Chern-Simons Supergravities with Off-Shell Local Superalgebras

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by  Ricardo Troncoso

Chern-Simons Supergravities with Off-Shell Local Superalgebras ∗
Ricardo Troncoso and Jorge Zanelli
†
Centro de Estudios Cient´ıficos de Santiago, Casilla 16443, Santiago 9, Chile
and
Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile.
The final answer to these questions is beyond the scope
Anewfamilyofsupergravitytheoriesinodddimensionsis
of this paper, howeverone can note a difference between
presented. TheLagrangian densitiesareChern-Simonsforms
YM and GR which might turn out to be an important
fortheconnectionofasupersymmetricextensionoftheanti-
clue: YM theory is defined on a fiber bundle, with the
deSitteralgebra. Thesuperalgebras are thesupersymmetric
connectionasthe dynamicalobject,whereasthe dynam-
extensions of the AdS algebra for each dimension, thus com-
ical fields of GR cannot be interpreted as components of
pleting the analysis of van Holten and Van Proeyen, which
9 a connection. Therefore, gravitation does not lend itself
9 was valid for N = 1 and for D = 2,3,4,mod 8. The Chern-
naturally for a fiber bundle interpretation.
9 Simons form of the Lagrangian ensures invariance under the
The closest one could get to a connection formulation
1 gaugesupergroupbyconstructionand,inparticular,underlo-
forGRisthe Palatiniformalism,with the Hilbertaction
cal supersymmetry. Thus, unlike standard supergravity, the
n
local supersymmetry algebra closes off-shell and without re-
a
J quiringauxiliaryfields. TheLagrangianisexplicitlygivenfor I[ω,e]= ǫabcdRab∧ea∧eb, (1)
D = 5, 7 and 11. In all cases the dynamical field content Z
0
3 iinticnliu(dψesµi)t,haenvdieslobmeiene(xetaµr)a,bthoesosnpiicn“cmonantteecrt”iofine(ldωsµabw),hiNchgvraarvy- weahiesreaRloacbal=ordtωhaobn+oωrmca∧alωfbcraismteh.eTcuhrivsaatcutrioentwisos-foomrmet,imaneds
1 from one dimension to another. The superalgebras fall into
v threefamilies: osp(m|N)forD=2,3,4,mod8,osp(N|m)for claimedto describe agaugetheoryforlocaltranslations.
3 D=6,7,8,mod8,andsu(m−2,2|N)forD=5mod4,with However, in our view this is a mistake. If ω and e were
0 m = 2[D/2]. The possible connection between the D = 11 the components of the Poincar´e connection, under local
0 case and M-Theory is also discussed. translations they should transform as
2
Abstract
0 δωab =0, δea =Dλa =dλa+ωa∧λb. (2)
9 b
9
Invariance of (1) under (2)would require the torsion-free
/ I. INTRODUCTION
h condition,
t
p- Agoodpartoftheresultspresentedinthislecturewere Ta =dea+ωa∧eb =0. (3)
alsodiscussedin[2]andalsopresentedattheJanuary’98 b
e
h meetinginBariloche[3]–wherethedetailedconstruction Thisconditionisanequationofmotionfortheaction(1).
v: ofthesuperalgebracanbefound–,butitwasatthemeet- Thismeansthattheinvarianceoftheaction(1)under(2)
i ingcoveredbytheseproceedingswheretheseresultswere couldnotresultfromthetransformationpropertiesofthe
X first presented. fieldsalone,butitwouldbeapropertyoftheirdynamics
r Threeofthefourfundamentalforcesofnaturearecon- aswell. The torsion-freecondition, being oneofthe field
a
sistently described by Yang-Mills (YM) quantum theo- equations, implies that local translational invariance is
ries. Gravity, the fourth fundamental interaction, resists at best an on-shell symmetry, which wouldprobably not
quantization in spite of several decades of intensive re- survive quantization.
search in this direction. This is intriguing in view of the The contradiction stems from the identification be-
factthatGeneralRelativity(GR)andYMtheorieshave tween localtranslations in the base manifold (diffeomor-
adeepgeometricalfoundation: thegaugeprinciple. How phisms)
cometwotheoriesconstructedonalmostthesamemath-
ematical basis produce such radically different physical xµ →xµ =xµ+ζµ(x), (4)
′
behaviours? What is the obstruction for the application
of the methods of YM quantum field theory to gravity? –which is a genuine invariance of the action (1)–, and
local translations in the tangent space (2).
Since the invariance of the Hilbert action under gen-
eralcoordinatetransformations(4)isreflectedintheclo-
∗TalkpresentedattheSixthMeetingonQuantumMechan- sureofthefirst-classhamiltonianconstraintsintheDirac
formalism, one could try to push the analogy between
ics of Fundamental Systems: Black Holes and the Structure
of the Universe,Santiago, August 1997. the Hamiltonian constraints Hµ and the generators of a
†John Simon Guggenheim fellow gauge algebra. However, the fact that the constraint al-
gebra requires structure functions, which depend on the
1
dynamical fields, is another indication that the genera- II. SUPERGRAVITY
tors of diffeomorphism invariance of the theory do not
form a Lie algebra but an open algebra (see, e. g., [4]). For some time it was hoped that the nonrenormaliz-
More precisely, the subalgebra of spatial diffeomor- ability of GR could be cured by supersymmetry. How-
phismsisagenuineLiealgebrainthesensethatitsstruc- ever, the initial glamour of supergravity (SUGRA) as
tureconstantsareindependentofthedynamicalfieldsof a mechanism for taming the wild ultraviolet divergences
gravitation, of pure gravity,was eventually spoiled by the realization
thatittoowouldleadtoanonrenormalizableanswer[10].
[H ,H ]∼H δ −H δ . (5)
i j′ j′ i i′ j Again, one can see that SUGRA is not a gauge theory
| |
either in the sense of a fiber bundle, and that the lo-
In contrast, the generators of timelike diffeomorphisms
calsymmetryalgebraclosesnaturallyonly onshell. The
form an open algebra,
algebra can be made to close off shell at the cost of in-
[H ,H ]∼gijH δ . (6) troducingauxiliaryfields,buttheyarenotguaranteedto
⊥ ⊥′ j′ |i exist for all D and N [11].
