Nuclear Physics B346 (1990) 213—234 North-Holland

TOPOLOGICAL GRAVITY AND SUPERGRAVITY IN VARIOUS DIMENSIONS A.H. CHAMSEDDINE* Inst itut für Theoretische Physik, Universitàt Zurich, Schbnberggasse 9, CH 8001 Zurich, Switzerland Received 7 May 1990 (Revised 11 June 1990)

Topological theories of gravity are constructed in odd-dimensional space-times of dimensions 2n + 1, using the Chern—Simons (2n + 1)-forms and with the gauge groups ISO(1,2n) or SO(1,2n + 1) or SO(2,2n). In even dimensions the presence of a scalar field in the fundamental representation of the gauge group is needed, besides the gauge field. Supersymmetrization of the de Sitter groups can be performed up to a maximal dimension of seven, but there is no limit on the super-Poincaré groups. The different phases of the topological theory are investigated. It is argued that these theories are finite. It is shown that the graviton propagates in a perturbative sense around a non-trivial background.

1. Introduction

The problem of unifying gravity with the other fundamental interactions remains as an open problem. Presently there are two main directions for looking at this problem. The first is from the field theory point of view where the aim is to find a unified theory with gauge symmetries incorporating gravity which should also be renormalizable. The nearest in spirit to this approach are supergravity models [11, especially the ones which are extensions of the standard Weinberg—Salam model. These, however, are not renormalizable [2]. The second point of view is the string theory approach where the space-time graviton arises as the massless spin-2 excitation of the string field [31. A better understanding of how these phenomena arise must await the theory of strings. This paper will be mainly concerned with the first approach. However, the analysis of the two-dimensional quantum gravity will be given, due to the importance of this in quantizing non-critical strings. Recently some understanding has been gained by studying three-dimensional gravity [4,5]. Here, instead of considering the Einstein—Hubert action, one takes the topological action constructed from the dreibein and the spin connection. This *

Supported by the Swiss National Foundation (SNF).

0550-3213/90/$03.50© 1990

—

Elsevier Science Publishers B.V. (North-Holland)

214

A.H. Chamseddine

/

Topological gravity

is also invariant under the ISO(1, 2) gauge transformations [5]. This gauge invariance promotes the theory to become finite. In contrast, the metric theory is only renormalizable, although in the space of invertible dreibeins the two actions are classically equivalent. This phenomenon gives a motivation to look for a gaugeinvariant construction of the gravity action in higher dimensions. This was shown to be possible in all odd dimensions when the Chern—Simons action and the Poincaré or de Sitter groups of that dimension are used [61.The theory obtained is a generalization of the Einstein action and includes the Euler densities. We shall show here that this is also possible in even dimensions but requires in addition to the gauge fields a scalar multiplet in the fundamental representation of the gauge group. All these theories are expected to be finite. The renormalization analysis has not been performed yet but the expectation is based on the heuristic argument that no potential counterterms exist. A major difficulty is to incorporate matter interactions. This arises because the space-time metric cannot be used, because the vielbein is a part of the gauge multiplet and is not gauge covariant. However, it is still possible to write matter interactions that contain, in the lowest-order approximation, the standard terms, but these are non-renormalizable. It is possible that the only way to include matter interactions in a renormalizable way is to consider a unified theory without matter in higher dimensions. The four-dimensional theory is then obtained by compactification in the Kaluza—Klein approach [7]. An interesting question that is worth investigating is to see whether the renormalizability property of a theory is preserved by compactification. To include fermions, we can enlarge the gauge group to a supergroup, or enlarge the space-time manifold to a supermanifold, or both. To avoid the ~omplexities of superspace only the gauging of the supergroups will be considered with the gauge fields based on an ordinary space-time manifold. The supergroups ;hat are extensions of the de Sitter groups are limited, and exist in dimensions less han or equal to seven. These have been classified by Nahm [8]. No such limit exists on the supersymmetric Poincaré groups. In seven space-time dimensions he graded anti-de Sitter group is (0(6,2) ® SU(N, q), (8, 2N)). In this notation )(6, 2) ~ SU(N, q) is the bosonic part of the group while the fermions live in the 8, 2N) representation of the two groups and where SU(N, q) is the group of luaternionic N X N matrices. This theory does not give chiral fermions and cannot ~ive realistic models. The best hope for this is to consider gauging the super~oincaré groups. The article is organized as follows. In sect. 2 we review the known case of hree-dimensional gravity [5] and emphasize the properties that could be general~ed [6]. We truncate this theory to two dimensions and establish the action for wo-dimensional quantum gravity [9—111.In sect. 3 we first generalize the results to 11 odd dimensions and then to the more subtle case of even dimensions. In sect. 4 ie focus on the non-trivial but still manageable case of five dimensions, to westigate the issue of graviton propagation in topological theories. We point the existence of classical solutions that compactify the five-dimensional space to

A.H. Chamseddine

/

Topological gravity

215

four. Sect. 5 deals with supersymmetrization, and for illustration, the five-dimensional action is given. Sect. 6 contains some comments and the conclusion.

2. Gravity in two and three dimensions It has been shown that three-dimensional gravity can have a good quantum behaviour when considered in the first-order formalism [5].The reason is that the first-order action is identical to the Chern—Simons action of the gauge group ISO(1,2). Inclusion of a cosmological constant can be easily achieved by taking instead the group S0(1, 3) or S0(2, 2). If A denotes the Lie algebra valued gauge field A=A~dx~, =

where

~AB

A, B

~A~BJAB,

=

0,1,2,3,

(2.1)

is the group generator, then the action is given by I3=kf(AdA+~A3).

