenformsisunderstoodthroughout.
a CSsystem (forISO(2,1) orSO(2,2) groups,
respe-tively);anyordinarymehanialsysteminHamiltonian
form an be viewed as an abelian CS system in 0+1
dimensions[12℄. Thiswayoflookingatmehanial
sys-tems isnotompletely absurd andit evensheds some
lightintoanientproblemssuhasthejustiationfor
theoldquantizationruleofBohrand Sommerfeld.
In retrospet, we an see that the key to the
on-strution of theCS form (in three dimensions) is the
following: thePontryaginform
P =Tr[F^F℄; (3)
islosed
dP =0: (4)
ByPoinare'slemma,P isloallyexat,that is,itis
alwayspossibleto writeitinanopenneighborhoodas
theexteriorderivativeofa3-form
P =dL: (5)
Thus,the3-formListheChern-SimonsLagrangian.
Clearly,thisideaanbegeneralizedtohigher(odd)
di-mensions, andforother integraltopologialinvariants,
like the Euler harateristi. This is preisely where
the onnetion with the LL theory an be found: if
one asks, what isthe 3-form whose exterior derivative
is the 4-dimensional Euler density?, the answeris the
Einstein-Hilbertationwithnonzeroosmologial
on-stant.Wenowbrieyreviewafewfatsaboutstandard
3-dimensionalCSsystems.
Theideaistondathree-formL
CS
suhthat
dL
CS
=Tr[F^F℄; (6)
where
F=dA+A^A (7)
istheurvature(eldstrength)intheadjoint
represen-tationandAisaLiealgebra-valuedonnetion1-form.
LetGbethegaugegroupandGitsLiealgebra
gen-eratedbythematriesT
a
,suhthat[T
a ;T
b ℄=C
ab T
.
Undertheationofthegaugegroup,theonnetion
A=A a
T
a dx
transformsas
A!A 0
=g 1
Ag+g 1
dg; (8)
where the 0-form g(x) is an element of G. Then, the
urvaturehangesas
F!F 0
=g 1
Fg; (9)
anditiseasily shown, usingtheylipropertyofthe
trae, that Tr[F^F℄ is invariant under (8, 9). From
(6),theCSLagrangianisfoundtobe
L
CS
=Tr[A^dA+ 2
3
A^A^A℄: (10)
A. Gauge Invariane
It iseasily hekedthat the CSation isinvariant
undergaugetransformations. Firstobservethatunder
agaugetransformationoftheform(8), therighthand
sideof(6)doesnothangeandthereforeÆ(dL
CS )=0
dÆL
CS =0:
In other words, under a variation of the elds
thatleavesthe Pontryaginforminvariant, theCS
La-grangianhangesbyalosedform. Thus,providedthe
hange ÆL
CS
approahes zero suÆiently fast at the
spaetimeboundary,theCSationshouldbeinvariant
aswell. Substituting (8)in(10)onends
L
CS (A
0
)=L
CS
(A) dTr[dgg 1
A℄ 1
3 Tr[(g
1
dg) 3
℄: (11)
Thentheationhangesas
I
CS [A
0
℄=I
CS [A℄
Z
M Tr[dgg
1
A℄ 1
3 Z
M
Tr[(gdg 1
) 3
℄: (12)
d
This raises animportantissue: is theation really
in-variantunder thegauge transformation (8)? The
an-swerseems to beno, unless g ! 1 suÆientlyfast to
aneltheseondterm,andsomeothermiralemakes
thethird termalsovanish. Therstonditiononean
always demand beause it is part of the rules of the
gamein anyvariationalproblemthat theeldssatisfy
restritsthetypeofeldtransformationsallowedatthe
spaetimeboundary.
The last termin (11)is losed and therefore,
pro-vided there are no topologial ontrivanes, this term
an be expressed loally as the exterior derivative of
some2-formwhihdependsong(x). Itisobviousthat
thistermannotbemadetovanishsimplybyimposing
someasymptotionditiononthegaugetransformation
g. In fat,there aresomeinteresting simpleexamples
in whih the integral of this last term doesn't vanish.
For instane, if the manifold M is topologially a
3-sphereandthegaugegroupisSU(2), thelast termin
(12)is4 2
N,whereN isthewinding numberofthe
mappingg:M!SU(2).
Thetransformationlaw(12)tellsusthattheation
hangesbyasurfaetermandpossiblybyafuntional
ofg(x). Althoughnoneofthesetermsanaltertheeld
equations,theyanhangetheglobalpropertiesofthe
theory, like the denition of onserved harges. They
also providedierent weightfor topologiallydierent
ongurationsinthequantumtheoryasdenedbythe
path integral. At any rate, the ation I
CS
[A℄ is
gen-uinelygaugeinvariantifthemanifoldhasnoboundary
(M = 0), or the gauge transformation goes to zero
at the boundary fastenough, and thetopology of the
mapping g : M ! G is trivial. These onditionsare
metbygaugetransformationswhihareeverywhere
in-nitesimallylosetotheidentity,e.g.,g=1+ a
(x)T
a ,
with a
(x)<< 1. This issuÆient formost pratial
purposesin thestudy ofCSsystemsaseld theories.
B. Field equations
Now that we have a good ation priniple for the
onnetion A, it is natural to ask, what does it
de-sribe? One way to answer this is to study the eld
equations. Varyingtheationyields
ÆI
CS
[A℄=2 Z
M Tr[T
a T
b ℄F
a
^ÆA b
Z
M Tr[T
a T
b ℄A
a
^ÆA b
:
d
Hereweseethattheonditionofhavinganextreme
underarbitraryvariationsÆA b
implies
F a
=0; (13)
provided thegroupalgebraissuh that
ab
Tr[T
a T
b ℄;
isanon-singularmatrix,whihisalwaystheasefora
semisimpleLie algebrain the adjointrepresentation
and
ab
istheKillingmetri.
Semisimplealgebrasarethosewhihdonotontain
invariantabeliansubalgebras(roughly,thosethat
an-not be written as G = G
0
U, where U is abelian).
Semisimplealgebrasorrespondto thelassialgroups
SO(n),SU(n), Sp(2n),and other moreexotihoies
suhasOSp(n;m),USp(p;q),E
8
,et. Thereisan
im-portantexeptionalLiealgebrawhihis not
semisim-pleand yet there isafaithfulrepresentationforwhih
ab
isnondegenerate. This isthe aseof thePoinare
groupin 2+1dimensions, ISO(2;1), whose algebrais
so(2;1)R 2;1
,whereR 2;1
isthegroupoftranslationsin
2+1dimensions. Thisexeptiontotheruleallows
writ-ingtheEinstein-HilbertLagrangianasaCS3-formfor
thePoinaregroup,andisthekeytothequantizability
ofgravityin2+1dimensions[11℄.
