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Comformal higher spin in (2+1) dimensions
Volume 225, number 3
PHYSICS LETTERS B
20 July 1989
C O M F O R M A L H I G H E R S P I N I N (2+1) D I M E N S I O N S C.N. P O P E and P.K. T O W N S E N D Centerfor Theoretical Physics, Texas A&M University, College Station, TX 77843-4242, USA Received 10 April 1989
The conformal algebra in (2 + 1) dimensions allows an infinite-dimensional extension. We show that the Chern-Simons action for this algebra yields a generalisation of conformal gravity which contains an infinite tower of gauge fields of all integer spins greater than two, consistently coupled to conformal gravity. A similar extension of the N= 1 conformal superalgebra yields a conformally-invariant gauge theory of all integer and half-integer spins starting with spin-3.
The theorem of Haag, Lopusanski and Sohnius [ 1 ] effectively forbids the existence o f interacting, local, relativistic, four-dimensional field theories for massless particles of spin greater than two. The theorem assumes that the n u m b e r o f fields is finite. Thus it might be possible to circumvent it in a field theory with an infinite number o f fields. Most authors who have attempted to do this have reached the conclusion that a consistent massless "higher-spin" field theory would necessarily describe particles of arbitrarily-high spin. Fradkin and Vasiliev [ 2 ], in particular, have argued that a crucial ingredient for the consistency o f such a theory is the existence of an infinite-dimensional "gauge" group containing the spacetime symmetry group in question (the anti-de Sitter group in their case). Some progress has been made towards the construction, along these lines, of a gauge-invariant action, but the results are, as yet, inconclusive. There is, of course, no evidence for an infinite number of massless fields in nature, so if such a higher-spin field theory were to be constructed, we would wish to arrange for a spontaneous breakdown of the infinite-dimensional gauge invariance. The resulting field theory might then describe an infinite tower ofmassiveparticles of ever-increasing spin. This reminds us of string theory; indeed it is conceivable that string theory is a spontaneously-broken higherspin field theory. One motivation for the study o f the Permanent address: DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
latter is that it might shed light on a possible "unbroken phase" o f string theory. It was recently suggested by Blencowe [ 3] that a spacetime of (2 + 1 ) dimensions, rather than (3 + 1 ) dimensions, could be a useful arena for the study o f consistent higher-spin field theories. His starting point was an infinite-dimensional algebra extending the three-dimensional anti-de Sitter algebra. The action is just the Chern-Simons term for this algebra and it contains the Chern-Simons term for the anti-de Sitter algebra, which was shown in ref. [4] to be equivalent to the usual Einstein-Hilbert action for (2 + 1 ) gravity with a cosmological constant. It also contains an infinite sequence o f higher-spin fields. Their equations o f motion are consistent since they are equivalent to the vanishing of the field strengths of the higher-spin gauge algebra. In this letter, we report on some investigations into conformally-invariant consistent higher-spin field theories in (2 + 1 ) dimensions. The action for (2 + 1 ) conformal gravity is the Chern-Simons term for the conformal algebra Sp (4, R) _~SO (3, 2) [ 5 ]. The original higher-spin algebra of ref. [ 2 ] was in fact an infinite-dimensional extension o f SO (3, 2), viewed in that context as the anti-de Sitter algebra of four dimensions. The same infinite-dimensional algebra can, however, be considered to be an extension of SO(3, 2 ) viewed instead as the conformal algebra o f three dimensions. The main point of this paper is to show that the (2 + 1 ) C h e r n - S i m o n s action for this infinite-dimensional algebra generalises the results o f 245