Here k is a dimensionless coupling constant and the quadratic form case of S0(1, 3) or S0(2,2) satisfies (JAB JCD)

=

=

ea(doj~+

kEabef M

~bfwC

K )

in the

(2.3)

EABCD.

Connection with gravity is made with the identification 3=etA, a=0,1,2. A~=w~, Aa 13 then becomes 13

(2.2)

+ ~Aebec),

(2.4)

(2.5)

3

where A 0, + 1, 1 for ISO(1,2), S0(2, 2) or S0(1, 3) respectively. The w’~ equation of motion is =

—

=

dea

+ wbe,

(2.6)

and provided that det(e~)‘‘ 0 the field ~ab can be solved for completely in terms of e~.When this is substituted into the field strength of the S0(1, 2) subgroup =

dw~’ + ~ac~b

(2.7)

this will coincide with the curvature tensor. On shell the action 13 coincides with

216

A.H. Chamseddine

/

Topological gravity

the Einstein action

f

3x.

(2.8)

M~%/~(R(g)+A)d

At the quantum level the two actions differ, and it was shown that 13 is finite [5]. Witten has argued that the topological gravity action undergoes a phase transition from an unbroken phase of quantum gravity, where the fluctuations are around (e 0, w 0), to a broken phase [51. This simple theory will serve as a prototype for a quantum theory of gravity in higher dimensions. A major difficulty in this approach is introducing renormalizable matter interactions. This is due to the fact that the space-time metric cannot be used and the is a gauge field and can only be introduced in a gauge covariant way. As an example consider a scalar field çb~in the fundamental representation of the gauge group SO(1,3) or S0(2,2) (the case of ISO(1,2) can be covered by performing an Inönu—Wigner contraction). A possible matter action is =

=

f

(2.9)

EABCD43ADF, M~

where D44

=

d4~+AA 8çb~~

(2.10)

and FAB is the field strength of AAB. Using the decomposition (2.4) for AAB and writing ~A (~a,~) it can be easily seen that, to lowest orders in and in the space where det(e~)~ 0, the action (2.9) contains the terms =

f (det e) [2~a~

—

~

2+ +

3A~

. .

.1.

(2.11)

This is a first-order formalism for the kinetic term of 4~i in the presence of a cosmological constant A. Unfortunately, it can be shown that the action (2.9) is not renormalizable and that counterterms of the form (~E )‘~EABcDcf~DçtBFCD must be added. It is possible that the only consistent matter interactions are the ones that result from the compactification of the topological theories of gravity in higher dimensions. We next turn our attention to the topological theory of gravity in two dimensions. This is important because of its relation to non-critical strings [9]. By adopting the gauge principle, the two-dimensional gravity action was taken to be [10] I2_kfABC~F,

(2.12)

A.H. Chamseddine

/

Topological gravity

217

where FAB is the field strength of the SO(1,2) group, the de Sitter group in two dimensions. The result for ISO(1, 1) could be obtained through an Inonu—Wigner contraction. The action (2.12) was shown to be anomaly free, and finite [101.Connection with gravity is made through the identification 2

mat,

=

Aa

=ea,

a

=

0,1.

(2.13)

The 4r~equation of motion implies that FAB

=

0

and this, in the space where det(e~)* 0, is equivalent to the Liouville equation [101. At this point, one may wonder whether the scalar multiplet 4A is necessary, and whether a pure geometrical action for two-dimensional gravity can be found. To answer this question, we assume that the three-dimensional gravity contains the information about the two-dimensional gravity. First, we dimensionally reduce the three-dimensional theory to two dimensions to obtain

f

EABC(4~AFBC+eADXBC),

(2.14)

M 2

where ~A

_=XAJJ.

~AB

(2.15)

To preserve only the S0(1,2) SL(2, l~)symmetry, one possible truncation is to set e~to zero, and the action coincides with (2.12). possible truncation 2 to zero,(2.14) and the resulting action canAnother be shown to be equivalent is to set 4~ andsome e to (2.12), after reparametrization. The form of the action (2.12) can also be supported from a different direction, by looking at the reduction and truncation of the three-dimensional metric action

f ~/~Rd~x. This is done by setting g~p=(h~

~

(2.16)

(2.17)

where h~ is the two-dimensional metric. The action reduced from (2.16) takes the form f~1~R(h)d2x.

(2.18)

218

AN. Chamseddine

/

Topological gravity

In dimensions other than two, the scalar factor 4 can be absorbed by rescaling the metric hap. This, however, is not possible in two dimensions because rescaling the metric hap by h~

—~

e~’hap

(2.19)

results into the rescaling of (2.18) according to (2.20)

f~/i~[4)R(h)+y4)Ll4)]d2x.

Thus the scalar field 4) cannot be separated from the metric hap. It is amusing to note that if one takes the action (2.20) as the candidate action for two-dimensional gravity in the metric approach, then in the path integral both hap and 4) enter. Equivalently,

z =f~h D4) DXexp{—ft~[4)R(h) + y4) u 4)] where DX refers to the matter fields. The explicitly and the result is

z

fDh DXexp{~fR

4)

d2x}e~natt~r,

(2.21)

integration can be performed

n ‘R d2x} (det u)’~2e. -

(2.22)

Thus the effect of the 4) field is to change the local action (2.20) into the non-local “induced” gravity action considered by Polyakov [11]. The factor (det LI) 1~~2is usually the contribution of one scalar field. Thus the gravity action (2.20) can be quantized exactly in the same way as for the induced gravity action, with the minor modification that the contributions of d scalar fields be replaced by d + 1. From these considerations it should be clear that including the scalar multiplet 4r4 in the topological action (2.12) is not only inescapable but also desirable. The problem of introducing matter interactions for the three-dimensional topological theory is also present in the two-dimensional case.