The eld equations(13)look strikinglysimpleand
isawellknownresultaordingtowhihifona
homo-topiallytrivialopensetB,
dA+A^A=0;
then Aanbewritten asagaugetransformationof a
trivialonnetion,
A=g 1
dg everywherein B: (14)
Inother words, by agaugetransformation itis
al-wayspossibletotakeA!0onanysmallpathofM.
Thus, the eld ongurations desribed by the
equa-tion (13) would betrivial unless there are topologial
obstrutionswhihprevent(14)from beingvalid
glob-ally throughout the entire manifold M even if it
re-mains valid on any small open set B. This is in fat
what happens with gravity in 2+1 dimensions, where
theeldequationsareoftheform(13)andstillonean
havenontrivialsolutionssuh asblakholesand
grav-itational ollapse. At any rate, what remains true is
thefatthataCSsystemin2+1dimensionshasno
lo-al degreesoffreedomthatouldpropagate. Inhigher
dimensions,however,CS systemspossesspropagating
C. Generalizationto Higher Dimensions
Inspiteoftheirinterestingmathematialstruture,
these three-dimensionaltheoriesmight seemtoo
unre-alistiasmodelsforourworld. Wearenowgoingtosee
howtheseideasareextendedtohigherdimensions. The
essentialingredient in theonstrution of CS theories
in higherdimensionsistheexisteneofa2n form
Q
2n
(A)=
a1:::an F
a
1
^F a
2
^^F a
n
; (15)
whihislosed,
dQ
2n =0;
andinvariantunderagaugetransformationA!A 0
=
g 1
Ag+g 1
dg,
Q
2n (A
0
)=Q
2n (A):
Itis straightforwardto showthat theinvariantsof
theform
Q
2n
(A)=hF^F^^Fi
| {z }
n-times
satisfytheserequirements,where wehavedened
a1:::an hT
a1 T
a2 T
an i;
and h:::i stands for a trae operation in an
appropri-aterepresentationoftheLiealgebraG. Theinvariants
oftheform(15)areinone-to-oneorrespondenewith
thenth rank invarianttensors
a1:::an
whih ouldbe
onstruted for a given gauge group. The number of
suhtensorsisrathersmallingeneralandisrelatedto
thenumberofCasimirinvariantsofthegroup. (In the
nexttwosetionswewilldisussspeirealizationsof
thesebrakets,somostofthemysterywillbedissipated
shortly.) Fromnowon,wewillnotexpliitlywritethe
wedge symbol unless thereis an ambiguity, sowewill
alsowriteQ
2n as
Q
2n
(A)=hF n
i: (16)
These invariants belong to the family of
hara-teristi lasses known asthe Chern-Weil invariants.
These lasses themselves dene integralinvariants
re-latedtothetopologialpropertiesofthemapsthatan
beestablishedbetweenmanifoldM andthegroupG.
Theequationanalogousto(6)nowreads
dL 2n 1
CS =hF
n
i; (17)
anditssolutionanbewrittenas
L 2n 1
CS =
1
(n+1)! Z
1
0 dt
A(tdA+t 2
A 2
) n 1
+;
(18)
whereisanarbitrarylosed(2n 1)-form(d=0).
Under agaugetransformation of theform (8), the
Chern-Simonsform(18)hangesas
L 2n 1
CS (A
0
)=L 2n 1
CS
(A)+d+( 1) n 1
n!(n 1)!
(2n 1)!
(g 1
dg) 2n 1
; (19)
d
where the(2n 1)-form is a funtion of A and
de-pendsong throughtheombinationg 1
dg. Thus, the
ation
I 2n 1
CS
[A℄= Z
M L
2n 1
CS ;
desribesagaugetheoryforthegroupG,whihunder
a nite gauge transformation hanges as the integral
of (19). The seond term in the RHS of (19) gives
riseto aboundaryterm, andthethird is proportional
to thewinding number. Again,oneanseethatunder
aninnitesimalgaugetransformation(onnetedtothe
identity)of theform
ÆA= r; where<<1;
theationhangesbyasurfaeterm,whihanbeset
III General Relativity as a
Chern-Simons Theory
Sofar,thebestdesriptionofouruniverseatlargesale
isgeneralrelativitywiththeEinstein-Hilbert(EH)
a-tiondened onafour-dimensionalspaetime
I = Z
M4 p
g(R 2)d 4
x; (20)
whereRistheRiisalarurvatureandisthe
os-mologialonstant. Morethansixtyyearsof frustrated
eortstoquantizethistheoryanexplainthe
immedi-ate attention drawn by Witten's lassial observation
thatgravityin2+1spaetimedimensionsisanexatly
quan-thatis,allorrelationfuntionsor,alternatively,its
en-tire Hilbert spae, an be known. This is remarkable
sineinjustonemorespatialdimension,the
quantiza-tion problem beomes intratable. It ould beargued
that quantization of 2+1 gravity is no big deal sine
thetheory hasnopropagatingdegreesof freedomand
thereforeitsquantumdesriptionislikethatofasystem
ofpointpartiles. Althoughthisisertainlyan
impor-tantsimpliation,thekeytotheproofofsolvabilityis
thefatthat2+1gravityisaCSsystemandhene,it
hasallthe niefeatures ofagaugetheory. Gravityin
3+1dimensions,ontheotherhandannotbeonstrued
asagaugesystemofthePoinareor(A-)dSgroups,and
thisisaseriouslimitationforitquantization. Inwhat
followswewillast2+1gravityasaCSsystemandwill
seehowthisanbegeneralizedto higherdimensions.
Thethree-dimensionalEHationanalogousto(20)
an beast as arst order theory, in whih onlyrst
orderderivativesour,byusingformlanguage,
I[!;e℄= 1
2 Z
abd
R ab
1
6 e
a
e b
e
e d
: (21)
HereR ab
istheurvaturetwo-formR a
b =d!
a
b + !
a
!
b
(here the wedge produt symbol (^) has been
sup-pressed),andthedynamialeldsarethevielbeine a
=
e a
(x)dx
, and the Lorentz (spin) onnetion ! ab
=
! ab
(x)dx
. Note that the ation(21) depends onthe
elds and their rst derivatives only (this is a
onse-queneofthefatthatonlyexteriorderivativesareused
throughoutandthereforehigherorderderivativeswith
respettooneoordinateanneverour). Whenthis
ationisvariedwithrespettoe d
and! ab
,thefollowing
eldequationsarefound
abd
R ab
1
3 e
a
e b
e
= 0 (22)
abd T
e d
= 0: (23)
The rst expression are the usual Einstein's
equa-tion, whih relates the urvature to the metri, while
the seond implies vanishing torsion, T a
:= De a
=
de a
+! a
b e
b
= 0, whih is usually postulated in
gen-eral relativity. Solvingthis seondequation for! ab
as
afuntionofe,deande 1
andsubstituting! ab
(e)into
(22) the standard seond order form of the Einstein
equationsisobtained. Theation(21)istherstorder
formulationoftheEHtheoryanditisthemostgeneral
loal 4-form invariant under Lorentz rotations in the
tangentspae,onstrutedoutofthevielbein,thespin
onnetion and their exterior derivatives only [13, 4℄.