This comment is particularly relevant in a CHern- Whether the lack of fiber bundle structure is the ul-
Simonstheory,wherespatialdiffeomorphismsarealways timate reason for the nonrenormalizability of gravity re-
part of the true gauge symmetries of the theory. The mains to be proven. However, it is certainly true that if
generators of timelike displacements (H ), on the other GR could be formulated as a gauge theory, the chances
hand, are combinations of the internal g⊥auge generators for proving its renormalizability would clearly grow.
and the generatorsof spatialdiffeomorphism, and there- In three spacetime dimensions both GR and SUGRA
fore do not generate independent symmetries [5]. define renormalizable quantum theories. It is strongly
suggestivethatpreciselyin2+1dimensionsboththeories
Higher D The minimal requirements for a consistent canalsobeformulatedasgaugetheoriesonafiberbundle
theorywhichincludesgravityinanydimensionare: gen- [12]. Itmightseemthattheexactsolvabilitymiraclewas
eral covariance and second order field equations for the due to the absence of propagating degrees of freedom in
metric. For D > 4 the most general action for gravity three-dimensionalgravity,butthe powercountingrenor-
satisfying this criterion is a polynomial of degree [D/2] malizabilityargumentrestsonthe fiberbundle structure
inthe curvature,firstdiscussedby Lanczosfor D =5 [6] of the Chern-Simons form of those systems.
and, in general, by Lovelock [7,8]. There are other known examples of gravitation theo-
riesinodddimensionswhicharegenuine(off-shell)gauge
First order theory theories for the anti-de Sitter (AdS) or Poincar´e groups
If the theory contains spinors that couple to gravity, [13–16]. These theories, as well as their supersymmetric
it is necessary to decouple the affine and metric proper- extensions have propagating degrees of freedom [5] and
ties of spacetime. A metric formulation is sufficient for areCSsystemsforthecorrespondinggroupsasshownin
spinless point particles and fields because they only cou- [17].
ple to the symmetric part of the affine connection, while
a spinning particle can “feel” the torsion of spacetime.
A. From Rigid Supersymmetry to Supergravity
Thus,itisreasonabletolookforaformulationofgravity
in which the spin connection (ωab) and the vielbein (ea)
µ µ
are dynamically independent fields, with curvature and Rigid SUSY can be understood as an extension of the
torsion standing on a similar footing. Thus, the most Poincar´ealgebrabyincludingsuperchargeswhicharethe
generalgravitationalLagrangianwouldbe ofthe general “square roots” of the generators of rigid translations,
form L=L(ω,e) [9]. {Q¯,Q}∼Γ·P. Thebasicstrategytogeneralizethisidea
Allowing an independent spin connection in four di- tolocalSUSYwastosubstitutethemomentumPµ =i∂µ
mensions does not modify the standard picture in prac- bythegeneratorsofdiffeomorphisms,H,andrelatethem
ticebecauseanyoccurrenceoftorsionintheactionleaves tothesuperchargesby{Q¯,Q}∼Γ·H. Theresultingthe-
the classical dynamics essentially intact. In higher di- ory has on-shell local supersymmetry algebra [18].
mensions, however, theories that include torsion can be An alternative point of view –which is the one we ad-
dynamically quite different from their torsion-free coun- vocate here– would be to construct the supersymmetry
terparts. onthe tangentspaceandnotonthe basemanifold. This
As we shall see below, the dynamical independence of approach is more natural if one recalls that spinors pro-
ωab andea alsoallowsdefining these gravitationtheories videabasisofirreduciblerepresentationsforSO(N),and
in 2n+ 1 dimensions on a fiber bundle structure as a notfor GL(N). Thus, spinorsare naturallydefined rela-
Yang-Millstheory,afeaturethatisnotsharedbyGeneral tive to a local frame on the tangent space rather than in
Relativity except in three dimensions. the coordinatebasis. Thebasicpointistoreproducethe
2+1 “miracle” in higher dimensions. This idea has been
successfully applied by Chamseddine in five dimensions
2
[14],andby us for pure gravity[15,16]andin supergrav- [21], the counting of degrees of freedom in CS theories is
ity[2,17]. TheSUGRAconstructionhasbeencarriedout completelydifferentfromtheoneforthesameconnection
for spacetimes whose tangent space has AdS symmetry one-forms in a YM theory.
[2], and for its Poincar´econtraction in [17].
In [17], a family of theories in odd dimensions, invari-
ant under the supertranslation algebra whose bosonic III. LANCZOS–LOVELOCK GRAVITY
sector contains the Poincar´e generators was presented.
The anticommutator of the supersymmetry generators A. Lagrangian
gives a translation plus a tensor “central” extension,
For D > 4, assumption (ii) is an unnecessary restric-
{Qα,Q¯ }=−i(Γa)αP −i(Γabcde)αZ , (7)
β β a β abcde tion on the available theories of gravitation. In fact, as
The commutators of Q,Q¯ and Z with the Lorentz gen- mentioned above, the most general action for gravity –
generallycovariantandwithsecondorderfieldequations
erators can be read off from their tensorial character.
forthemetric–istheLanczos-LovelockLagrangian(LL).
All the remaining commutators vanish. This algebra is
TheLLLagrangianinaD-dimensionalRiemannianman-
thecontinuationtoallodd-dimensionalspacetimesofthe
ifold can be defined in at least four ways:
D = 10 superalgebra of van Holten and Van Proeyen
(a)Asthemostgeneralinvariantconstructedfromthe
[19], and yields supersymmetric theories with off-shell
metric and curvature leading to second order field equa-
Poincar´e superalgebra. The existence of these theories
tions for the metric [6–8].
suggeststhatthereshouldbesimilarsupergravitiesbased
(b) As the most general D-form invariant under local
on the AdS algebra.
Lorentz transformations, constructed with the vielbein,
the spin connection, and their exterior derivatives, with-
out using the Hogde dual (∗) [22].
B. Assumptions of Standard Supergravity
(c) As a linear combination of the dimensional con-
tinuation of all the Euler classes of dimension 2p < D.
