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PHYSICAL REVIEW D 70, 044042 ~2004!
Born-Infeld gravity in any dimension
J. A. Nieto*Facultad de Ciencias Fı´sico-Matema´ticas de la Universidad Auto´noma de Sinaloa, Av. de las Ame´ricas y Blvd. Universitarios,
Ciudad Universitaria, C.P. 80010, Culiaca´n Sinaloa, Me´xico~Received 10 February 2004; published 26 August 2004!
We develop a Born-Infeld type theory for gravity in any dimension. We show that in four dimensions ourformalism allows a self-dual~or anti-self-dual! Born-Infeld gravity description. Moreover, we show that sucha self-dual action is reduced to both the Deser-Gibbons and the Jacobson-Smolin-Samuel action of Ashtekarformulation. A supersymmetric generalization of our approach is outlined.
DOI: 10.1103/PhysRevD.70.044042 PACS number~s!: 04.60.2m, 04.65.1e, 11.15.2q, 11.30.Ly
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I. INTRODUCTION
More than 80 years ago, Eddington@1# introduced theaction
SEDD5E d4xAdet~Rmn~G!!, ~1!
as a possibility to incorporate affine invariance in a gravtional context. HereRmn5Rnm is the Ricci tensor andGmn
a
5Gnma becomes the Christoffel symbol, after using the Pa
tini method.In 1934 Born and Infeld~BI! @2# for different reasons than
Eddington proposed a similar action for electrodynamics
SBI5E d4xAdet~gmn1Fmn!, ~2!
wheregmn is the space-time metric andFmn is the electro-magnetic field strength. It turns out that the actionSBI re-duces to Maxwell action for small amplitudes. Some yeago, the action~2! became of very much interest becauserelation with SUSY ~see @3# and references therein! andstring theory~see@4–7# and references therein! and D-branephysics@8#. At present, the BI theory has been generalizedthe supersymmetric non-Abelian case~see@9# and referencestherein! and it has been connected with noncommutativeometry@10# and Cayley-Dickson algebras@11#. Moreover, itturns out intriguing that the BI action can be derived fromatrix theory@12# and D9-brane@13#. For interesting com-ments and observation about the BI theory the reader isferred to the works of Schwarz@14# and Ketov@15#.
Inspired by the interesting properties of the BI theoDeser and Gibbons@16# proposed in 1998 the analogueSBI for a gravity,
SDG5E d4xAdet~agmn1bRmn1cXmn!. ~3!
Here,Xmn5Xnm stands for higher order corrections in cuvature anda,b andc are appropriate coupling constants. Dser and Gibbons determined some possible choices forXmn
*Electronic address: [email protected]
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by imposing three basic criteria to the action~3!, namelyfreedom of ghosts, regularization of singularities and supsymmetrizability. In fact, they found that the expressiXmn;Rm
aRan may provide one interesting choice forXmn ,allowing the action~3! to describe gravitons but no ghosts
From the point of view of string theory both the photonwell as the graviton should be part of the spectrum ostring. Therefore one should expect that just asSBI is used toregularizep-branes theSDG action may be used with similapurpose. However, for this possibility to be viable it is neessary to generalize theSDG action to higher dimensionsMoreover, Wohlfarth@17# has pointed out that the supersymmetrizability requirement presumably implied by M-theormay provide another reason to be interested in such a higdimensional extension ofSDG .
In this work, we use similar technics as the one usedMacDowell-Mansouri formalism@18# ~see also@19# and ref-erences therein! in order to generalize theSDG action tohigher dimensions. In four dimensions, our approach isduced to the theory of Deser-Gibbons. We also show thafour dimensions our action admits self-dual~anti-self-dual!generalization which is reduced to the Ashtekar actionproposed by Jacobson-Smolin-Samuel@20#. Since Born-Infeld type action has shown to be also useful to study dferent aspects of the brane world scenario, our work mappear interesting in this area of research.
The plan of this paper is as follows: In Sec. II, usingMacDowell-Mansouri’s type formalism we establish our prposed action for Born-Infeld gravity in any dimension.Sec. III, we discuss the relationship between our propoaction and the Deser and Gibbons action. In Sec. IV,consider a self-dual~anti-self-dual! version of our proposedaction in four dimensions and we show that it is reducedthe Jacobson-Smolin-Samuel action for the Ashtekar formism. Finally, in Sec. V we outline a possible supersymmegeneralization of our formalism and we make some finalmarks.