This suggeststhat in other dimensions, thesame
pre-sription ould be applied to onstrut the ation for
thegravitationaleld.
A. Beyond the Einstein-Hilbert ation
Inordertodesribethegravitationaleldford>4
oneould assumethespaetime geometry asgiven by
the sameEH ation (20) {withor without
osmolog-ialonstant{ integratednowoverddimensions. One
ofthereasonsfortheuniversalappealoftheEHation
isthatityieldsseondordereld equations,butaswe
already mentioned, other attrative alternatives exist
in higherdimensionsoftheform(1).
TheLanzos-Lovelok(LL)Lagrangianisthemost
generalloal d-form invariant under Lorentz rotations
of the tangent spae, onstruted out of the vielbein,
thespinonnetionandtheirexteriorderivatives
with-outusingotherstrutures[13℄. Inpartiular, the
met-ri,theinversevielbeinortheHodge-*dual 4
arenever
usedin theonstrution,ensuringthat onlyrstorder
eld equations for e and ! an be produed. If the
torsion-freeonditionisassumed,theequationsforthe
metribeomeseondorder.
B. Gravityasan (anti-)deSitterCS
The-ory
TheoeÆients in theLL ation an be made
di-mensionlessby theredenition
p
!
p l
d 2p
, where l
is a parameter with dimensions of length. Then, the
LLLagrangianreads
L= [d=2℄
X
p=0
p l
2p d
a1:::ad R
a
1 a
2
R a
2p 1 a
2p
e a
2p+1
e a
d
:
(24)
Theembarrassingfreedomto hoosethe
p
's
arbitrar-ily anbedrastiallyutbythefollowingobservation:
Theeldequationsforeand! involveR ab
,T a
ande a
.
Takingtheovariantexteriorderivativesofthese
equa-tions and using the Bianhi identities, new algebrai
relationsbetweentheurvatureandtorsiontensorsare
produed, whih would generiallyintrodue
nonholo-nomirestritionsonthedegreesoffreedomofthe
the-ory . This is so for all hoies of
p
's, exept in two
ases, for whih no new onstraints on the geometry
aregenerated[14℄:
4
TheHodge-operationrelatesap-formanda(d p)-formthroughitsationonthebasis
(dx
1
dx
p
) = 1
(d p)! p
jgjg
1
1
g
p
p
1
d dx
p+1
dx
d
d=2n: TheBorn-Infeld(BI)ase,
p =
n
p
;opn (25)
d=2n 1: TheAdSChern-Simons(AdS-CS)
ase,
p =
d 2p
n 1
p
;0pn 1: (26)
Therefereneto BIintherstasestemsfromthefatthat, inthat ase,theLagrangianreads
L =
a1:::ad
R a
1 a
2
+ 1
l 2
e a
1
e a
2
R a
d 1 a
d
+ 1
l 2
e a
d+1
e a
d
;
= pfaff
R ab
+ 1
l 2
e a
e b
; (27)
=
s
det
R ab
+ 1
l 2
e a
e b
;
d
whihis reminisentof LagrangianfortheBorn-Infeld
eletrodynamis
Theodd-dimensionalaseisreferredtoasAdS-CS
beausethat LagrangianisoftheCSfamilyanditan
be ast in aform whih is manifestly invariant under
loalAdStransformations. Thisanbeseenasfollows.
Considerthearrayof1-forms
W AB
=
! ab
l 1
e a
l 1
e b
0
; (28)
where the indies a;b;::: = 1;2;::d and A;B;::: =
1;2;::d+1. It is a simple exerise to show that this
onnetiondenesaurvature2-formgivenby
R AB
= dW AB
+W A
C W
CB
=
R ab
+l 2
e a
e b
l 1
T a
l 1
T b
0
; (29)
wheretheA;B;:::indiesareraisedandloweredusing
theAdSmetri 5
,
AB
=diag ( 1; 1;+1;::::;+1): (30)
TheAdSurvatureR AB
anbeused to onstrutthe
Eulerformin 2ndimensionsas
E
2n =
A
1 :::A
2n R
A1A2
R
A2nd 1A2n
; (31)
whih, by virtue of the Bianhi identity (DR ab
= 0),
an beshownto be losed, dE
2n
= 0. [HereR AB
de-notesthe(d+1=2n)-dimensionalurvature,nottobe
onfusedwiththe (d=2n 1)-dimensionalone,R ab
.℄
Substituting (29)into (31) onendsthe expliitform
forE
2n
intermsofthe(d=2n 1)-dimensionaltensors
undertheSO(2n 2;1)(Lorentz)group,
E
2n =n
a1:::a2n
1 (R
a1a2
+l 2
e a1
e a2
)(R
a2nd 3a2n
2
+l 2
e a2nd
3
e a2n
2
)T a2n
1
:
Now, thesameargumentsapplied tond theCSLagrangianfrom thePontryagindensityanbeapplied now
to the Euler form. Thus, one may ask, what is the (2n 1)-formwhose exteriorderivative gives (31)?. Diret
omputation yieldstheanswerintheform(24),butwith xedoeÆients:
E
2n
= d 0
n [d=2℄
X
p=0 l
2p d
d 2p
n 1
p
a1:::ad R
a
1 a
2
R a
2p 1 a
2p
e a
2p+1
e a
d 1
A
: (32)
= dL AdS
2n 1
: (33)
5
Thatis, theoeÆients
p
givenby(26).
C. Gauge invariane
Wenow show that the LagrangianL AdS
2n 1
is really
invariant under innitesimalSO(d 1;2) (AdS)
rota-tions. Under SO(d 1;1)e a
and! a
b
transformas
! ab
!! 0ab
=d ab
+! a
b
+!
b
a
D
!
ab
e a
!e 0a
= a
b e
b
; (34)
thenthenewurvatureR AB
transformsasatensor
un-deralargergroup. Infat,(34)isapartiularaseof
theSO(d 1;2)transformationsoftheform
W AB
!W 0
AB
=d AB
+W A
C
CB
+W B
C
AC
D
W
AB
;
(35)
wherewehavedened
AB
=
ab
l 1
a
l 1
b
0
: (36)
These transformations inlude, besides the Lorentz
transformations (determined by ab
), also \AdS
boosts" dened by a
. Setting a
= 0 in (36), (34)
isobtained. Therefore,the(d+1)-formE
2n
isinvariant
under the full Anti-de Sitter group. This in turn
im-plies that theCS Lagrangianobtainedfrom theEuler
formisalsoinvariantunderloalAdStransformations.
Itisthemagiofthehoie(26)thatthegroupofloal
invarianesof theation hasgrownfrom SO(d 1;1)
toSO(d 1;2).