Three implicit assumptions are usually made in the
[8,23]
construction of standard SUGRA:
(d) As the most general low energy effective gravita-
(i)The fermionicandbosonic fields inthe Lagrangian
tional theory that can be obtained from string theory
shouldcomeincombinationssuchthattheirpropagating
[24].
degrees of freedom are equal in number. This is usually
Definition (a) was historically the first. It is appropri-
achieved by adding to the graviton and the gravitini a
ateforthemetricformulationandassumesvanishingtor-
number of lower spin fields (s < 3/2) [18]. This match-
sion. Definition(b)isslightlymoregeneralthanthefirst
ing, however, is not necessarily true in AdS space, nor
and allows for a coordinate-independent first-order for-
in Minkowski space if a different representation of the
mulation,andevenallowstorsion-dependenttermsinthe
Poincar´e group (e.g., the adjoint representation) is used
action [9]. As a consequence of (b), the field configura-
[20].
tionsthatextremizetheactionobeyfirstorderequations
Theothertwoassumptionsconcernthepurelygravita-
for ω and e. Assertion (c) gives directly the Lanczos–
tionalsector. TheyareasoldasGeneralRelativityitself
Lovelocksolution as a polynomial of degree [D/2] in the
andaredictatedbyeconomy: (ii)gravitonsaredescribed
curvature of the form
by the Hilbert action (plus a possible cosmological con-
stant),and,(iii)thespinconnectionandthevielbeinare [D/2]
not independent fields but are related through the tor- I = α Lp, (8)
G p
Z
sion equation. The fact that the supergravitygenerators Xp=0
do not form a closedoff-shell algebracan be tracedback
to these asumptions. where αp are arbitrary constants and1
The procedure behind (i) is tightly linked to the idea
thatthe fieldsshouldbe inavectorrepresentationofthe LpG =ǫa1···aDRa1a2···Ra2p−1a2pea2p+1···eaD, (9)
Poincar´e group [20] and that the kinetic terms and cou-
wherewedgeproductofformsisunderstoodthroughout.
plings are such that the counting of degrees of freedom
Statement (d) reflects the empirical observation that
works like in a minimally coupled gauge theory. This
the vanishing of the superstring β-function in D = 10
assumption comes from the interpretation of supersym-
gives rise to an effective Lagrangianof the form (9) [24].
metric states as represented by the in- and out- plane
waves in an asymptotically free, weakly interacting the-
ory in a minkowskian background. These conditions are
not necessarily met by a CS theory in an asymptotically
AdS background. Apart from the difference in back- 1Forevenandodddimensionsthesameexpression(9)canbe
ground,whichrequiresacarefultreatmentoftheunitary used,butforoddD,Chern-SimonsformsfortheLorentzcon-
irreduciblerepresentationsofthe asymptoticsymmetries nectioncouldalsobeincluded(thispointisdiscussedbelow).
3
Inevendimensions,thelastterminthesumistheEu- C. The vanishing of Classical Torsion
lercharacter,whichdoesnotcontributetothe equations
ofmotion. However,in the quantumtheory,this termin Obviously Ta = 0 solves (12). However, for D > 4
the partition function would assign different weights to this equation does not imply vanishing torsion in gen-
nonhomeomorphic geometries. eral. In fact, there are choices of the coefficients α and
p
The large number of dimensionful constants αp in the configurations of ωab, ea such that Ta is completely ar-
LLtheorycontrastswiththetwoconstantsoftheEHac- bitrary. On the other hand, as already mentioned, the
tion (G and Λ) [25,26,15]. This feature could be seem as torsion-freepostulateis atbestagooddescriptionofthe
an indication that renormalizability would be even more classical dynamics only. Thus, an off-shell treatment of
remote for the LL theorythan inordinarygravity. How- gravityshouldallowfordynamicaltorsioneveninfourdi-
ever,this is not necessarily so. There are some very spe- mensions. In the first order formulation, the theory has
cialchoicesofαp suchthatthe theorybecomes invariant second class constraints due to the presence of a large
underalargergaugegroupinoddspacetimedimensions, number of “coordinates” which are actually “momenta”
which could actually improve renormalizability [12,16]. [30],thuscomplicatingthedynamicalanalysisofthethe-
ory.
On the other hand, if torsion is assumed to vanish, ω
B. Equations could be solved as a function of e 1 and its first deriva-
−
tives,butthis wouldrestrictthe validityofthe approach
Consider the Lovelock action (8), viewed as a func- to nonsingular configurations for which det(ea) 6= 0.
µ
tional of the spin connection and the vielbein, In this framework, the theory has no second class con-
straintsandthenumberofdegreesoffreedomisthesame
ILL =ILL ωab,ea . (10) as in the Einstein-Hilbert theory, namely D(D−3) [23].
2
(cid:2) (cid:3)
Varying with respect to the vielbein, the generalized
Einstein equations are obtained,
D. Dynamics and Degrees of Freedom
n 1
−
α (D−2p)ǫ Ra1a2···Ra2p−1a2p × Imposing Ta = 0 from the start, the action is
Xp=0 p a1···aD I=ILL[ea,ω(e)] and varying respect to e, the “1.5 order
ea2p+1···eaD−1 =0. (11) formalism” [18] is obtained,
Varying with respect to the spin connection, the torsion δI δILLδea+ δILLδωbcδea. (13)
equations are found, = δea δωbc δea
n 1 Assuming δILL = 0 the equations of motion consist
− δωbc
α p(D−2p)ǫ Ra3a4···Ra2p−1a2p × of the Einstein equations (11), defined on a restricted
Xp=0 p aba3···aD configuration space.
ea2p+1···eaD−1TaD =0. (12) For D ≤ 4, Ta = 0 is the unique solution of eqn.(12).