3. Topological gravity in dimensions higher than three It has been known for some time that the Einstein—Hilbert action can be derived in the first-order formalism using the vielbein and the spin connections [12]. Alternatively, it can be obtained by gauging and constraining the Poincaré group or its de Sitter generalization [13]. This action is not then a standard gauge theory, and the non-renormalizability properties are analogous to those of the metric theory [14]. Motivated by the encouraging results of the three-dimensional

A.H. Chamseddine

/

Topological gravity

219

theory, it is natural to attempt the generalization of these properties to higher dimensions. The use of the Chern—Simons form was essential in having a gauge invariant action without constraints. These forms exist only in odd dimensions [15]. The metric on the space-time manifold is not used and the actions constructed this way are topological. The gauge group must be chosen to be the Poincaré, de Sitter or anti-de Sitter groups in that dimension. These are ISO(1, 2n), SO(1, 2n + 1) or SO(2, 2n) in 2n + 1 dimensions. We shall deal with the de Sitter or anti-de Sitter cases and indicate how to recover the Poincaré case by Inonu—Wigner group contraction. It is essential to use the (n + 1) invariant form (JA

1BIJA2B,

...

J~

1B1)

where ~AB is the group generator and A, B The action is taken to be [6]

‘2n+1

CAB

=

=

=kf

(3.1)

AD

0, 1,..., 2n

+

1.

(3.2)

W2n+l~ M2,, ±

where

w2,~1

is the (2n

+

1)-form [15] 1~t(A(tdA+t2A2)n),

(3.3)

w2~+1=(n+1)f where A =AA~~JABis the Lie algebra valued one-form. Under a gauge transformation the gauge field A transforms as Att=g_IAg~i~g_l dg,

(3.4)

and the Chern—Simons form transforms as [15] n!(n+1)! (2n+1)!

2 ((g_1dg)fl+l~.

(3.5)

Here a 2~is a 2 n-form which istoa the function of number. A and g’ the ISO(1, integral2n), of 2’~1)is proportional winding Fordg, the and groups ((g’ dg) SO(1,2n + 1) and SO(2,2n) the homotopy H 2~~1 of these groups is proportional to the torsion, and the winding number being insensitive to torsion, vanishes*. To illustrate this consider the five-dimensional case where the relevant groups are 5 is related to the fifth cohomology SO(2,4)H5(M or SO(1, 5). The integral5(M of (g~ dg) groups 5, 3r5(SO(2, 4)) or H 5, ir5(SO(1, 5))). The homotopy groups were *

R. Bott, private communication.

220

AN. Chamseddine

/

Topological gravity

computed by Hilton [24]: ‘7T5(SO(1,5)) =ir5(SO(5)) =Z2, ir5(SO(2,4)) —3T5(SO(4)) =Z2+Z2, and are known as torsion [25]. (An element in a Z module is said to be torsion if some integral multiple of it is zero.) However, since the cohomology of a manifold is computed using de Rahm theory 5with real coefficients, these not sensitive to vanishes. This explains whyare homotopy groups torsion and the integral of (g’ dg) such as ii5(S0(4)) took longer to be computed. The above analysis implies that the coupling constant k is not quantized and this makes the theory more acceptable to describe gravity. Thus for manifolds without boundary aM2~±1 is zero and the action ‘2n + is gauge invariant. For manifolds with boundary to achieve gauge invariance one must require that either A or g~ dg vanish at the boundary 3M2~~1. For simplicity we shall assume that the manifold M2~~1is without boundary. In analogy with the three-dimensional case, connection with gravity is made through the identification 2~t=ea

a=0,1,...,2n.

(3.6)

Aa,

Aab=~~

The surprise here is that the action (3.2) can be written in such a way that the e” appear without derivatives and the ~ab enter only through the field strength of the SO(1, 2n) subgroup. This is a very appealing result, and is possible because of the special nature of the Chern—Simons form. For manifolds M 2~+ without boundary the action (3.2) takes the form ‘2n+1

=kf

E

‘~ 10

~aa 2,,+I

xR”’~2A

. . .

A

1 A 21+1

(~)

Ra2~_2,_1,~~2,_2, A e’~2~_2/+

A

eQ2~d1 ,

(3.7)

where

A=

1 —1 0

forSO(2,2n), forSO(1,2n+1), for ISO(1,2n),

(3.8)

and =

dw~th+ ~/ww~b.

(3.9)

A.H. Chamseddine

/

221

Topological gravity

Notice that in the ISO(1, 2n) case only the first term A

EaaRa1~~2

. . .

A

(3.10)

R’~2~_’~2~ A e~2~±1

remains. The Einstein term is only present in the de Sitter cases. In particular, the action 15 in five dimensions reads 1~ kfM

Ca

=

as(ealRa2a3Ra4as + ~Aea1ea2ea3Ra4a5

2ea1.