Finally, the Lagrangian in d = 2n 1 dimensions
reads
L
2n 1 =
n 1
X
p=0 1
D 2p
n 1
p
l 2p D
a
1 a
2n 1 R
a1a2
R a2p 1a2p
e a2p+1
e a2n
1
: (37)
d
Hereisthegravitationalonstantsimilarto
New-ton'sonstantind=4and anbeshown to be
quan-tizedin this theory [15℄. Aremarkablefeature of this
Lagrangianisthat ifwrittenin termsof theAdS
on-netion W AB
, it has no dimensionful onstants, and
thismakesitagoodandidateforarenormalizableeld
theory. However,thereisanobsurepointherebeause
the vauum of the theory seems to be dened by the
AdS-at ongurationW AB
=0andthismeansthat
both! ab
and e a
mustvanish, asituation hardto
re-onilewith ameaningfulspaetimeinterpretation.
D. Field Equations
The eld equations obtained varying with respet
to W AB
are
A
1 :::A
2n R
A3A4
R
A2nd 1A2n
=0;
or,equivalentlyvaryingwithrespetto! ab
ande a
Æ! ab
:
aba3:::a2n
1 (R
a3a4
+l 2
e a3
e a4
)(R a
2nd 3 a2n
2
+l 2
e a
2nd 3
e a2n
2
)T a2n
1
=0
Æe a
:
aa2:::a2n
1 (R
a2a3
+l 2
e a2
e a3
)(R
a2nd 2a2n
1
+l 2
e a2nd
2
e a2n
1
) =0
: (38)
d
Congurations with vanishing loal AdS urvature
(R AB
= 0) are obvious solutions of these equations,
while torsion-free spaes with at Lorentz urvature
(T a
=0=R ab
)arenot. Otherlesstrivialsolutionsare
torsion-freespaeswith(2n 2)-dimensional
submani-foldsofonstanturvaturefoliatedalongonediretion,
suhasblakholesorosmologialsolutions[16℄.
E. Gravity in 2+1 Dimensions
As in all CS systems, the 2+1dimensional ase is
CS Lagrangian whose exteriorderivativeis the Euler
density in 4 dimensions is the standard EH ation in
threedimensions,
L
2+1 =
l
ab (R
ab
+ 1
3l 2
e a
e b
)e
: (39)
Theorrespondingeldequationsare
ab (R
ab
+ 1
l 2
e a
e b
) = 0; (40)
T a
= 0: (41)
that the spaetime must have onstant urvature at
eahpoint. Inviewofthefat that (2+1) gravityhas
nopropagatingdegreesoffreedom,onstanturvature
spaetimes were thought to be rather dull
ongura-tions. However,oneanbesurprised by thefat that
solving(40)expliitlyforspheriallysymmetri,stati
ongurations doesnot neessarilyprodue a globally
AdS spaetime, but an AdS spaetime with
identia-tionsaswell. This isbeausetheeldequations only
refertoloalpropertiesofspaetimeanddonotrestrit
theglobaltopologyfurther. Thus,thespaetime
mani-foldanbeutandpasted,identifyingpointsonneted
by anite isometryalongaKillingvetor{aswhen
onemakes aylinder outof aplane{, oneshould still
haveasolution 6
. Infat,oneanprodueablakhole
inthis fashion. Whatisevenmoreremarkable,isthe
fatthatoneangenerateasolutionbyaLorentzboost
thatsetstheblakholeinrotationaboutitssymmetry
axis. [17℄
A good example of a nontrivial solution of (40) is
the2+1blakholegeometry[18,19℄,
ds 2
= N(r) 2
f(r) 2
dt 2
+f(r) 2
dr 2
+r 2
(d'+N '
(r)dt) 2
;
0r<1; 0'<2; t
1
tt
2
: (42)
d
SolvingtheEinstein equationsyields
f 2
= r 2
M+ J
2
4r 2
(43)
N = N(1) (44)
N '
= J
2
1
r 2
+N '
(1)
; (45)
whihdenesablakholeofmassM,angular
momen-tum J,providedJ 2
M
2
.
F. Exoti Gravity
ThePontryaginform anbedened foranygroup,
and in partiular this is also true of the Lorentz and
AdS groupstoo. The Lorentz-Pontryaginform is
de-nedas
P Lor
=R a1
a2 R
a2
a3 R
an
a1
; (46)
andtheAdS-Pontryaginformas
P AdS
=R A
1
A
2 R
A
2
A
3 R
A
n
A
1
: (47)
Beausetheseurvaturetwo-formsare
antisymmet-riin theirindies, itisobviousthattheyanonlybe
dened for even n. This means that they are
natu-rallyonstrutedin4kdimensions. Thisinturnmeans
that bothAdSandLorentzChern-Simonstheories
as-soiated with thePontryaginfamily an only exist in
4k _
1dimensions, these are the so-alled exoti
La-grangians, whih are independent from theEulerCS
forms disussedaboveind=2n 1.
IV Chern-Simons Supergravity
Supersymmetry is the only nontrivial way to extend
spaetimesymmetries. Thisresultwaswellknown
be-forethe era of supersymmetryand it statesthat in a
loal eld theorywhose S-matrix relatesin- and
out-eigenstates of energy and momentum, all other
\in-ternal"quantum numberssuh asolor,avor,
hyper-harge,et.,mustbespaetimesalars. Inotherwords,
under spaetime transformations these labels do not
transform. Inmathematialtermsthis meansthatthe
groupofinvarianesoftheS-matrixmustbeoftheform
G=SI,whereS isthegroupofspatial
transforma-tions(e.g., Poinare,AdS, et.), and I is theinternal
symmetrygroup.
Forsometimeitwashopedthatthe
nonrenormaliz-abilityofGRouldbeuredbyitssupersymmetri
ex-tension.However,theinitialhopesraisedby
supergrav-ity(SUGRA)asamehanismfortamingthe
ultravio-letdivergenesofpuregravityeventuallyvanishedwith
therealizationthat SUGRAswouldbe
nonrenormaliz-able as well [20℄. Again, one an see that, like GR,
SUGRA isnotagaugetheory foragroupora
super-group, and that the loal (super-) symmetry algebra
loses naturally only on shell. The algebra ould be
madetoloseoshellbyfore,attheostof
introdu-ingauxiliaryelds {whih arenotguaranteed toexist
foralldandN and[21℄{,andstillthetheorywouldnot
6
Careshouldbetaken,howevernot toproduelosedtimelikeurvesthatouldgenerateparadoxialspaetimessenarios(where
haveaberbundlestruturesinethebasemanifoldis
identiedwithpartoftheber.
Whetherit isthelakofberbundlestruturethe
ultimate reason for the nonrenormalizability of
grav-ityremainsto beproven. Itisertainlytrue,however,
that ifGRouldbeformulatedasagaugetheory,the
hanes forprovingitsrenormalizabilitywouldlearly
grow.
In three spaetime dimensions, onthe other hand,
bothGR andSUGRA dene renormalizablequantum
theories. Itisstronglysuggestivethat preiselyin 2+1
dimensions both theories an also be formulated as
gaugetheories ona berbundle. It ould bethought
thattheexatsolvabilitymiraleisdue totheabsene
ofpropagatingdegreesoffreedominthree-dimensional
gravity, but nal the power ounting
renormalizabil-ityargumentrests ontheberbundlestrutureofthe
Chern-Simonsform ofthosesystems.