In those dimensions, the different variational principles
The presence of the arbitrary coefficients α in (first-,second-and1.5-thorder)areclassicallyequivalent
p
the action implies that static, spherically symmetric in the absence of sources. On the contrary, for D > 4,
Schwarzschild-like solutions possess a large number of Ta = 0 is not logically necessary and is therefore unjus-
horizons [27], and time-dependent solutions have an un- tified.
predictable evolution [23,28]. However, as shown below, The LL–Lagrangians (9) include the Einstein-Hilbert
for a particular choice of the constants α the dynamics (EH) theory as a particular case, but they are dynami-
p
is significantly better behaved. cally very different in general. The classical solutions of
Additional terms containing torsion explicitly can be the LL theory are not perturbatively related to those of
included in the action. It can be shown, however, that the Einstein theory. For instance, it was observed that
the presence of torsional terms in the Lagrangian does the time evolution of the classical solutions in the LL
not change the degrees of freedom of gravity in four di- theory starting from a generic initial state can be un-
mensions. Indeed, the matter-free theory with torsion predictable, whereas the EH theory defines a well-posed
terms is indistinguishable (at least classically) from GR, Cauchy problem.
[29]. However,inhigherdimensions,thesituationiscom- It can also be seen that even for some simple minisu-
pletely different [9]. perspacemodels,thedynamicscouldbecomequitemessy
because the equations of motion are not deterministic in
the classical sense, due to the vanishing of some eigen-
values ofthe Hessianmatrix oncriticalsurfacesin phase
space [23,28].
4
E. Choice of Coefficients The resulting Lagrangian is the Euler-CS form. Its
exterior derivative is the Euler form in 2n dimensions,
Atleastforsomesimpleminisuperspacegeometriesthe
dLAdS =κǫ RA1A2···RA2n−1A2n (18)
indeterminate classical evolution can be avoided if the G2n−1 A1···A2n
coefficients are chosen so that the Lagrangian is based =κE2n,
on the connection for the AdS group,
where RAB = dWAB +WAWCB is the AdS curvature,
C
n−1 which contains the Riemann and torsion tensors,
(D−2p) 1 , D =2n−1
 − (cid:18) p (cid:19)
αplD−2p = (cid:18)np (cid:19), D =2n. RAB =(cid:20) Rab−+Tbl12/elaeb Ta0/l (cid:21). (19)
 (14) Theconstantκis quantized[16](inthefollowingwewill
set κ=l=1).
This corresponds to the Born-Infeld theory in even di- In general, a Chern-Simons Lagrangian in 2n−1 di-
mensions [26], and to the AdS Chern-Simons theory in mensions is defined by the condition that its exterior
odd dimensions [15,13,14], derivative be an invariant homogeneous polynomial of
degree n in the curvature, that is, a characteristic class.
Inthe caseabove,(??)defines the CSformforthe Euler
1. D=2n: Born-Infeld Gravity class 2n-form.
A generic CS Lagrangian in 2n−1 dimensions for a
In even dimensions the choice (14) gives rise to a La- Lie algebra g can be defined by
grangian of the form
dLg =hFni, (20)
2n 1
ea1ea2 eaD−1eaD −
L=κǫ (Ra1a2 + )···(RaD−1aD + ).
a1···aD l2 l2 whereh istandsforamultilinearfunctionintheLiealge-
brag,invariantundercyclicpermutationssuchasTr,for
(15)
an ordinaryLie algebra,or STr, in the case of a superal-
ThisisthePfaffianofthetwo–formRab+ 1eaeb,and,in gebra. In the caseabove,the only nonvanishingbrackets
l2 in the algebra are
this sense it can be written in the Born-Infeld-like form,
J ,···,J =ǫ . (21)
1 A1A2 AD−1AD A1···AD
L=κ det(Rab+ eaeb). (16) (cid:10) (cid:11)
r l2
The combinations Rab+ 1eaeb are the components of
l2 3. D=2n−1: Poincar´e Gauge Gravity
theAdScurvature(c.f.(19)below). Thisseemstosuggest
thatthe systemmightbe naturallydescribedintermsof
Starting from the AdS theory (??) in odd dimensions,
anAdSconnection[31]. However,thisisnotthecase: In
a Wigner- Ino¨nu¨ contraction deforms the AdS algebra
even dimensions, the Lagrangian (15) is invariant under
into the Poincar´e one. The same result is also obtained
local Lorentz transformations and not under the entire
choosing α =δn. Then, the Lagrangian(8) becomes:
AdS group. As will be shown below, it is possible, in p p
odd dimensions, to construct gauge invariant theories of
LP =ǫ Ra1a2···RaD−2aD−1eaD. (22)
gravity under the full AdS group. G a1···aD
Inthiswaythelocalsymmetrygroupof(8)isextended
fromLorentz(SO(D−1,1))toPoincar´e(ISO(D−1,1)).
2. D=2n−1: AdS Gauge Gravity
Analogously to the anti-de Sitter case, one can see that
the action depends on the Poincar´e connection: A =
Theodd-dimensionalcasewasdiscussedin[13,14],and eaP + 1ωabJ . It is straightforward to verify the in-
lateralsoin[15]. Considerthe action(8)withthe choice a 2 ab
variance of the action under local translations,
given by (14) for D =2n−1. The constant parameter l
hasdimensions oflengthandits purpose isto renderthe δea =Dλa, δωab =0, (23)
action dimensionless. This also allows the interpretation
of ω and e as components of the AdS connection [26], Here D stands for covariant derivative in the Lorentz
A= 1ωabJ +eaJ = 1WABJ , where connection. If λ is the Lie algebra-valuedzero-form,λ=
2 ab aD+1 2 AB
λaP , the transformations (23) are read from the gen-
a
ωab ea/l eral gauge transformation for the connection, δA=∇λ,
WAB = , A,B =1,...D+1. (17)
(cid:20) −eb/l 0 (cid:21) where ∇ is the covariant derivative in the Poincar´e con-
nection.