+ ~A

. .

e~). (3.11)

5

The first term is a generalization of the Gauss—Bonnet form, the second term is the Einstein action in five dimensions while the third term is a cosmological constant. The general action ‘2n + is the sum of Euler densities. Such a sum has been considered before as a generalization of the Einstein action [16]. The important difference here is that the full action is fixed by gauge invariance. This term was also encountered before in the low-energy limit of string theory. The appearance of the higher derivatives was shown not to generate ghosts [17]. The classical equation for AAB is given by CA1B

ABF22

A

...

(3.12)

A FA,,±1~±I

Decomposing this equation according to (3.6) results into the two equations Ca

Ea

a2(Raltl2

a2Ta(

+ AeaIe~~2) . . . (Ra2~_Ia2~ + ~

R’~2’~3 + Ae~2a3).

. .

=

0,

(Ra2~_2d12~_l+ Ae~I2a_2e22~_1) =

(3.13a) 0, (3.13b)

where T’~is the torsion given by =

de’~+ w2be~~~.

(3.14)

The full moduli space of eq. (3.12) is not known but contains the moduli space of flat ISO(1, 2n), SO(1, 2n + 1) or SO(2, 2n) connections according to the value of A. More generally, if we allow only T~to be zero, then the second equation of (3.13) will be satisfied. If we further restrict ourselves to the space of invertible e, then ~ab is completely determined in terms of e, ~ab

=

~

+

—

Q~~)e~ePb,

(3.15)

where =

(a.~e~ 3~e~)eap. —

(3.16)

222

A.H. Chamseddine

/

Topological gravity

With ~ab as given in eq. (3.15), the field strength Riemann tensor of the metric g~ eepa [181:

Rd”

becomes related to the

=R”~(g).

(3.17)

=

eaepbR~(w(e))

Eq. (3.13a) becomes a generalization of the Einstein equation. Such a system of equations has been studied before in connection with spherically symmetric black hole solutions [16]. We have thus constructed a theory of gravity in odd dimensions, which is explicitly gauge invariant, and is a generalization of the Einstein theory. It is extremely important to address the question of renormalizability of these theories. Since in dimensions higher than three no quadratic terms appear, the background field method must be used [19]. By expanding around a non-zero classical solution a

(e0 ,w0

ab

ea

=

eoa +

ea,

~ab

=

ab +

(3.18)

~,

where ~a and i~i’~ are the quantum fluctuations, a quadratic piece is generated. This is similar to the situation in Einstein gravity where one expands the metric g~,around a background metric. The propagators can be read from the quadratic part of the action, and the perturbative analysis can be performed just as in a standard gauge theory. The renormalization analysis is lengthy and must be carefully performed. This is presently under study and the results will be reported somewhere else. We expect these theories to be finite. This is based on the heuristic argument that there are no potential counterterms. Using both gauge invariance and the topological nature these actions are unique. In reality, to each action we constructed using the (n + 1) invariant form (3.1) there exists a corresponding action that uses the standard trace Ii TCD\_S~CD trkJABJ )UAB.

These, however, are of opposite parity and are not expected to arise as counterterms. This was already tested in the three-dimensional case [201. Up to this point we have only dealt with odd-dimensional spaces, but of equal interest are the even-dimensional spaces. Here there is no natural geometric candidate such as the Chern—Simons form. The wedge product of n of the field strengths can make the required 2 n-form in a 2 n-dimensional space-time. The natural gauge group is ISO(1, 2n 1), SO(1, 2n) or S0(2, 2n 1). With the last two, the group epsilon tensor EAA can be defined. The ISO(1,2n 1) case can be recovered by the group contraction. To form a group invariant 2n-form, the n-product thefundamental field strength is not enough, but will require in addition a scalar 4 inofthe representation. field 4r —

—

—

AN. Chamseddine

/

223

Topological gravity

The 2n-dimensional action is then

I2~=kf

A

A=0,1,...,2n,

~

(3.19)

M2,,

where F~dAAB+AACAcB.

(3.20)

This can also be inferred by reducing the (2n + 1)-dimensional action (3.2) to 2n dimensions. This should contain in addition to the 2n-dimensional gravity a scalar field (both described by the action (3.20)) and a vector field. We have seen a similar situation in the two-dimensional case. It is very illustrative to study the simple dimensional reduction of the five-dimensional theory to four dimensions, in order to be convinced of the form of the action (3.20). The action (3.11) when dimensionally reduced, reads 4x 14

=

3kf

[A2e,~

e~2e~a3e~a4e

E~~~PUEa a

d

5

~

4a5 + ~Ae~1 e~2e~a3R~4a5

+

~

+

~

,

(3.21)

where ji, ii, p, ci are the curved indices on M4, and ~ can be read from (3.9). It is easily seen that the SO(1,4) or S0(2,3) subgroup of S0(1,5) or S0(2,3) respectively, can be preserved in the truncation e~ 0. The surviving term is the last term in (3.21) and coincides with the four-dimensional action of (3.20). This is, however, not the natural truncation. By rewriting eq. (3.21) in terms of the four-dimensional fields we obtain =

4x 14

=

3kf

d

[A2e

E~i~VP~Eapya

e~e~~e~e 4A2eej3ep~eu4ela+ 4 ~ 4 —

+4Aee~e~4R~,

4R~

+ 4~ 2Aee~e4

+

+ ~4 4Ae e~e41~R~~

0.~ + ~ +

~

+2e~°Rf34R~~. 4R °‘~R~~ + 4C +e4 4~’ 4~~R1l]~RPffa4] (3.22) .