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ref. [ 5 ] to a conformally-invariant action of ever-increasing spins 2, 3, 4, .... The field equations for the higher-spin fields involve higher-spin generalisations of the Cotton tensor of the graviton field equation. The supersymmetric extension is obtained by consideration of the infinite-dimensional extension of OSp(1 [4). Unlike the suggestions for higher-spin field theories in four dimensions, there is no necessity in three dimensions to introduce an infinite-dimensional extension of the spacetime symmetry algebra; a finitedimensional extension can also be consistent. One simply takes the Chern-Simons action for a n y YangMills group containing the spacetime symmetry group as a subgroup. For example, in ref. [6] it was observed that the group SU(N, N) × S U ( N , N), which contains the anti-de Sitter group SO (2, 2) ~ SU ( 1, 1 ) × S U ( 1, 1 ), yields a consistent theory for spins 2, 3 .... , N. For our purposes a useful toy example of this type is obtained from the Chern-Simons action for SO(4, 2) Yang-Mills. The adjoint of this group decomposes into the 10 (adjoint) and 5 (vector) representations of the conformal group SO (3, 2). As we shall now show, this model describes the interaction of a pure spin-2, but non-gauge, field with conformal gravity. We choose the generators of the conformal group SO(3, 2) robe (Ma, Pa, Ka, D), where a = 0 , 1, 2, and the extra generators of the group SO(4, 2) to be (Ja, U, V). Given a real 2 × 2 representation ~'afor the 2 + 1 Dirac matrices, a convenient representation for SO (4, 2) is provided by
Mo=½
D=½"~®0"3 @~
f
R a ( M ) =doga+ ½eaocogb ^ ¢0c + ~abc(2e b ^ f c _ ½sb ^ s c) , R ( D ) = d b + 2ea ^ f b r l a b - - 2 U A V, Ra(p)=dea+e%~eb ^oS-ea R a ( K ) = d f a + (.abcO,)b^ f c + f a
Au,
^ b+s a ^ v,
R a ( J ) =dsa+eabcO)b A s C + 2ea ^ v - - 2 f a ^ U, R(U)=du+b^ R(V) =dv-b
u - - e a Asbrlab,
(4)
^ v+ f ~ ^ Sbtlab ,
where qab is the ( 2 + 1 ) Minkowski metric with "mostly + " signature. These two-forms in fact coincide with the curvatures of conformal gravity in (3 + 1 ) dimensions [ 7 ], but the interpretation is of course quite different here. One can see on inspection that the equations Ra,(U)=0, RJ(P)=0, Rj(M)=O and e U ~ R j (J) = 0 can be solved algebraically forfu a, ogua, stu~ 1and vu, where su, =suae~a. Note that we are interpreting eu ~ as the dreibein, and that we are assuming it to be invertible. The solutions are Stuv ] = - O[uu~ ] -btuUul , wua= -- ( e~aeub -- ½eSe,b )e~pa( Opeob+ bpe b + ups~b) ,
1 crp -- ~1 [SuAS Av--SuvS--~guv(SpaS --S 2 ) ] , 1 t' 1 v . = - ~e a~t~s.l a - ~1 u bc09l~b S ul c + !zfu ~ -- ~-zfu~u~.
(5) i ~ V = ~ ® ® a _ ®a2
(1)
where a+ =a~ +ia2, and 0.1, 0"2, 0"3 denote the Pauli matrices. We introduce the one-form Yang-Mills SO (4, 2 ) potential A=ogaM~+bD+e~Pa+f~K~+SaJ~+uU+vV,
and the Chern-Simons action 246
^b-sa
fu~ = - ½[Rl,,(o9) - ¼R(og)gu,]
,
i Ja = ~ Y~®0"3®0"2, i
(3)
t r ( A d A + ~ A 3) .
The field equations are simply the vanishing of the SO (4, 2 ) field strengths, which are
Pa=½ 7~®0"+®~, Ka = ½ ~'a®0"- ®~,
U = ~ "~®0"+ ® 0 " 2 ,
20 July 1989
(2)
where s = s~g. f = f u~ and g.~ = e.ae~a is the three-dimensional metric. Here R.~(og) = eo~aRu°~(o9). where R ~ ( o g ) - 28fuO9~+ ¢%cO9u%9~c. The remaining independent gauge fields are therefore e. a, bg. S(u~) and uu. Their gauge transformations follow from 6A=dA+[A,A]
,
(6)
where we expand the SO (4, 2)-valued parameter A as
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A =Aa( M)Ma + A ( D )D + Aa( p)P: +A:(K)Ka+A:(J)Ja+A(U)U+A(V)V.
(7)
Substituting (2) and (7) into (6), we can read off the component gauge transformation rules. In particular, one finds that
6(K)bu=2Au(K), 6(j)uu=-A~,(J), 6
(8)
where Au = A aeu:. Thus the gauge parameters A ~(K), Aa(J) a n d A ( V ) can be used to set
b~=uu=s=O ,
(9)
where we recall that s=sUu. Hence, the only new independent field introduced by the extension of SO (3, 2) to SO(4, 2) is a symmetric traceless tensor 1 s~ ~ = s(u~)- ~sgu~. In this gauge, (5) reduces to
stu~j = 0 ,
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is the Cotton tensor density. It is symmetric, traceless, covariantly conserved and conformally covariant. Eqs. (14) follow from varying the only remaining independent fields, namely gu, and gu~, in the action
I=
f
2 a ~aF2 ~F zp~ d 3 x ~.uup (F o uzO~F3.p~+ ~F
_ 2gu~V~gp~).