5
Moreover,theLagrangian(22)isaChern-Simonsform. which would be inconsistent with the transformation of
Indeed, with the curvature for the Poincar´ealgebra, F= the fields under local translations (2). Thus, the spin
dA +A∧A =21RabJab+TaPa, LPG satisfies connection and the vielbein –the soldering between the
base manifold and the tangent space– cannot be identi-
dLP = Fn+1 , (24) fiedasthecompensatingfieldsforlocalLorentzrotations
G
(cid:10) (cid:11) and translations, respectively.
where the only nonvanishing components in the bracket Inourconstructionω andeareassumedtobedynam-
are ically independent and thus torsion necessarily contains
propagating degrees of freedom, represented by the con-
Ja1a2,···,JaD−2aD−1,PaD =ǫa1···aD. (25) torsiontensorkµab :=ωµab−ω¯µab(e,...),whereω¯ isthespin
(cid:10) (cid:11) connection which solves the (algebraic) torsion equation
Thus, the Chern character for the Poincar´e group is
in terms of the remaining fields.
written in terms of the Riemman curvature and the tor-
The generalization of the Lovelock theory to include
sion as
torsion explicitly can be obtained assuming definition
(b). This is a cumbersome problem due to the lack of
F3 =ǫ Ra1a2···RaD−2aD−1TaD. (26)
a1···aD a simple algorithmto classify all possible invariants con-
(cid:10) (cid:11) structedfromea,RabandTa. InRef.[9]auseful“recipe”
Thesimplestexampleofthisisordinarygravityin2+1
to generate all those invariants is given.
dimensions, where the Einstein-Hilbert action with cos-
mological constant is a genuine gauge theory of the AdS
group, while for zero cosmological constant it is invari-
A. The Two Families of AdS Theories
ant under local Poincar´e transformations. Although this
gauge invariance of 2+1 gravity is not always empha-
sized, it lies at the heart of the proof of integrability of Similarly to the theory discussed in section III, the
the theory [12]. torsional additions to the Lagrangianbring in a number
ofarbitrarydimensionfulcoefficientsβ ,analogoustothe
k
α ’s. Also in this case, one can try choosing the β’s in
p
IV. ADS GAUGE GRAVITY suchawayastoenlargethelocalLorentzinvarianceinto
anAdSgaugesymmetry. Ifnoadditionalstructure(e.g.,
inversemetric,Hodge-∗,etc.) isassumed,AdSinvariants
As shown above, the LL action assumes spacetime
can only be produced in dimensions 4k and 4k−1.
to be a Riemannian, torsion-free, manifold. That as-
The proof of this claim is as follows: invariance under
sumption is justified a posteriori by the observationthat
Ta = 0 is always a solution of the classical equations, AdS requires that the D-form be at least Lorentz invari-
ant. Then,inorderforthesescalarstobeinvariantunder
and means that e and ω are not dynamically indepen-
AdS as well, it is necessary and sufficient that they be
dent. This is the essence of the second order or metric
expressible in terms of the AdS connection (17). As is
approachtoGR,inwhichdistanceandparalleltransport
well-known (see, e.g., [33]), in even dimensions, the only
arenotindependentnotions,butarerelatedthroughthe
D-forminvariantunderSO(N)constructedaccordingto
Christoffelsymbol. Thereisnofundamentaljustification
the recipe mentionedaboveare2 the Eulercharacter(for
for this assumptionand this was the issue of the historic
N = D), and the Chern characters (for any N). Thus,
discussion between Einstein and Cartan [32].
In four dimensions, the equation Ta = 0 is algebraic the only AdS invariant D-forms are the Euler class, and
linear conbinations of products of the type
and could in principle be solvedfor ω in terms of the re-
mainingfields. However,forD >5,CSgravityhasmore
P =c ···c , (28)
degrees of freedom than those encountered in the corre- r1···rs r1 rs
sponding second order formulation [5]. This means that
with 2(r +r +···+r )=D, where
1 2 s
theCSgravityactionhaspropagatingdegreesoffreedom
for the spin connection. This is a compelling argument c =Tr(Fr), (29)
r
toseriouslyconsiderthepossibilityofintroducingtorsion
terms in the Lagrangianfrom the start. defines the r-th Chern character of SO(N). Now, since
Another consequence of imposing a dynamical depen- the curvature two-formF in the vectorialrepresentation
dencebetweenωandethroughthetorsion-freecondition
is that it spoils the possibility of interpreting the local
translational invariance as a gauge symmetry of the ac-
tion. Consider the action of the Poincar´e group on the 2For simplicity we will not always distinguish between dif-
fields as given by (23); taking Ta ≡0 implies
ferent signatures. Thus, if no confussion can occur, the AdS
group in D dimensions will also be denoted as SO(D+1).
δωab = δωabδec 6=0, (27) The de Sitter case can be obtained replacing αp by (−1)pαp
δec in (14).
6
is antisymmetric in its indices, the exponents {r } are concentrateontheconstructionofthepuregravitysector
j
necessarily even, and therefore (28) vanishes unless D is asagaugetheorywhichisparity-odd. Thisconstruction
a multiple of four. Thus, one arrives at the following was discussed in [35], and also briefly in [2,3].
lemmas:
Lemma: 1 For D = 4k, the only D-forms built from
ea, Rab and Ta, invariant under the AdS group, are the B. Even dimensions
Chern characters for SO(D+1).
Lemma: 2ForD =4k+2,therearenoAdS-invariant In D =4, the the only local Lorentz-invariant4-forms
D-forms constructed from ea, Rab and Ta. constructed with the recipe just described are [9]:
In view of this, it is clear why attempts to construct
E =ǫ RabRcd
gravitation theories with local AdS invariance in even 4 abcd
dimensions have been unsuccessful [31,34]. L =ǫ Rabeced
EH abcd
boSuinndcaerythteerfmorsmisnP4rk1···drsimaernesicolnosse–dw,htihcehydaorenoatt cboenst- LC =ǫabcdeaebeced
C =RabR
tribute totheclassicalequations,butcouldassigndiffer- 2 cd
ent weights to configurations with nontrivial torsion in LT1 =Rabeaeb
the quantum theory. In other words, they can be locally L =TaT .