/ Topological gravity In this decomposition (e,~”,w~°0) will describe the four-dimensional graviton, (e, w,~”4)the vector, and (e 4, ~) the scalar. The standard truncation is to set 4 e~4,w’t, e 4 and w 4 4’~to zero which is consistent with the remaining SO(1, 4) or SO(2,3) de Sitter invariance. If we further relabel 224

AN. Chamseddine

—

ab_(

a

4

w~ ~ ap\

~

—

4

a

and integrate by parts, the truncated 14 can be rewritten as 14

=

3kf

E~P~~Eabcde43aRbcRde,

(3.24)

M 4

where =

dw~th+ ~ac~b

(3.25)

This coincides completely with the simple truncation and shows the uniqueness of the group invariant action (3.20). Again, a similar situation is encountered in ten-dimensional supergravity [21] where the dilaton field appears as a partner of the graviton. This is also true for the low-energy limit of string theory [20]. The analogy with ten-dimensional supergravity is amusing, because this was first constructed [21] from eleven-dimensional supergravity [22]. The dilaton field results from the truncation of the eleven-dimensional graviton to ten. Similarly here, the eleven-dimensional theory is a pure gauge theory and truncating it to ten dimensions would result in the dilaton being coupled to gravity.

4. Propagation and dynamical analysis The Chern—Simons gravitational action (3.2) contains the metric theory in the subspace where T’~is zero and det(e~)* 0, and is thus more general. The theory has different domains. In the unbroken phase, the fluctuations are around e’~ 0 w~. The concept of propagation, in the perturbative sense, does not apply here because there is no bare quadratic piece in the action. In the broken phase where the fluctuations are taken around a non-trivial classical background (eoa, ~0ah) the propagators exist and the perturbative analysis can be performed. For illustration we consider the simplest non-trivial example, the five-dimensional action (3.11). By expanding around a background (e0’~,~0ab) using eq. (3.18), the lagrangian ./ can be written =

=

~y

,

k

=

1)

(4.1)

A.H. Chamseddine

/

225

Topological gravity

where ~J7O =.j~~~(e

0a, oocth),

(4.2a)

=

3kEabcde[2T~(RObC +

=

3kEabcde[(RO’~ +

Aeobeoc)i~+

(R0~L~ + AeoaeotJ~)(Ro~I + Aeoceji)ënl

Aeoaeob)(2ëcDoi~~t~ + w~’w~’eof) + T

bCDO~de1.

(4.2b) (4.2c)

Here ë’~and ~ab are the quantum fluctuations and D0 is the covariant derivative with respect to a/i =

di.YTh +

ac—b + ~~jac~b

(4.3)

In the expansion (4.1), ../° is the background lagrangian and ~/1 is proportional to the equations of motion and vanishes when expanding around a classical 1’ solution. From ~/2 satisfying it is evident that the propagators for the fields ~a and w” exist for background Ro~~+Aeo~~eob*0,T~*0.

(4.4)

If only T 1~vanishes, the propagators correspond to those of a metric theory with a generalized Einstein action of the form (3.11) but as a function of the metric g~. If both R0’~”+ Aeo~~eob and T~vanish, then the quadratic part ~‘2 vanishes. This should come as no surprise since the vanishing of these quantities implies that the background gauge field is a flat SO(1, 5) or SO(2, 4) connection. This modulo gauge transformations, is locally equivalent to zero. Of course, globally there could be differences related to the non-trivial topology of the space-time manifold. Thus the expansion around non-trivial solutions shows that the broken phase of the topological theory occurs and this gives rise to a propagating graviton in five dimensions. To be more precise, in the broken phase there are different domains depending on the different classes of the background solution. One interesting domain occurs for the background solution T1~=0, RoaP+AeoaeoP=0, + Ae0”e0~*

0.

a=0,1,...,4 a=0,1,2,3 (4.5)

226

AN. Chamseddine

/

Topological gravity

An explicit solution satisfying the above properties is 1 e0~~11,

=

—

A2

~=0,1,...,4

(~~P_~P~a) 1— ~AX’~Xa

4 c (constant), (4.6) e04 where ~a is a curved index on the four-dimensional submanifold. The solution (4.6) satisfies the equations of motion (3.13) and compactifies the five-dimensional space to four. In this domain, it can be shown that only the four-dimensional graviton propagates with the other possible excitations locked into the topological phase. To see this, we explicitly write ~/2 of eq. (4.2c) using the background (4.6): =

=

— 1

—

6kAc ~Axax<~[e:Doc~v

~

—

i — 1

—

(4.7) To this order the ~aP~a(1

wi~’~equation

of motion gives

_+AXaXa)[(Bp~ay_3a~py)

—

(0y~pa3p~ya) +

(aaeyp—ayeap)], (4.8)

where (4.9)

(9a~py

This result can be compared with the one obtained from the standard Einstein action (in the first-order formalism) with a cosmological constant in four dimensions: 14

=

3kAc ~

f d4x

+

~Ae~~’e~4) ,

(4.10)

where R~

=

+

ay$

—

(4.11)

It can be easily checked that (e~,w~’~) as given in eq. (4.6) satisfy the equations

AN. Chamseddine

/

Topological gravity

227

of motion of the action (4.10). Also the quadratic part of (4.10) in the expansion around this solution coincides with eq. (4.7). When the group is ISO(1,4), the solution (4.6) becomes trivial (A 0). A non-trivial solution is given by =

(4.12)