Thereareotherknownexamplesofgravitation
the-ories in odd dimensions whih are genuine (o-shell)
gaugetheoriesfortheanti-deSitter(AdS)orPoinare
groups[22, 23,24,15℄. Thesetheories,aswellastheir
supersymmetriextensionshavepropagatingdegreesof
freedom[25℄andareCSsystemsfortheorresponding
groupsasshownin[26℄.
A.From RigidSupersymmetryto
Super-gravity
Rigid SUSY an be understood asan extension of
the Poinarealgebra by inluding superhargeswhih
arethe\squareroots"ofthegeneratorsofrigid
trans-lations, f
Q;Qg P. This idea an be extended
to loal SUSY substituting themomentum P
=i
bythe generatorsof dieomorphisms, H , andrelating
them to thesuperhargesbyf
Q;Qg H . The
re-sultingtheoryhason-shellloalsupersymmetryalgebra
[27℄.
Analternativeapproah{whihwewouldliketo
ad-voatehere{istoonstrutthesupersymmetryon the
tangentspaeandnotonthe basemanifold. Thispoint
ofviewis morenaturalifonereallsthat spinors
pro-vide abasis of irreduible representations for SO(N),
and not for GL(N). Thus, spinors are naturally
de-ned relative to a loal frame on the tangent spae
rather than on the oordinate basis. The basi point
is to reprodue the 2+1 \mirale" in higher
dimen-sions. This idea has been suessfully applied in ve
dimensions[23℄,andforpuregravity[24,15℄andto
su-pergravity[28,26℄. TheSUGRAonstrutionhasbeen
arriedoutforspaetimeswhosetangentspaehasAdS
symmetry[28℄,andforitsPoinareontrationin[26℄.
B. Assumptions of Standard
Supergrav-ity
Threeimpliitassumptionsareusuallymadein the
onstrutionofstandardSUGRA:
(i) The fermioni and bosoni elds in the
La-grangianshould ome in ombinationssuh that their
propagating degrees of freedom are equal in number.
Thisisusuallyahievedbyaddingto thegravitonand
thegravitinianumberoflowerspinelds(s<3=2)[27℄.
Thismathing,however,isnotneessarilytruein AdS
spae,norinMinkowskispaeifadierent
representa-tion of thePoinaregroup (e.g., the adjoint
represen-tation)isused[29℄.
Theothertwoassumptionsonernthepurely
grav-itationalsetor. TheyareasoldasGeneralRelativity
itself and are ditated by eonomy: (ii) gravitonsare
desribed by the Hilbert ation (plus a possible
os-mologialonstant),and, (iii)thespinonnetionand
thevielbein arenotindependentelds but arerelated
throughthetorsionequation. Thefatthatthe
super-gravitygeneratorsdonotformalosedo-shellalgebra
anbetraedbakto theseassumptions..
The proedure behind (i) is tightly linked to the
idea that the elds should be in a vetor
representa-tion of thePoinaregroup andthat thekineti terms
and ouplingsaresuh that theountingofdegreesof
freedom workslike in a minimally oupledgauge
the-ory. This assumption omes from the interpretation
ofsupersymmetristatesasrepresentedbythein-and
out-plane wavesin anasymptotiallyfree, weakly
in-terating theoryin aminkowskianbakground. These
onditionsarenotneessarilymetbyaCStheoryinan
asymptotially AdS bakground. Apart from the
dif-ferene in bakground, whih requires aareful
treat-ment of the unitary irreduible representations of the
asymptotisymmetries[30℄, theountingofdegreesof
freedominCStheoriesisompletelydierentfrom the
ountingforthesameonnetion1-formsinaYM
the-ory.
Thus, the only natural extension for a gauge
the-ory of the AdS spaetime issupergravity, onstruted
enlarging thegroupto somesupergroups that ontain
AdS but are otherwise semisimple. The onstrution
of thesetheories anbefound in[31℄. Theruial
ob-servation is that theDira matries provide anatural
representation of the AdS algebra in any dimension,
thus, the onnetionW AB
anbewritten in this
rep-resentationasW =e a
J
a +
1
! ab
J
J a = 1 2 ( a ) 0 0 0 ; (48) J ab = 1 2 ( ab ) 0 0 0 : (49)
Thisspinorialrepresentationfortheonnetion
nat-urally leads to a representation for a superalgebra
whose generatorshave entries in the remainingbloks
of similar matries. The non zero bloks in (48) and
(49) are mm where m = 2 [d=2℄
is the number of
omponents of aspinor in d dimensions. The algebra
is ompleted by the(pseudo-)Majoranageneratorsof
supersymmetry, Q k = 0 Æ Æ k j C u k i 0 ; (50)
whih arein avetorrepresentation(desribed by the
index k)ofsomeinternal symmetrygroupwith
gener-ators M k l = 0 0 0 (m k l ) i j : (51)
Following[31℄,onearrivesatallthepossible
superalge-brasindimensionsd,exeptford=5mod _
4. Inthose
asestherepresentationsareneessarilyomplex. Then
the pseudo-Majorana ondition hasto be relaxedand
thegeneratorsofsupersymmetrytaketheform
Q l = 0 Æ Æ l j 0 0 ; (52) Q k = 0 0 G Æ i k 0 ; (53)
instead of (50). With this, onean write the algebra
foranyd. Theonlynontrivialonditionforthelosure
ofthesuperalgebraomesfromtheantiommutatorof
twosupersymmetrygenerators. Insomeasesthis
an-tiommutator inludes all generators of the Pauli
al-gebra, (k ) = 1 k ! Æ a 1 a k b1bk b1 bk
; 0 k d (for
d = 5 mod 4), sometimes it ontains only the
sym-metrisubalgebra, (C
(k ) )
T
=C
(k )
(ford=2;6;7;8
mod 8), and sometimes it inludes only the
antisym-metriones,(C
(k ) )
T
= C
(k )
(ford=2;3;4;6mod
8). The fat that d = 2 and 6 appear in both
fami-liesis duetothefat thatin thoseasestherearetwo
inequivalent hoies of harge onjugationmatrix (C)
withC T
=C.
Wenowonsider thesupersymmetriextensionsof
the loally AdS theories dened above. Inpartiular,
oneanwritethe\exoti"Lagrangian,
d b L 4k 1 = 1 4k Tr[(R AB AB ) 2k ℄: (54)
whihisapartiularformofthePontryaginform(47).