II. PROPOSED BORN-INFELD-GRAVITY „BIG … ACTION
Consider the extended curvature
R mnab5Rmn
ab1Smnab , ~4!
where
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J. A. NIETO PHYSICAL REVIEW D 70, 044042 ~2004!
Rmnab5]mvn
ab2]nvmab1vm
acvncb2vn
acvmcb ~5!
and
Smnab52~em
a enb2en
aemb !. ~6!
Here,vmab is a SO(d21,1) connection andem
a is a vierbeinfield. It turns out that Eq.~4! can be obtained using thMacDowell-Mansouri formalism@18# ~see also Ref.@19# andreferences therein!. In fact, in such a formalism the gravitational field is represented as a connection one-form assated to some group which contains the Lorentz group asubgroup. The typical example is provided by theSO(d,1)de Sitter gauge theory of gravity. In this case, theSO(d,1)gravitational gauge fieldvm
AB52 vmBA is broken into the
SO(d21,1) connectionvmab and thevm
da5ema vierbein field,
with the dimensiond fixed. Thus the de Sitter~or anti–deSitter! curvature
R mnAB5]mvn
AB2]nvmAB1vm
ACvnCB2vn
ACvmCB ~7!
leads to the curvature~4!.Let us now introduce the definition
R ma [eb
nR mnab , ~8!
whereebn is the inverse vierbien field.
Our proposed action is
S521
d! E ddx«m1 . . . mdea1 . . . adRm1
a1 . . . Rmd
ad , ~9!
-
04404
ci-a
where«m1 . . . md is the completely antisymmetric tensor assciated to the space-time, with«0 . . .d2151 and «0 . . .d2151, whileea1 . . . ad
is also the completely antisymmetric ten
sor but now associated to the internal groupS(d21,1), withe0 . . .d21521. We assume that the internal metric is givby (hab)5diag(21, . . . ,1). So, wehavee0 . . .d2151.
Using Eq.~6! we get
ebnSmn
ab5lema , ~10!
wherel512d. Therefore, Eqs.~4! and~8! and~10! lead to
R ma 5Rm
a 1lema , ~11!
where
Rma [eb
nRmnab . ~12!
Substituting Eq.~11! into Eq. ~9! we obtain
S521
d! E ddx«m1 . . . mdea1 . . . ad~Rm1
a1 1lem1
a1 ! . . . ~Rmd
ad
1lemd
ad !. ~13!
From this action we get
S52ld
d! E ddx«m1 . . . mdea1 . . . adem1
a1 . . . emd
ad 2dld21
d! E ddx«m1 . . . mdea1 . . . adem1
a1 . . . emd21
ad21 Rmd
ad
2d~d21!ld22
2!d! E ddx«m1 . . . mdea1 . . . adem1
a1 . . . emd22
ad22 Rmd21
ad21 Rmd
ad 2•••21
d! E ddx«m1 . . . mdea1 . . . adRm1
a1 . . . Rmd
ad .
~14!
in
Since
«m1 . . . mdea1 . . . ad52e~ea1
[m1 . . . ead
md]!, ~15!
where the bracket@m1 . . . md# means completely antisymmetric ande5det(em
a ), we find that the action~14! can bewritten as
S5ldE ddxe1ld21E ddxeR
1ld22
2! E ddxe~R22RmnRmn!1•••1E ddx det~Rma !.
~16!
Here, R[eamRm
a and Rmn[enaRma ,while Rmn[gmagnbRab ,
wheregab is the matrix inverse ofgab[eaaeb
bhab . We rec-ognize in the first and second terms in the action~16! theEinstein-Hilbert action with cosmological constantd-dimensions.
III. RELATION WITH DESER-GIBBONS TYPE ACTION
Let us define the tensor
Gmn[1
l2R m
a R nbhab . ~17!
Using Eq.~11! we get
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BORN-INFELD GRAVITY IN ANY DIMENSION PHYSICAL REVIEW D 70, 044042 ~2004!
Gmn51
l2~Rm
a 1lema !~Rn
b1lenb!hab
5gmn1LRmn1L2
4Rm
aRan , ~18!
whereL52/l. Here, we used the fact thatRmn5Rnm .From Eq.~17!, it is not difficult to see that
det~Gmn
s
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the
J. A. NIETO PHYSICAL REVIEW D 70, 044042 ~2004!
S5l4E d4xe1l3E d4xe6R1l2
2!E d4xe~ 6R2
2 6Rmn6Rmn!1l
3!E d4xe~ 6R3236R6Rmn6Rmn
126Rma6Rab6Rm
b !1E d4xedet~ 6Rmn!. ~31!