4~a

e0~’=6~,

eo4=l~~x’~xa,

~04a

with all other components vanishing. Repeating the same steps as from (4.7) to (4.11) shows that the four-dimensional graviton propagates. Having specified the conditions for which the propagators can be defined, the remaining issue is to determine the dynamical degrees of freedom. This is necessary since the existence of propagators does not guarantee that the theory is non-trivial, free of ghosts and propagates only the massless graviton. For example in the three-dimensional case, although a propagator is present, there are no dynamical degrees of freedom because the equation of motion is F 0. This implies that locally, modulo gauge transformation, the gauge field can be set to zero. In higher dimensions the equations of motion (3.13) are highly non-trivial. In the absence of a known general solution satisfying eqs. (3.13) and (4.4) I shall determine here the dynamical degrees of freedom in the general gravitational phase (defined for vanishing torsion and e~invertible). As is standard in general relativity this can be done by looking at the linearized equations. To avoid the complications of dealing with the general case and for illustration I shall analyze the five-dimensional case. Substituting the solution (3.15) for vanishing torsion and using eq. (3.17), into the action (3.11) gives =

I~ kf~/~~ d5x[ =

~

+ 2A~1~R,~,7>’(g) + A2(4!)],

where the Kronecker delta is completely anti-symmetric in K, A, ~j, r, and similarly for 5~. The equation of motion with respect to the metric g~ is ~

E~~VP~YCKA,,To(~R/AVKA( g)

R~1(g)

~t,

+ AR~WKA(g)&,~?~ + A2~~&~)

=

v, p,

(4.13)

ci

and

0. (4.14)

We now linearize eq. (4.14) by expanding ~ where

i~

(4.15)

is the Minkowski metric 71~~=diag(1,—1,—1,—1,—1),

(4.16)

228

A.H. Chamseddine

/

Topological gravity

and keeping only terms linear in hg,,. This is permissable although ~ does not solve eqs. (4.14). Nevertheless it has always been used in analyzing de Sitter gravity. The linearized equation now reads 6A2~~

_a2h)

(a~a”h_a,~0Uh~~ _a~aUh~+32h;)J

+

,

(4.17)

where h ~ and indices are raised and lowered with the Minkowski metric and where we have used [181 =

=

~

—

a~a’-’h~~ —

+

~

(4.18)

From eq. (4.18) it is very simple to see that no derivatives higher than two appear in the action (3.13). Working in the harmonic gauge [261 g~T~(g) =0,

(4.19)

which for the linearized field takes the form =

+a~h,

(4.20)

eq. (4.17) then simplifies to —

~~92h

=

6A~,

(4.21)

or equivalently, =

(4.22)

—4A~L~.

Note that in the form (4.13) the only invariance left is that of general coordinate transformations. Here we see that eq. (4.22) is still invariant under the residual gauge transformation —~

h~+

(4.23)

+

where ~ 0, and this respects the harmonic gauge (4.20). Counting the degrees of freedom, h~, being a symmetric tensor, has 15 components. The condition (4.20) eliminates five components and the gauge transformation (4.23) eliminates another five leaving five degrees of freedom. This conforms with the standard counting of gravity where the dynamical degrees of freedom of a graviton in D dimensions is known to be ~(D 2)(D — 1) — 1. The above analysis can be repeated for higher dimensions without obstacles. On the other hand, determining the degrees of freedom in the non-gravitational case =

—

/

AN. Chamseddine

Topological gravity

229

not an easy task since here e and are independent fields and in the absence of explicit solutions no certain statements can be made. This is the domain where non-perturbative effects are expected to play a role. is

5. Supersymmetrization Up to this point we have dealt only with bosons, and we must also discuss how to incorporate fermions. This can be achieved by considering the gauging of supergroups, or by extending the space-time manifold to a supermanifold, or both. To avoid the complexities of superspace, we shall limit our considerations to graded groups which are extensions of the Poincaré, de Sitter or anti-de Sitter groups. The supergroups that are extensions of de Sitter or anti-de Sitter groups have been classified by Nahm [8]. The choice here is limited and the list stops at the supersymmetric extension of 0(6,2) where the space-time dimension is seven. No such limitation exists on the supersymmetric extension of the Poincaré groups. We shall list here the de Sitter or anti-de Sitter groups relevant to space-time dimensions of four or more. The supergroups are [8]: (0(3,2) eO(N),(4,N)),

N= 1,2,...

(o(4,1)~U(1),4+~) (0(4,2) ~U(N),(4,N) (0(4,2) ~ SU(4),(4,4)

(51)

—

+ +

D—4

(4,N)),

N*4 D—5

(4~))

(0(6,1)®SU(2),(8,2)) (0(5,2)GSU(2),(8,2)) (0(6,2) ~ SU(N,q),(8,2N)),

N= 1,2,...

52

-

(

D—6

.)

53

-

(.)

D

=

7.

(5.4)

SU(N, q) is the group of unitary quaternionic N x N matrices. The notation used is such that the bosonic part of the supergroup is shown, and the representation of the fermions with respect to the bosonic part is indicated. The most interesting models in the de Sitter or anti-de Sitter cases are for the six- and seven-dimensional spaces, as these correspond to N 4 supergravity in four dimensions. The super Poincaré groups have to be investigated separately. In this paper our aim is to establish the method of construction of the theories leaving the more difficult task of finding a realistic model to future investigations. To this end a good example illustrating the construction is the five-dimensional theory. =

/

AN. Chamseddine

230

Topological gravity

The supergroup (5.2) is isomorphic to SU(2, 2 IN), the group of graded matrices leaving the quadratic form Z~FABZA invariant, where [20]

0 I

~‘4

0

=

The gauge field denoted by matrix representation:

diagonal(1, 1, —1,

4)AB

—

1).