T2 a
expressed as
P =dLAdS (W). (30) The first three terms are even under parity and the
r1···rs {r}4k−1 rest are odd. Of these, E and C are topological invari-
4 2
Thus,foreachcollection{r},the(4k−1)-formLAdS antdensities (closedforms): the Eulercharacterandthe
r 4k 1
defines a Lagrangian for the AdS group in 4k −{ }1 d−i- second Chern character for SO(4),respectively. The re-
mensions. It takes direct computation to see that these mainingfourtermsdefinethemostgeneralgravityaction
Lagrangians involve torsion explicitly. These results are in four dimensions,
summarized in the following
I = [αL +βL +γL +ρL ]. (31)
EH C T1 T2
Z
Theorem: Therearetwofamiliesofgravitationalfirst M4
orderLagrangiansforeandω,invariantunderlocalAdS Itcanalsobeseen,thatbychoosingγ =−ρ,thelasttwo
transformations: terms are combined into a topological invariant density
a: Euler-Chern-SimonsforminD =2n−1,whose ex- (theNieh-Yanform). Thus,withthischoicetheoddpart
terior derivative is the Euler character in dimension 2n, of the action becomes a boundary term. Furthermore,
which do not involve torsion explicitly, and C , L andL canbe combinedinto the secondChern
2 T1 T2
character of the AdS group,
b: Pontryagin-Chern-Simons forms in D = 4k − 1,
RaRb +2(TaT −2Rabe e )=RA RB. (32)
whoseexteriorderivativesaretheCherncharactersin4k b a a a b B A
dimensions, which involves torsion explicitly. ThisistheonlyAdSinvariantconstructedwithea,ωab
It must be stressed that locally AdS-invariant gravity
andtheirexteriorderivativesalone,confirmingthatthere
theories only exist in odd dimensions. They aregenuine
arenolocallyAdSinvariantgravitiesinfourdimensions.
gauge systems, whose action comes from topological in-
Ingeneral,theonlyAdS-invariantfunctionalsinhigher
variants in one dimension above. These topological in-
dimensionscanbe writtenintermsoftheAdScurvature
variants can be written as the trace of a homogeneous
as [9]
polynomialofdegreen inthe AdS curvature. Obviously,
for dimensions 4k−1 both a- and b-families exist. The I˜ = C ···C , (33)
mostgeneralLagrangianof this sortis a linear combina- r1···rs ZM r1 rs
tion of the two families.
or linear combinations thereof, where C =Tr[(RA)r] is
An important difference between these two families is r B
ther-thCherncharacterfortheAdSgroup. Forexample,
thatunderaparitytransformationthe firstisevenwhile
the second is odd 3. The parity invariant family has en D =8 the Chern characters for the AdS group are
beenextensivelystudiedin[13–15,26]. Inwhatfollowswe Tr[(RA )4]=C ,
B 4
(34)
Tr[(RAB)2]∧Tr[(RAB)2]=(C2)2.
3Parityisunderstoodhereasaninversionofonecoordinate, Similar Chern classes are also found for D = 4k. (As
binostthaninceththeetaEnugleenrtcshpaarcaectaenrdisininthvearbiaansetmunadneifrolpda.riTtyh,uws,hfioler oaldrdea,dwyhimchenistitohneedc,asI˜er1i·n··r4skv+an2isdhiemseinfsoionnes.o)f the r’s is
theLorentzCherncharactersandthetorsionaltermsarepar- Thus, there are no AdS-invariant gauge theories in
ity violating. even dimensions.
7
C. Odd dimensions V. EXACT SOLUTIONS
The simplest example is found in three spacetime di- As stressed here, the local symmetry of odd-
mensions where there are two locally AdS-invariant La- dimensionalgravitycanbeextendedfromLorentztoAdS
grangians, namely, the Einstein-Hilbert with cosmologi- byanappropriatechoiceofthefreecoefficientsintheac-
cal constant, tion. TheresultingLagrangians(withorwithouttorsion
terms), are Chern-Simons D-forms defined in terms of
1
LAdS =ǫ [Rabec+ eaebec], (35) the AdS connection A, whose components include the
G3 abc 3l2 vielbein and the spin connection [see eqn. (17)]. This
impliesthatthefieldequations(11,12)obtainedbyvary-
and the “exotic” Lagrangian
ingthevielbeinandthespinconnectionrespectively,can
LATd3S =L∗3(ω)+2eaTa, (36) be written in an AdS-covariant form
<Fn 1J >=0, (39)
where − AB
2 where F= 1RABJ is the AdS curvature with RAB
L ≡ωadωb + ωaωbωc, (37) 2 AB
∗3 b a 3 b c a given by (19) and JAB are the AdS generators.
It is easily checked that any locally AdS spacetime is
istheLorentzChern-Simonsform. Notethatin(36),the a solution of (39). Apart from anti-de Sitter space it-
localAdSsymmetryfixestherelativecoefficientbetween self, some interesting spacetimes with this feature are
L∗3(ω), and the torsion term eaTa. The most general the topological black holes of Ref. [37], and some “black
action for gravitationin D =3, which is invariant under branes”withconstantcurvatureworldsheet[38]. Forany
SO(4) is thereforealinear combinationαLAGd3S+βLATd3S. D, there is also a unique static, spherically symmetric,
For D = 4k−1, the number of possible exotic forms asymptotically AdS black hole solution [15], as well as
grows as the partitions of k, in correspondence with their topological extensions which have nontrivial event
the number of composite Chern invariants of the form horizons [39].
P{r} = jCrj. The most general Lagrangian in 4k−1 Exact solutions of the form AdS4 × SD−4 have also
dimensioQns takes the form αLAdS +β LAdS , been found [40] 4 as well as alternative four-dimensional
G4k 1 r T r 4k 1
where dLAdS = P , with− r {=} 4k.{ }Th−ese cosmologicalmodels.
T r 4k 1 r j j
Lagrangians{h}ave−proper{d}ynamicsPand, unlike the even All ofthe above geometriescanbe extended into solu-
tionsofthe gravitationalBorn-Infeldtheory(16)ineven
dimensionalcases,they are notboundary terms. For ex-
dimensions. Friedmann-Robertson-Walker like cosmolo-
ample, in seven dimensions one finds [35,36]
gies have been shown to exist in even dimensions [26],
and it could be expected that similar solutions exists in
LAdS =β [Ra Rb +2(TaT −Rabe e )]LAdS odd dimensions as well.