We recognize in the second term of this expression the Atekar action as proposed by Jacobson-Smolin-Samuel@20#.
V. FINAL COMMENTS
In this work, we proposed the action~9! as an alternativeto generalize to higher dimensions the Born-Infeld-gravaction of Deser and Gibbons. We proved that our actionfour dimensions allows a generalization to self-dual~anti-self-dual! action. Moreover, we showed that such a self-daction is reduced to the Jacobson-Smolin-Samuel actiothe Ashtekar formulation.
It is remarkable that the only requirement for the prposed action~9! is the general covariance implied by thtransition SO(d,1)→SO(d21,1). It remains to checkwhether the action~9! satisfies the criteria of ghosts freedoand regularization of singularities used by Deser and Gbons @16# in four dimensions and Wohlfarth@17# in higherdimensions. However, these requirements should be appvery carefully to the action~9! because under compactification ~9! should lead not only to gravity but also to YanMills and scalar fields. From the point of view of M-theorone may even think in a phase in which matter fields aghosts are mixed implying a generalization of the action~9!to some kind of topological theory.
As we mentioned in the Introduction, supersymmetrability provides to Wohlfarth@17# with one of the main mo-tivation for considering BI gravity in higher dimensionWohlfarth’s theory is constructed by using the symmetriesthe Riemann tensor, considering tensors with pair of asymmetrized indices. Specifically, the Wohlfarth’s formution is based on the action*A2g@(det(dB
A1jRBA)§21#,
whereRAB5R[mn][ ab] , dBA5dm
adnb2dm
bdna andj, § are con-
stants~see Ref.@17# for details!. Although interesting, it doesnot seem evident how to supersymmetrize this theory.
In contrast, our treatment admits a straightforward genalization to the supersymmetric version in four dimensioIn fact, in the action~9! we may replace the curvatureR mn
ab
by its supersymmetric extension
R mnab5Rmn
ab1Smnab1Qmn
ab ,
R mni 5Rmn
i 1Smni , ~32!
R mna 5Rmn
a 1Smna ,
where
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n
lof
-
-
ed
d
-
fi-
r-.
Qmnab5
1
2f i j
abcmi cn
j , ~33!
Smna 5
1
2f i j
4acmi cn
j , ~34!
Smni 5 f 4a j
i ~ema cn
j 2enacm
j !, ~35!
and
Rmni 5]mcn
i 2]ncmi 1
1
2f cd j
i ~vmcdcn
j 2vncdcm
j !. ~36!
Here,cmi is a spinor field and the quantitiesf cd j
i and so onare the structure constants of algebra associated to thesupergroupOsp(1u4). It may be interesting for further research to see if the resulting supersymmetric Born-Infgravity theory is related to theD54, N51 Born-Infeld su-pergravity proposed by Gates and Ketov@21#.
Since some authors@22,23# have found interesting connections between the non-Abelian Born-Infeld theory aD-branes it may also be interesting for further researchinvestigate the relation of the action~9! with string theoryand withp-branes physics.
Recently, Liu and Li@24# have found an interesting applcation of Born-Infeld and brane worlds and Garcı´a-Salcedoand Breton@25# have studied the possibility that Born-Infelinflates the Bianchi cosmological models. It may be intereing to pursue an application of the action~9! in these direc-tions.
Pilatnik @26# has found spherical static solutions of thBorn-Infeld gravity theory which may also correspond to ttheory described by the action~9!. Feigenbaum@27# foundthat Born regulates gravity in four dimensions, so it appeattractive to see if the action~9! can be used to regulatgravity in higher dimensions. Furthermore, Vollick@28# usedthe Palatini formalism in Born-Infeld Einstein theory in foudimensions and identify the antisymmetric part of the Ritensor with the electromagnetic field. So, if we use simiPalatini’s technics in connection with the action~9! oneshould expect to identify part of the Ricci tensor with geeralized electromagnetic field, perhaps Yang-Mills field.
Finally, it has been shown~see @29#! that dualities inM-theory ~see@30# and references therein! are deeply relatedto Born Infeld theory. Since eleven dimensional supergravmust be part of M-theory one should expect that Born-Infgravity theory in any dimensions may play an important roin this context. In this direction the derivations of BornInfeld theory from supergravity@31# and the Nambu-Gotoaction from Born-Infeld theory@32# may be of particularinterest to continue a further research in connection withaction ~9!.
ACKNOWLEDGMENT
I would like to thank to M. C. Marı´n for helpful com-ments.
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