(5.5)

belongs to the superalgebra and has the following

~

M~ ~ a ~a a A1’

(5.6)

t(Y4)~

(5.7)

where (~‘)

=

and the indices i, j are in the fundamental representation of SU(N). The matrices ‘~B satisfy the property cP= —Fcl’F,

(5.8)

which, in terms of components becomes Mt= —y 4My4,

At= —A.

(5.9)

We can express M,j~in a basis of 4 x 4 Dirac y-matrices: Ma~

=

where the

Ya

[~-iô~v+

+

~(ya)a’~~

~(yab)a~O-~~bl ,

(5.10)

matrices are defined by 2&~b’

5

a=1

{Ya,Yb}

_!f

I-—

Ya — Ya e4

=

‘

ie°

Yab — 2 1

~4a

=

Ya’ Yb

j~Oa

(5.11)

We have used, and only for this section, the notation of ref. [23] to avoid the cluttering of i in the component form of the action.

AN. Chamseddine

/

231

Topological gravity

The action is given by the five-dimensional Chern—Simons form, but with supergroup SU(2, 21N) as a gauge group: 1s

=

ikf

Str[tli dcP d’P + ~cI~ diP + M 5

Str[4)AB] =0,

(5.12)

where the supertrace is defined as StrXAB =Xaa ~

(5.13)

In view of the property (5.8) the factor i is introduced to insure that 15 is real. 0, 4’a’ The action be expanded in The termsresults of Mafor the and A,’. The (5.13) traces can of the y-matricesand cancomputed be furtherexplicitly evaluated. pure gravity part will naturally agree with (3.11) (apart from the i factor that has to do with the e4 notation). We simply quote the result of a straightforward but tedious computation: ./=iktrap(MdMdM+ ~-M3dM+ ~M5) —iktr,

3dA 1(AIAdA

+ ~ik(D~ij0X;/i/ia’

+

+

+

~A5)

~A

ç1JJ0X~/D~/Ja’),

(5.14)

where X (dM

2)a0~3j’+

0”/

=

(dM

+ M2)a0

=

3-i dv ~

~,f’(t4

+A2),’

—

+M + ~Ta(ya)aP

+

3-(R~+ eaeb)(yab)aP

(5.15)

and the covariant expressions are D~

=

dlP•a’

+ Ma0 Ill~’+

=

d~

+

ip

0M 0~+A1j~,

=

de”

+

waet~ (5.16)

Rab=dW~~h+~acWct5.

The bosonic part of (5.14) can be expanded further in terms of v,

ea

and

~ab

to

232

AN. Chamseddine

/

Topological grat’ity

give

~bosonic

=

1jk

[eaR~R~

+

3-eaebecR~]+ ~eaehecedee +k(~

— 3-k(TaTa _RahRab)

v

+

—k tr

—

~)vdvdv

2)(dA’ +A’2))v 11((dA’ +A’

—iktr,

3dA’+3-A5),

(5.17)

1(A’dA’dA’+3-A’ where

=

1 -~~~‘ +A’~, 3v

A”,

=

0.

(5.18)

In the special case when N 4 we can consistently set v to zero [2]. The gravitino ~ is a Dirac spinor with 4N components. When viewed from four dimensions this corresponds to 2N supergravity. This theory differs from the usual Poincaré supergravities in the field content. The nearest theory to this is the N 1 supergravity in five dimensions [23] which has an identical field content (for N 1), but the two actions are different. The main difference being the role of the vector field A. Here, the vector field, unlike the graviton and gravitino does not seem to propagate. We were unable to find a phase where the propagation occurs. It is possible that the role of the vectors is only to extend the symmetry of the fermions. In this respect, gravitation is special because the fields e” and ~ah conspire to give the necessary second-order propagator for e”. The gravitino, being a fermion, has the standard kinetic energy to lowest order in e’~.The vector, being a boson, and having no partner cannot acquire a second-order propagator, even in the non-topological phase. For the seven-dimensional theory, the maximum for de Sitter symmetries, we can give a description of the matrix representation of the gauge fields. The easiest representation is the one using quaternionic matrices. This is due to the isomorphism SO(6,2) S0~(8) SO(4, q). The relevant group is thus OSP(4/N) over the quaternionic field. Equivalently, we can take (8 + 2N) x (8 + 2N) real matrices by using the 2 x 2 matrix representation of quaternions. The relevant theory is the seven-dimensional Chern—Simons form. The prospects for this model to become realistic are minimal. This is due to the difficulty of obtaining chiral fermions with de Sitter symmetry. Theories based on the super-Poincaré groups are more promising. To search for a realistic model, it will be necessary to analyze the possible compactification schemes. =