T 7 2,2 b a a a b T 3
+β [L (ω)+2(TaT +Rabe e )Tae +4T Ra Rb ec],
4 ∗7 a a b a a b c
VI. CHERN-SIMONS SUPERGRAVITIES
where L is the Lorentz-CS (2n-1)-form,
∗2n 1
−
dL (ω)=Tr[(Ra)n]. (38) Wenowconsiderthesupersymmetricextensionsofthe
∗2n 1 b
− locallyAdStheoriesdefinedabove. Theideaistoenlarge
Summarizing: TherequirementoflocalAdSsymme- the AdS algebra incorporating SUSY generators. The
try is rather strong and has the following consequences: closure of the algebra (Jacobi identity) forces the addi-
tion of further bososnic generators as well [19]. In order
• Locally AdS invariant theories of gravity exist in to accomodate spinors in a natural way, it is useful to
odd dimensions only. cast the AdS generators in the spinor representation of
SO(D+1). In particular, one can write,
• ForD =4k−1therearetwofamilies: oneinvolving
only the curvature and the vielbein (Euler Chern- −1
Simons form), and the other involving torsion ex- dLATd4Sk−1 = 24kTr[(RABΓAB)2k]. (40)
plicitly in the Lagrangian. These families are even
and odd under space reflections, respectively.
• ForD =4k+1onlytheEuler-Chern-Simonsforms
4The de-Sitter case (Λ > 0) was discussed in [41] for the
exist. These ar parity even and don’t involve tor-
torsion-free theory. Changing the sign in the cosmological
sion explicitly.
constant has deep consequences. In fact, the solutions are
radically different,andlocally supersymmetricextensionsfor
positive cosmological constant don’t exist in general.
8
which is a particular form of (20) where hi has been re- D S-Algebra Conjugation Matrix Internal Metric
placed by the ordinary trace over spinor indices in this 8k−1 osp(N|m) CT =C uT =−u
representation. 8k+3 osp(m|N) CT =−C uT =u
Other possibilities of the form Fn−p hFpi, are not 4k+1 su(m|N) C† =C u† =u
necessary to reproduce the minima(cid:10)l supe(cid:11)rsymmetric ex-
tensions of AdS containing the Hilbert action. In the In each of these cases, m = 2[D/2] and the connection
supergravity theories discussed below, the gravitational takes the form
sector is given by ± 1 LAdS − 1LAdS . The ± sign
2n G2n 1 2 T 2n 1 1 1
corresponds to the two choic−es of inequiv−alent represen- A= ωabJ +eaJ + b[r]Z +
2 ab a r! [r]
tations of Γ’s, which in turn reflect the two chiral repre-
1 1
sentationsinD+1. Asinthethree-dimensionalcase,the (ψ¯iQ −Q¯iψ )+ a Mij. (42)
i i ij
2 2
supersymmetricextensionsofL oranyoftheexoticLa-
G
grangianssuchasLT,requireusingbothchiralities,thus The generators Jab,Ja span the AdS algebra and the
doubling the algebras. Here we choosethe + sign,which Qi’s generate (extended) supersymmetry transforma-
α
gives the minimal superextension [35]. tions. TheQ’stransforminavectorrepresentationunder
The bosonic theory (40) is our starting point. The theactionofM andasspinorsundertheLorentzgroup.
ij
idea now is to construct its supersymmetric extension. Finally, the Z’s complete the extension of AdS into the
For this, we need to express the adjoint representation largeralgebrasso(m), sp(m)orsu(m),and[r] denotes a
in terms ofthe Dirac matrices of the appropriatedimen- set of r antisymmetrized Lorentz indices.
sthioenD. iTrahcisaligsebarlwa,a{yIs,pΓoas,siΓbaleb,.b..e}c,apursoevitdheeagbenaesirsatfoorrsthoef whIenre(4C2)aψn¯id=uψajrTeCguijvien(ψ¯iin=thψej†tCabuljei afobrovDe.=T4hkes+e 1a)l-,
spaceofsquarematrices. Theadvantageofthisapproach gebrasadmit(m+N)×(m+N)matrix representations
is that it gives an explicit representation of the algebra [31],where the J andZ haveentriesinthe m×m block,
and writing the Lagrangiansis straightforward. the M ’s inthe N×N block,while the fermionicgener-
ij
ThesupersymmetricextensionsoftheAdSalgebrasin ators Q have entries in the complementary off-diagonal
D = 2,3, 4,mod8, werestudied by vanHoltenandVan blocks.
Proeyenin[19]. TheyaddedoneMajoranasupersymme- Under a gauge transformation, A transforms by δA=
trygeneratortotheAdSalgebraandfoundalltheN =1 ∇λ,where∇isthecovariantderivativeforthesamecon-
extensionsdemandingclosureofthefullsuperalgebra. In nection A. In particular, under a supersymmetry trans-
spite of the fact that the algebra for N = 1 AdS super- formation, λ=ǫ¯iQ −Q¯iǫ , and
i i
gravity in eleven dimensions was conjectured in 1978 to
be osp(32|1) by Cremer, Julia and Scherk [42], and this ǫkψ¯ −ψkǫ¯ Dǫ
δ A= k k j , (43)
wasconfirmedin[19],nobodyconstructedasupergravity ǫ (cid:20) −Dǫ¯i ¯ǫiψj −ψ¯iǫj (cid:21)
action for this algebra in the intervening twenty years.
Onereasonforthelackofinterestintheproblemmight where D is the covariant derivative on the bosonic con-
have been the fact that the osp(32|1) algebra contains nection,Dǫj =(d+12[eaΓa+21ωabΓab+r1!b[r]Γ[r]])ǫj−aijǫi.
generatorswhichareLorentztensorsofrankhigherthan
two.In the past, supergravityalgebras were traditionally
B. D=5 Supergravity
limited to generators which are Lorentz tensors up to
second rank. This constraint was based on the observa-
tion that elementary particle states of spin higher than Inthiscase,asineverydimensionD =4k+1,thereis
two would be inconsistent [43]. However, this does not no torsional Lagrangians LT due to the vanishing of the
rule out the relevance ofthose tensor generatorsin theo- Pontrjagin4k+2-formsfor the Riemann cirvature. This
ries of extended objects [44]. In fact, it is quite common factimplies thatthe localsupersymmetric extensionwill
nowadays to find algebras like the M−brane superalge- be of the form L=LG+···.