OtherpossibilitiesoftheformTr[F n p
℄Tr[F p
℄,arenot
neessarytoreproduetheminimalsupersymmetri
ex-tensionsof AdS ontainingthe Hilbert ation. In the
supergravitytheoriesdisussedbelow,thegravitational
setorisgivenby
dL
2n 1
=STr[F
dL Grav 2n 1 = 1 2 4k Tr[(R AB AB ) 2k ℄ (55) = 1 2 n L AdS
G2n 1 1
2 L
AdS
T2n 1
: (56)
whihisapartiularformof (47)wherethetraeover
spinorindiesinthisrepresentation. Otherpossibilities
oftheform F n p h F p
i,arenotneessarytoreprodue
theminimalsupersymmetriextensionsofAdS
ontain-ingtheHilbert ation. Thesigns orrespondto the
twohoiesofinequivalentrepresentationsof 's,whih
in turn reet thetwohiralrepresentations in d+1.
As in the three-dimensional ase, the supersymmetri
extensions of L
2n 1
or any of the exoti Lagrangians
suhas b
L
2n 1
,requireusingbothhiralities,thus
dou-bling the algebras. Here wehoose the+ sign, whih
givestheminimalsuperextension.
Under a gauge transformation, A transforms by
ÆA=r, wherer isthe ovariantderivativefor the
sameonnetionA. Inpartiular, under a
supersym-metrytransformation,= i Q i Q i i ,and Æ A= k k k k D j D i i j i j ; (57)
whereDistheovariantderivativeonthebosoni
on-netion,
D
j =(d+
1 2 [e a a + 1 2 ! ab ab + 1 r! b [r℄ [r℄ ℄) j a i j i :
Twointerestingasesanbementionedhere:
C. d=5 SUGRA
InthisasethesupergroupisU(2;2jN). The
asso-iatedonnetionanbewrittenas,
where the generators J
a , J
ab
, form an AdS algebra
(so(4;2)), T
( = 1;N 2
1) are the generators
of su(N), Z generatesa U(1) subgroup and Q;
Q are
the supersymmetry generators, whih transform in a
vetor representation of SU(N). The Chern-Simons
Lagrangianforthisgaugealgebraisdened bythe
re-lationdL=iSTr[F 3
℄,whereF=dA+A 2
isthe
(anti-hermitean)urvature. Usingthisdenition,oneobtains
theLagrangianoriginallydisussedbyChamseddinein
[23℄,
L=L
G (!
ab
;e a
)+L
su(N) (a
r
s )+L
u(1) (!
ab
;e a
;b)+L
F (! ab ;e a ;a r s ;b; r ); (58) with L G = 1 8 abde R ab R d e e =l+ 2 3 R ab e e d e e =l 3 + 1 5 e a e b e e d e e =l 5 L su(N)
= Tr a(da) 2 + 3 2 a 3 da+ 3 5 a 5 L u(1) = 1 4 2 1 N 2 b(db) 2 + 3 4l 2 T a T a R ab e a e b l 2 R ab R ab =2 b + 3 N f r s f s r b L F = 3 2i r Rr r + s F r s r r +:: ; (59) wherea r s a ( ) r s
isthesu(2;2)onnetion,f r
s
isitsurvature,andthebosonibloksofthesuperurvature:
R= 1 2 T a a + 1 4 (R ab +e a e b ) ab + i 4 dbI 1 2 s s ;F r s =f r s + i N dbÆ r s 1 2 r s : d
Theosmologialonstantis l 2
;andtheAdS
o-variantderivativer atingon
r is r r =D r + 1 2l e a a r a s r s +i 1 4 1 N b r : (60)
whereDistheovariantderivativeintheLorentz
on-netion.
The aboverelationimpliesthat the fermionsarry
a u(1) \eletri" harge given by e = 1 4 1 N . The
purely gravitational part, L
G
is equal to the
stan-dard Einstein-Hilbert ation with osmologial
on-stant,plusthedimensionallyontinuedEulerdensity 7
.
The ation is by onstrution invariant {up to a
surfaeterm{undertheloal(gaugegenerated)
super-symmetry transformationsÆ
A = (d+[A;℄) with
= r Q r Q r r ,or Æe a = 1 2 r a r r a r Æ! ab = 1 4 r ab r r ab r Æa r s = i r s r s Æ r = r r Æ r = r r
Æb = i
r r r r :
Asan be seenfrom (59) and(60), for N =4the
b eld looses its kineti term and deouples from the
fermions(thegravitinobeomesunhargedwithrespet
totheU(1)eld). Theonlyremnantoftheinteration
withthebeldisadilaton-likeouplingwiththe
Pon-tryaginfour formsfor theAdSand SU(N)groups(in
thebosonisetor). AsitisalsoshownintheAppendix
A,theaseN =4isalsospeialatthelevelofthe
alge-bra,whih beomesasuperalgebrawithau(1)entral
extension.
Inthebosonisetor,forN =4,theeldequation
obtainedfromthevariationwithrespettobstatesthat
the Pontryagin four form of AdS and SU(N) groups
are proportional . Consequently, ifthe urvatures
ap-proah zero suÆiently fast at spatial innity, there
is a onserved topologial urrent whih states that,
for the spatial setion, the seond Chern haraters
of AdS and SU(4) are proportional. Consequently, if
the spatial setion has no boundary, the
orrespond-ing Chern numbers are related. Using the fat that
4
(SU(4))=0,theaboveimpliesthat theHirzebruh
signatureplustheNieh-Yannumberofthespatial
se-tionannothangein time.
d=11 SUGRA
In this ase, the smallest AdS superalgebra is
osp(32j1)andtheonnetionis
A= 1 2 ! ab J ab +e a J a + 1 5! b abde J abde + Q ; 7
TherstterminL
G
isthedimensionalontinuationoftheEuler(orGauss-Bonnet)densityfromtwoandfourdimensions,exatly
asthethree-dimensionalEinstein-HilbertLagrangianistheontinuationofthethetwodimensionalEulerdensity. Thisistheleading
where b is a totally antisymmetri fth-rank Lorentz
tensor one-form. Now, in terms of the elementary
bosoniandfermionields,theCSformin(15)reads
L osp(32j1)
11
(A)=L sp(32)
11
()+L
F
(; ); (61)
where
1
2 (e
a
a +
1
2 !
ab
ab +
1
5! b
abde
abde )
isan sp(32)onnetion. Thebosonipartof (61)an
bewrittenas
L sp(32)
11
()=2 6
L AdS
G11 (!;e)
1
2 L
AdS
T11
(!;e)+L b
11
(b;!;e):
ThefermioniLagrangianis
L
F
= 6(
R 4
D ) 3
(D
D )+(
R )
(
R 2
D ) (62)
3
(
R 3
)+(D
R 2
D )
(
D )+ (63)
2
(D
D ) 2
+(
R ) 2
+(
R )(D
D )
(
D ); (64)
d
where R =d+ 2
is the sp(32) urvature. The
su-persymmetrytransformations(57)read
Æe a
= 1
8
a
Æ! ab
= 1
8
ab
Æ =D Æb
abde
= 1
8
abde
:
Standard eleven-dimensional supergravity [32℄ is
an N=1 supersymmetri extension of Einstein-Hilbert
gravitythat annot aommodate aosmologial
on-stant [33, 34℄. An N > 1 extension of this theory is
not known. In our ase, the osmologial onstantis
neessarily nonzeroby onstrutionand the extension
simplyrequiresinludinganinternalso(N)gaugeeld
oupled to thefermions,and theresultingLagrangian
isanosp(32jN)CSform [35℄.