(16)
This action is invariant under the conformal transformation gu,~t2Zgu,, gu~--,t2gu, for an arbitrary function g2, as guaranteed by its construction. Note that the symmetric tensor fields gu~ is not a gauge field; its gauge transformation involved the parameter Aa (J), which has been used to set u = 0. Since gu~ is traceless, it can be represented by a totally-symmetric four-index multi-spinor of SL (2, • ), g,~py6,for which the linearised action can be written as
v~=-¼V~/,
1 a w a=__ (e,, eub-- ~e~, e,,b)Eppa Opeob ,
L,, = -- ½(Ru,, - ¼Rgm,) - ½( ~ j ~ ~ - ~ g ~ ) ,
(lO)
where V~ is the usual covariant derivative defined with the Levi-Civita connection F%p = ½gUO( O,g~z+ 0pg,~- 0~g~a) and Ru,-RZup~ is the usual (symmetric) Ricci tensor. The Riemann tensor is given by
RPauv=OuFP~a-OvFPu~+FP~F'~va-FPvaF'~uo.
( 11 )
In three dimensions, it can be expressed in terms of the Ricci tensor, the Ricci scalar R=RUu and g - det(gu~ ) as follows: R % , , = I g l - 'e%aeu~( R ~ - ½Rg ~) •
(12)
Taking into account Bianchi identities on the curvatures (4), the only remaining equations of motion are
e~°~Rp/(K)e~=O,
eUP~Rpaa(J)e~=O.
(13)
These are equivalent to
(where ~u~= ~#~u# for any spinor ~#). This contrasts with the action for the graviton field h~,~-gu,,-~1~,~, for which, in two-component spinor notation, the linearised action is
f d3x h~flYaO,~'(D6~6f+O~Orq)h,an6,
,
Of Og
Of Og ,
(14)
{f, g}_=
(15)
w h e r e f a n d g are any pair of functions of q" and p,. The algebra of SO ( 3, 2) can now be realised in terms of Poisson brackets for the generators
where
CU~= euPOVp(R° u ~R6~)
( 18 )
where [] - 0"0u= - 10~P0~# is the d'alembertian. This action has the gauge invariance 6h(~pya= 0~zClya) (P). In general, the linearised gauge-invariant irreducible spin-s action (s>_- 1 ) will involve ( 2 s - 1 ) derivatives on a totally-symmetric 2s-index multispinor potential. Such fields will play a role in the infinite-dimensional extension of the conformal algebra to which we now turn our attention. We shall introduce this infinite-dimensional algebra through a realisation in terms of the coordinates (q~, p#) of a four-dimensional phase space, which we take to be two commuting two-component spinors of SL (2, R ). The Poisson bracket relation for this phase space is defined to be
C up+ e~(~Vpg~)~g~ + C°(ug~)~V~%= 0 , ~_P~('UVpSV)cr:O
17
3x
a
0q ~ Op,~
Op,~ Oq'~
(19)
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M ' ~ = q'~pp - ½(i~'~lypy , D=qYp,~,
P~'P=q~qP,
Ko~=p,~pp,
(20)
corresponding to the generators M,, D, P~ and K, that we introduced earlier. An obvious infinite-dimensional extension of SO (3, 2) can now be obtained by considering the Poisson-bracket algebra of arbitrary functions on phase space, which is the algebra of symplectic diffeomorphisms of phase space. We shall be interested in the infinite-dimensional subalgebra for which the generators are the even functions on phase space. (By an even function, we mean one for which f ( q , p) = f ( - q, - p ) . ) We denote these generators by • (a, b) '~t. . . . . fll...flb=q°"'"q"aPa, ""PBb,
(21)
where a + b is taken to be even. They can be organised into the subspaces G ( r ) = {qb(a, b ) [ a + b = r ) ,
(22)
in terms of which the infinite-dimensional algebra is G(2)~G(4)~G(6)~
....