bra [45,46], As shown in the previous table, the appropriate AdS
superalgebrainfive dimensionsissu(2,2|N),whosegen-
{Q,Q¯}∼ΓaP +ΓabZ +ΓabcdeZ . (41) erators are K,J ,J ,Qα,Q¯ ,Mij, with a,b = 1,...,5
a ab abcde a ab β
and i,j = 1,...,N. The connection is A= bK +eaJ +
a
1ωabJ +a Mij +ψ¯iQ −Q¯jψ , so that in the adjoint
2 ab ij i j
representation
A. Superalgebra and Connection
Ωα ψα
A= β j , (44)
The smallest superalgebra containing the AdS alge- (cid:20)−ψ¯βi Aij (cid:21)
bra in the bosonic sector is found followingthe same ap-
with Ωα = 1(ibI +eaΓ +ωabΓ )α, Ai = i δib+ai,
proachasin[19],butliftingtherestrictionofN =1[35]. β 2 2 a ab β j N j j
The result, for odd D >3 is (see [3] for details) and ψ¯βi = ψ†αjGαβ. Here G is the Dirac conjugate (e.
g., G=iΓ0). The curvature is
9
R¯α Dψα ea → 1ea
F= β j (45) α
(cid:20) −Dψ¯i F¯i (cid:21) ωab → ωab
β j
b → 1 b
3α
where ψ → 1 ψ (53)
i √α i
Dψα =dψα+Ωαψβ −Aiψα, ψi → 1 ψi
j j β j j i √α
R¯α =Rα−ψαψ¯i, (46) aij → aij.
β β i β
F¯i =Fi−ψ¯iψβ. Then,ifthegravitationalconstantisalsorescaledasκ→
j j β j
ακ, in the limit α→∞ the action becomes that in [17],
Here Fi =dAi +AiAk+ i dbδi is the su(N) curvature, plus a su(N) CS form,
j j k j N j
andRα =dΩα+ΩαΩσ is the u(2,2)curvature. Interms
β β σ β
of the standard (2n−1)-dimensional fields, Rβα can be I = 1 [ǫabcdeRabRcdee−RabRabb− (54)
written as 8Z
Rα = idbδα+ 1 TaΓ +(Rab+eaeb)Γ α. (47) 2Rab(ψiΓabDψi+DψiΓabψi)+Lsu(N)].
β 4 β 2 a ab β
(cid:2) (cid:3) The rescaling (53) induces a contraction of the su-
In six dimensions the only invariant form is per AdS algebra su(m|N) into [super Poincar´e]⊗su(N),
where the second factor is an automorfism.
P =iStr F3 , (48)
(cid:2) (cid:3)
which in this case reads
C. D=7 Supergravity
P =Tr R3 −Tr F3 (49)
+3 D(cid:2)ψ¯((cid:3)R¯+F¯)(cid:2)Dψ(cid:3)−ψ¯(R2−F2+[R−F](ψ)2)ψ , The smallest AdS superalgebra in seven dimensions is
osp(2|8). The connection (42) is A =1ωabJ +eaJ +
(cid:2) (cid:3) 2 ab a
where (ψ)2 = ψ¯ψ. The resulting five-dimensional C-S Q¯iψi+ 21aijMij, where Mij are the generators of sp(2).
density can de descompossed as a sum a a gravitational In the representation given above, the bracket h i is the
part, a b-dependent piece, a su(N) gauge part, and a supertraceand,intermsofthe componentfieldsappear-
fermionic term, ing in the connection, the CS form is
L=LAGdS+Lb+Lsu(N)+LF, (50) Lo7sp(2|8)(A)=2−4LAGd7S(ω,e)− 21LATd7S(ω,e)
with −L7∗sp(2)(a)+LF(ψ,ω,e,a). (55)
LAdS = 1ǫ (RabRcdee+ 2Rabecedee+ 1eaebecedee)
G 8 abcde 3 5 Here the fermionic Lagrangianis
L =−( 1 − 1 )(db)2b+ 3(TaT −Rabe e − 1RabR )b
b N2 42 4 a a b 2 ab L =4ψ¯j(R2δi +Rfi+(f2)i)Dψ
+3bfifj F j j j i
N j i +4(ψ¯iψ )[(ψ¯jψ )(ψ¯kDψ )−ψ¯j(Rδk+fk)Dψ ]
j k i i i k
L =−(aidajdak+ 3aiajakdal + 3aiajakal am) −2(ψ¯iDψ )[ψ¯j(Rδk+fk)ψ +Dψ¯jDψ ],
su(N) j k i 2 j k l i 5 j k l m i j i i k i
LF = 23 ψ¯(R¯+F¯)Dψ− 12(ψ)2(ψ¯Dψ) . wherefji =daij+aikakj,andR= 14(Rab+eaeb)Γab+21TaΓa
are the sp(2) and so(8) curvatures,respectively. The su-
(cid:2) (cid:3)
(51)
persymmetry transformations (43) read
The action is invariant under local gauge transforma- δea = 1ǫ¯iΓaψ δωab =−1ǫ¯iΓabψ
tions, which contain the local SUSY transformations 2 i 2 i
δea = −1(ǫiΓaψ −ψiΓaǫ ) δψi =Dǫi δaij =ǫ¯iψj −ψ¯iǫj.
2 i i
δωab = 1(ǫiΓabψ −ψiΓabǫ ) Standard seven-dimensional supergravity is an N = 2
4 i i theory (its maximal extension is N=4), whose gravi-
δb = i(ǫiψ −ψiǫ )
i i (52) tational sector is given by the Einstein-Hilbert action
δψ = Dǫ
i i with cosmological constant and with an osp(2|8) invari-
δψi = Dǫi ant background [47,48]. In the case presented here, the
δai = i(ǫiψ −ψiǫ ). extension to larger N is straighforward: the index i is
j j j
allowed to run from 2 to 2s, and the Lagrangianis a CS
As in 2+1dimensions, the Poincar´esupergravitythe- form for osp(2s|8).
ory is recovered contracting the super AdS group. Con-
sider the following rescaling of the fields
10