E. Summary
Thesupergravitiespresentedherehavetwo
distin-tivefeatures: Thefundamental eldis alwaysthe
on-netionAand,intheirsimplestform,theyarepureCS
systems (matterouplings are disussed below). As a
result,thesetheoriespossessalargergravitational
se-tor, inluding propagating spin onnetion. Contrary
to what one ould expet, thegeometrial
interpreta-tion isquitelear, theeld strutureissimpleand, in
ontrast with the standard ases, the supersymmetry
transformationsloseoshellwithoutauxiliaryelds.
Torsion. It an be observed that the torsion
La-grangians (L
T
)are odd while the torsion-free terms
(L
G
)areevenunderspaetimereetions. Theminimal
supersymmetri extension of theAdS groupin 4k 1
dimensionsrequiresusinghiralspinorsofSO(4k)[30℄.
This in turn impliesthat the gravitational ation has
no denite parity,but requires the ombinationof L
andL
G
asdesribed above. In D =4k+1this issue
doesn'tariseduetothevanishingofthetorsion
invari-ants,allowingonstrutingasupergravitytheorybased
onL
G
only, asin [23℄. If one tries to exlude torsion
termsin4k 1dimensions,oneisforedtoallowboth
hiralitiesforSO(4k)dupliatingtheeldontent,and
theresultingtheoryhastwoopiesofthesamesystem
[36℄.
Field ontent and extensions with N>1. The
eldontentompareswiththatofthestandard
super-gravitiesin D=5;7;11asfollows:
D Standardsupergravity CSsupergravity
5 e a
e a
!
ab
b
7 e a
A
[3℄ i
a
i
j
e
a
!
ab
i
a
i
j
11 e a
A
[3℄
e a
!
ab
b
abde
Standardsupergravityinvedimensionsis
dramat-iallydierent from the theory presented here, whih
wasalsodisussedbyChamseddinein [23℄.
Standardseven-dimensionalsupergravityisanN =
2theory(itsmaximal extensionisN=4), whose
gravi-tationalsetorisgivenbyEinstein-Hilbertgravitywith
osmologialonstantandwithabakgroundinvariant
under OSp(2j8) [37,38℄. Standard eleven-dimensional
supergravity[32℄ isanN=1 supersymmetri extension
ofEinstein-Hilbertgravitythatannotaommodatea
osmologialonstant[33,34℄. An N >1extension of
thistheoryisnotknown.
Intheasepresentedhere,theextensionstolarger
N are straightforward in any dimension. In D = 7,
theindex iisallowedto runfrom2to2s,andthe
mustinludeaninternalso(N)eldandtheLagrangian
is an osp(32jN) CS form [28℄. The osmologial
on-stantisneessarilynonzeroin allases.
Spetrum. Thestabilityand positivity ofthe
en-ergyforthesolutionsofthesetheoriesisahighly
non-trivialproblem. As shown in Ref. [25℄, thenumberof
degrees of freedom of bosoni CS systems for D 5
is not onstant throughout phase spae and dierent
regionsanhaveradiallydierentdynamialontent.
However,in aregion wheretherankofthe sympleti
formismaximalthetheorybehavesasanormalgauge
system, and this ondition is stable under
perturba-tions. As it is shown in [39℄, there exists a
nontriv-ialextensionoftheAdSsuperalgebrawithoneabelian
generatorforwhihanti-deSitterspaewithoutmatter
elds isabakgroundof maximalrank,andthegauge
superalgebra isrealizedin the Dirabrakets. For
ex-ample, forD=11and N =32, theonlynonvanishing
antiommutatorreads
fQ i
;
Q j
g =
1
8 Æ
ij
C a
J
a +C
ab
J
ab +C
abde
Z
abde
(65)
M ij
C
; (66)
d
whereM ij
arethegeneratorsofSO(32)internalgroup.
On this bakground the D = 11 theory has 2 12
fermioniand2 12
1bosonidegreesof freedom. The
(super)hargesobeythesamealgebrawithaentral
ex-tension. This fat ensuresalowerbound forthemass
asafuntionoftheotherbosoniharges[25℄.
Classialsolutions. Theeld equationsforthese
theories in termsof theLorentz omponents (!, e, b,
a, )arespread-outexpressionsfor<F n 1
G
(a) >=0,
where G
(a)
are the generators of thesuperalgebra. It
is rather easy to verify that in all these theories the
anti-de Sitter spae is a lassial solution , and that
for = b =a = 0 there exist spheriallysymmetri,
asymptotially AdS standard[16℄, as well as
topolog-ial [40℄ blak holes. Inthe extreme ase these blak
holesanbeshownto beBPSstates.
Matter ouplings. It is possible to introdue
a minimal ouplings to matter of the form AJ. For
D = 11, the matter ontent is that of a theory with
(super-)0,2,and5{branes,whoserespetive
worldhis-toriesoupletothespinonnetionandtheb elds.
Standard SUGRA. Some setor of these
theo-ries might be related to the standard supergravities
if one identies the totally antisymmetri part of the
ontorsiontensor inaoordinatebasis,k
,withthe
abelian 3-form, A
[3℄
. In11 dimensions oneould also
identifytheantisymmetripartofbwithanabelian
6-form A
[6℄
, whoseexteriorderivative, dA
[6℄
, is thedual
ofF
[4℄ =dA
[3℄
. Hene,in D=11theCStheory
possi-blyontainsthestandardsupergravityaswellassome
kindofdualversionofit.
V Dynamial Contents of
Chern Simons Theories
Thephysialmeaningofatheoryisdenedbythe
dy-namisitdisplaysbothatthelassialaswellasatthe
quantum levels. In order to understand this question
oneshould beableto separate thephysialdegreesof
freedomfromthosewhihareredundant. Inpartiular,
itshouldbepossible{atleastinpriniple{toseparate
thepropagatingmodesfrom thegaugedegreesof
free-dom,andfromthosewhihdonotevolveindependently
at all(seond lassonstraints). The standardwayto
proeedisDira'sonstrainedHamiltoniananalysisand
hasbeenstudiedin CSsystemsin[25℄. Herewe
sum-marizethisanalysisbutreferthereadertotheoriginal
papersforthedetails.