(23)
the general structure of this algebra is {G(r), G ( s ) } = G ( r + s - 2 ) ,
(24)
which shows that whilst the SO ( 3, 2 ) generators G (2) alone form a subalgebra, the inclusion of any of the higher-order generators in (23) then requires the inclusion of all of them. With respect to the SO (3, 2) subgroup, the representations in the sequence (23) are 1 0 ~ 3 5 ~ 8 4 ~ .... We shall now show that the Chern-Simons action for this infinite-dimensional algebra is equivalent to an action for an infinite sequence of gauge fields describing irreducible spins 2, 3, 4 ..... We write the gauge field A as A=
~
~ ( a, b) °'~..... a,....a~,
{a, bla+ b even}
× A (a, b) a ' a " ....... .
(25)
The equations of motion, following from the ChernSimons action, are the vanishing of the complete set of field strengths. For the purposes of determining the spin content it is sufficient to examine the linearised equations, for which we may expand the dreibein e~,a about the flat background ~ . Thus the gauge poten248
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tial one-form A (2, 0),,p-e~p is assigned a preferred role in the theory in that those terms in a given field strength of the form e ^ A (a, b) will contribute to the linearised equations. Specifically, using the Poissonbracket relation {PY6, q)(a, b) ....... a,...ab} =2b~l~ l qO(a+ 1, b - 1 )a) ....... a2...ab),
(26)
we find that the linearised field strengths are given by R ~ ( a, b ) u, a,...a~ ....... = 20 o,A ( a, b ) ~j a'p~
.......
+ 4 ( b + 1 )et~ty,~lA(a- 1, b + 1 )~1 ~a' a~(~2..... 6a-~ ). (27) Here, ca, a is just eu" with the Lorentz index replaced by the symmetric pair of spinor indices c~fl; i.e. e~,,a=eua(Ta)jEya. As explained previously for the SO (4, 2) model, the linearised equations of motion R ~ " ( a , b) = 0 will allow some oftheA ( a - 1, b + 1 ) to be algebraically solved for. To determine which these are, we need to identify those SL(2, ~) representations of a given field strength that occur in the nonderivative term in (27). To see how this works, we shall fist concentrate on the 35 representation of SO (3, 2) which corresponds to the G ( 4 ) generators qb(4, 0); (/)(3, 1 ); q~(2, 2); qo( 1, 3); qb(0, 4). These contain the SL(2, R) representations
5; 5~3; 5~3~1; 5~3; 5,
(28)
respectively. The corresponding gauge potentials, which carry an additional vector index, therefore contain the above representations multiplied by a 3 of SL (2, ~). Similarly, because we are in three dimensions, the antisymmetric-tensor field strengths also contain the above representations multiplied by the 3. Thus starting from the Rain(4, 0 ) = 0 field equation, which contains the 3 ® 5 = 7 @ 5 ~ 3 , we see that we can solve algebraically for the corresponding representations of A(3, 1), which contains the 3® ( 5 ~ 3) = 7 G 5 @ 3 ~ 5 ~ 3 ~ 1 . We are left with the 5~33G1 representations in A(3, 1 ). Continuing this process for the remainder of the field equations, we find that we are left with the following representations inA(4, 0);A(3, 1);A(2, 2);A(1, 3);A(0, 4) respectively: 7 ® 5 0 3 ; 5 G 3 ~ 1 ; 3;
.;
(29)
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The potentials A(a, b) are still subject to gauge transformations. In particular, because of (26), a gauge parameter A(a, b) in a given SL(2, •) representation will give rise to a shift symmetry of the corresponding SL(2, R) representation in (29) of the potential A(a+ 1, b - 1 ). Hence any representation in a member of the sequence (29) that appears also in the member to its immediate right in the sequence (28) corresponds to a field that can be gauged away. For our example, this leaves only one irreducible SL(2, R) representation, namely the 7 in A(4, 0) at the left-most end of the sequence (29). It follows from the scale invariance of the lagrangian that the equation of motion for this field is the vanishing of the corresponding 7 representation in the field strength R ~in(0, 4 ) for an auxiliary gauge field at the right-most end of the sequence. This auxiliary field is solved for in terms of a single derivative of a potential appearing in R'in( 1, 3). In turn, this field is auxiliary, and is solved for in terms of a single derivative of a potential appearing in R l~n(2, 2 ). This process is continued until the physical gauge field at the left-most end of the sequence is reached. Thus the auxiliary field at the right-most end of the sequence will be expressed in terms of four derivatives on the physical field. Since the curvature itself contains one derivative, it follows that the field equation for the physical spin-3 field is of fifth order in derivatives. This is precisely what is required of an irreducible spin-3 gauge field, i.e. one for which all the lower-spin components can be gauged away. In SL(2, R) spinor notation, the linearised irreducible spin-3 action for a six-index totallysymmetric multispinor h,a.., 6 takes the form