= =

A.N. Chamseddine

/

233

Topological gravity

6. Summary and conclusions In this paper we have constructed topological actions for gravity in all dimensions. The odd-dimensional theories are based on the Chern—Simons forms with the gauge groups taken to be ISO(1,2n) or SO(1,2n + 1) or SO(2,2n) depending on the sign of the cosmological constant. The even-dimensional theories use in addition to the gauge fields, a scalar multiplet in the fundamental representation of the gauge group. The relevant groups here are ISO(1,2n — 1) or SO(1,2n) or S0(2,2n —1). We gave a supporting analysis for the necessity of the scalar multiplet. In particular, we showed that if in the metric approach to two-dimensional gravity a scalar field is introduced, then the effective theory is equivalent to the one introduced by Polyakov [11]. We argued that these theories when considered in perturbation theory, should be finite. The proof of this is presently under study. The issue of propagation in these topological theories is then considered, and it is shown that there are different domains. In the unbroken phase, the fluctuations are around A 0, and there is no propagation in the perturbative sense. When the fluctuations are around a non-trivial background, then the graviton propagates. In the region where the fluctuations are around a non-trivial background but with vanishing torsion, the propagators agree with those of a metric theory. We also found an interesting background where the five-dimensional theory compactifies to a four-dimensional one, and only the four-dimensional graviton propagates. Supersymmetrization can be carried out by simply generalizing the gauge group to a supergroup. For the de Sitter groups, the maximal group is OSP(4/N) over the quaternionic field and the relevant dimension is seven. No such limitation exists on the super-Poincaré groups, but whether these can result in realistic models remains to be seen. In short, we have found a possible new avenue for unifying gravity with the other interactions. We are hopeful that these theories are finite. The interesting questions are to check this explicitly and to search within this framework for realistic models. The difficulty of introducing matter interactions in this approach suggests that we look for the unified theory in higher dimensions, and obtain the fourdimensional theory by compactification. We hope that we have given enough motivation to look further into this new direction. =

I would like to thank J.-P. Derendinger for helpful discussions.

References 11] H.P. Nilles, Phys. Rep. 110 (1984) 1, and references therein [2] P. van Nieuwenhuizen, Phys. Rep. 68 (1981) 189 [31MB. Green, J.H. Schwarz and E. Witten, Superstring theory, 2 Vols. (Cambridge Univ. Press, Cambridge, 1987)

234

AN. Chamseddine

/

Topological gravity

[4] S. Deser, J. Jackiw and G. ‘t Hooft, Ann. Phys. (N.Y.) 152 (1984) 220; S. Deser and R. Jackiw, Ann. Phys. (N.Y.) 153 (1984) 405 [5] E. Witten, Nuci. Phys. B311 (1988) 96; B323 (1989) 113 [6] A.H. Chamseddine, Phys. Lett. B233 (1989) 291 [7] M. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rep. 130 (1986) 1 [8] W. Nahm, Nucl. Phys. B135 (1978) 149 [9] A.M. Polyakov, Phys. Lett. B103 (1981) 207, 211 [10] A.H. Chamseddine and D. Wyler, Phys. Lett. B228 (1989) 75; NucI. Phys. B340 (1990) 595 [11] A.M. Polyakov, Mod. Phys. Lett. A2 (1987) 899; V.G. Knizhnik, A.M. Polyakov and A.A. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819 [12] R. Utiyama, Phys. Rev. 101 (1956) 1597; T.W.B. Kibble, J. Math. Phys. 2 (1961) 212 [13] A.H. Chamseddine and P.C. West, Nucl. Phys. B129 (1977) 39; SW. MacDowell and F. Mansouri, Phys. Rev. Lett. 38 (1977) 739 [14] M. Grisaru and P. van Nieuwenhuizen, in New pathways in Theoretical Physics, ed. B.B. Kursunoglu and A. Perlmutter (Coral Gables, 1977) [15] B. Zumino, in Relativity, Groups and Topology II, Proc. Les Houches Summer School (1983), ed. R. Stora and B. de Witt (North-Holland, Amsterdam, 1984) [16] D. Lovelock, J. Math. Phys. 12 (1971) 498; 13 874 (1972); B. Zumino, Phys. Rep. 137 (1986) 109; F. MüIler-Hoissen, Phys. Lett. B163 (1985) 106; D.G. Boulware and S. Deser, Phys. Rev. Lett. 55 (1985) 2656; J.T. Wheeler, NucI. Phys. B268 (1986) 737; R.C. Myers and J.Z. Simon, Phys. Rev. D38 (1988) 2434 [17] B. Zwiebach, Phys. Lett. B156 (1985) 315 [18] M. Veltman, in Les Houches Summer School (1976), ed. R. Balian and J. Zinn-Justin (North-Holland, Amsterdam) [19] B. Dc Witt, Phys. Rev. 162 (1967) 1195, 1239; J. Honkerkamp, NucI. Phys. B48 (1972) 269; G. ‘t Hooft and M. Veltman, Ann. Inst. Henri Poincaré 20 (1974) 69 [20] L. Alvarez-Gaumé, J. Labastida and A. V. Ramallo, NucI. Phys. B334 (1990) 103; E. Guadagnini, M. Martellini and M. Mintchev, Phys. Lett. B227 (1989) 111; C.P. Martin, Phys. Lett. B241 (1990) 513; F. Delduc, C. Lucchesi, 0. Piguet and S. Sorella, Geneva University preprint UGVA-DPT 1990 12-653 [21] A.H. Chamseddine, NucI. Phys. B185 (1981) 403 [22] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B76 (1978) 409; E. Cremmer and B. Julia, NucI. Phys. B159 (1979) 141 [23] A.H. Chamseddine and H. Nicolai, Phys. Lett. B96 (1980) 89; E. Cremmer in Superspace and supergravity, ed. S. Hawking and M. Rocek (Cambridge Univ. Press, Cambridge) (1981) [24] P.J. Hilton, An introduction to homotopy theory (Cambridge Univ. Press, Cambridge) p. 88 [25] R. Bott and L.W. Tu, Differential forms in algebraic topology (Springer, Berlin) [26] 5. Weinberg, Gravitation and Cosmology: principles and applications of the general theory of relativity (Wiley, New York)