The BGH onstrution
Fromthedynamialpointofview,aCSsysteman
bedesribedbyaLagrangianoftheform 8
L
2n+1 =l
i
a (A
b
j )
_
A a
i A
a
o K
a
; (67)
wherethe(2n+1)-dimensionalspaetimehasbeensplit
into spaeandtime, and
K
a =
1
2 n
n
aa1::::an
i
1 :::i
2n
F a1
i1i2 F
an
i2n 1i2n :
Theeldequationsare
ij
ab (
_
A b
j D
j A
b
0
) = 0; (68)
K
a
= 0; (69)
where
ij
ab =
Æl j
b
ÆA a
i Æl
i
a
ÆA b
j
(70)
= 1
2 n 1
aba
2 ::::a
n
iji
3 :::i
2n
F a2
i
3 i
4 F
an
i
2n 1 i
2n
is thesympletimatrix. Thepassageto the
Hamilto-nianhastheproblemthattheveloitiesappearlinearly
in the Lagrangian, ant therefore there is a numberof
primaryonstraints
i
a p
i
a l
i
a
0: (71)
Besides these there are the seondary onstraints
K
a
0,whih anbeombinedwith the's into the
expressions
G
a
K
a +D
i
i
a
; (72)
whihtogetherwiththe'sformalosedalgebra,
f i
a ;
j
b
g =
ij
ab
f i
a ;G
b
g =C
ab
i
fG
a ;G
b g =C
ab G
;
where C
ab
are thestrutureonstantsofthegauge
al-gebra of thetheory. Clearly theG's form arstlass
algebrawhih reetsthegaugeinvarianeofthe
the-ory, while some of the 's are seond lass and some
are rst lass, depending onthe rank of the
symple-ti form. Here we fae the rstserious diÆulty with
CS theory: the matrix ij
ab
depends on the eld
on-gurations, and therefore its rank annot be thought
ofasaonstant,but it anhange fromoneregion of
phasespaetoanother. Thisissuehasbeenanalyzedin
theontext of somesimplied mehanial models and
theonlusionisthatthedegenerayofthesystem
o-ursatsurfaesoflowerdimensionalityinphasespae,
whih an be viewed as end sets of (unstable) initial
pointsorsets of (stable) end points forthe evolution.
Unless the system is haoti, it an be expeted that
generiongurations where the rank of ij
ab
is
maxi-malshouldllmostofphasespae[12℄.
Thereisaseondmoretratableproblemandthat
ishowtoseparatetherstandseondlassonstraints
among the 's. In Ref.[25℄ the following results are
shown:
The maximal rank of ij
ab
is 2nN 2n , where
N is the number of generatorsin the gauge Lie
algebra.
Inonsequene,thereare2nrstlassonstraints
amongthe'swhihorrespondtothegenerators
ofspatialdieomorphisms(H
i ).
ThegeneratoroftimelikereparametrizationsH
?
isnotanindependentrstlassonstraint.
Puttingallthesefatstogether, oneonludesthat
in a generi onguration, the number of degrees of
freedomof thetheoryis
g = (n Æ
ofoordinates) (n Æ
of1 st
lassonstraints) 1
2 (n
Æ
ofseondlassonstraints)
= 2nN (N+2n) 1
2
(2nN 2n)=nN N n: (73)
d
Thisresultissomewhatperplexing. Forthegravity
theory as a CS system for the AdS group, this result
gives,in d=2n+1dimensionshas
N =
(2n+1)(2n+2)
2
=(2n+1)(n+1);
andtherefore,
g CS
AdS
= n(2n+1)(n+1) (2n+1)(n+1) n
= 2n 3
+n 2
3n 1: (74)
Thisnumberofpropagatingdegreesoffreedomismuh
ofgravityin dimensiond.
g metri
d
=
d(d 3)
2 =
(2n+1)(2n 2)
2
= 2n 2
n 1: (75)
Thus, for d 3the number ofdegrees of freedom
oftheCStheory ismuhlargerthanthat ofthe
stan-dardmetritheory. Inpartiular,ford=5,(74)is13,
while(75)equals5. Obviously,theextradegreesof
free-dommustorrespondto propagatingmodesontained
in the torsion, whih here are independent from the
metridegreesoffreedom. Thepreiseidentiationof
sep-theDiramatrixandeliminatingtheseondlass
on-straintsonsistently. Thisobjetiveseemsunattainable
ingeneralforanarbitrarygaugegroup.
As it is also shown in [25℄, an important
simpli-ationourswhen thegrouphasaninvariantabelian
fator. Inthat asethesympletimatrix ij
ab
takesa
partiallyblok-diagonalform wherethekernelhasthe
maximal size allowed by a generi onguration. For
example, in ve dimensions, those authors show that
foragroupG=G
0
U(1), onehasageneri
ongu-rationiftheurvaturefortheG
0
onnetionvanish,but
theU(1)urvature,f =db,isanondegenerate2-form,
ij
ab
F a
=0 =
0 0
0 1
2
ab
ijk l
f
k l
:
Then, the lowerblok of ij
ab
is the Diramatrix and
the onstrained Hamiltonian analysis an proeed in
thestandardway.
ItisaniesurpriseintheasesofCSsupergravities
disussed above that it seems that for ertain unique
hoies extendedsupesymmetriesthealgebrasdevelop
anabeliansubalgebraandmaketheseparationofrst
and seondlass onstraintspossible. It isremarkable
that insomeases(e.g., ford=5,N =4)thealgebra
isnotadiretsumbutanalgebrawithanabelian
en-tralextension. Inotherases(e.g.,ford=11,N=32),
the algebrais adiret sum, but theabeliansubgroup
is not put in by hand but it is asubset of the
gener-atorsthat forthat partiularextendedsupersymmetry
spontaneouslydeouplesfrom therest ofthealgebra.
Final Comments
Chern-Simonstheoriesontainawealthofother
in-terestingfeatures, startingwith its relation to
geome-tryandeldtheoriesandknotinvariants. The
higher-dimensional CS systemsremain somewhat mysterious
espeially beause of the diÆulties to treat them as
quantum theories. However, they have many
ingre-dients that makethem likely models to be quantized:
Theyarrynodimensionfulouplings;theonly
param-eterstheyanhavemustbequantized;ingravitythese
aretheonly theoriesoftheLovelokfamilythat gives
riseto blakholes with positive spei heat [41℄ and
hene, apable to reah thermal equilibrium with an
externalheatbath.
Eorts to quantize CS systems seem promising at
leastin theasesinwhihthespaeadmitsaomplex
struturesothat thesympletiformanbeastasa
Kahlerform[42℄.
Aknowledgments
The author is grateful to R. Aros, M. Henneaux,
Tronoso,formanyenlighteningdisussionsandhelpful
omments. Thisworkwassupportedin partbygrants
1980788 and 1990189 from FONDECYT (Chile), and
27.953/ZI-DICYT (USACH). Institutional support to
CECSfromFuerzaAereadeChile,I.Muniipalidadde
LasCondes,I.MuniipalidaddeSantiago,andagroup
of Chilean ompanies(AFP Provida, Business Design
Assoiates, CGE, Codelo, Cope, Empresas CMPC,
Gener SA, Minera Collahuasi, Minera Esondida,
No-vagasandXerox-Chile)isalso reognized. CECS isa
Millennium SieneInstitute.
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