f d3x
(30,~, ~ ...0,~,~5h~l...~,,6
+ 10[] O~,,eL..OmB3hpl...,e3 ....... + 3 ( [] ) 23,, a,hp~,2...~6),
(30)
where ( [] )2 is the square of the d'alembertian. In summary, we have seen that the 35 representation of SO ( 3, 2 ), corresponding to the G (4) generators of the infinite-dimensional algebra, ultimately describes just a single irreducible spin-3 gauge field. Analogous arguments lead to the conclusion that the subsequent 84 representation, corresponding to the G ( 6 ) generators, produces just a seventh-order equation for an irreducible spin-4 gauge field, and so
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on. Thus the final physical spin-content of the infinite dimensional theory is simply the set of all integer-spin>~ 2 irreducible gauge fields, each occuring exactly once. In fact, the linearised action is I=l
d3x
2 ~2r+/~2s1"/a...... 2sll-"l~r~ ii ] s=2 r=0
× ~,,~...~,~,_~_~ ms-~-~ha,...~_~_,,~. >...~.,
(31) where the first summation is over all integer spins s>~ 2, and ( [ ] ) r denotes the rth power of the d'alembertian. The symbol C~ is the combination factorp!/ [q!(p-q)!]. The action 31) has the gauge invariances ~ha'"'az'= 0 (ma2a ~3...azO --~s)
(32)
for the set of 2s-index multispmor totally-symmetric spin-s gauge fields. Variation of (31 ) with respect to the spin-2 field h ....... yields the linearised Cotton tensor. Variation with respect to the other fields yields the linearisations of the higher-spin analogues of the Cotton tensor. Note that the sum over spins in (31 ) could be extended to include s = 1. This would amount to the addition of U ( 1 ) factor in the algebra, corresponding to an electromagnetic field that does not couple to any of the higher spins. (In three dimensions, it is not true that everything must couple to gravity. ) One might suppose that the supersymmetric extension of the above model is obtained simply by extending the generators qb(a, b) to include those for which a + b is odd. Let us call these the odd generators. The problem is that the Poisson bracket of two odd generators is antisymmetric, whereas for a superalgebra one needs a symmetric product for odd generators. One can circumvent this problem by introducing the product [ , ] + defined by
[f+,g+]--{f+,g+}, [f_,g_]+=-2f_g_,
[f+, g_ ]_ - {f+, g_ }, (33)
where f+ and g+ are even functions on phase space, f_ and g_ are odd functions, and {, } is the Poisson bracket defined in (19). (Recall that the coordinates q" a n d p , of phase space are commuting spinors. ) One can readily verify that this product satisfies the super Jacobi identities. With the product defined above, the algebra gen249
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erated by G ( 1 ) @ G (2) is the N = I superconformal algebra OSp(114). Its infinite-dimensional extension is simply {G(1)~G(2)}~{G(3)~G(4)}~{G(5)0)G(6)} ~3....
(34)
where the bracketed terms are irreps of OSp (114). The Chern-Simons action for this infinite dimensional superalgebra yields, after elimination of all auxiliary gauge fields, in the manner described previously, an action for irreducible spins 3, 2, ~, 3, 7, etc., i.e. all integer and half-integer spins t> 3, each occurring once. This action can be consistently truncated to the N = 1 conformal supergravity action of van Nieuwenhuizen [5 ], by setting to zero all fields with spins greater than two. As we have mentioned, the infinite-dimensional extensions of SP (4, R) and OSp (114) considered here cannot be consistently truncated to any other finite-dimensional (super)algebra. However, this does not preclude the existence of a finite-dimensional algebra containing Sp(4,R) for which the Sp(4, ~) content reproduces that of th~ infinite-dimensional algebra up to some level. For example, as shown in ref. [ 6 ], the first N representations of the infinitedimensional extension of the anti-de Sitter algebra SO (2, 2) are reproduced by Chern-Simons action based on the finite-dimensional algebra SU(N, N) × SU(N, N), even though the latter is not a subalgebra of the infinite-dimensional algebra. Similarly, there may be a finite-dimensional algebra containing SO (3, 2) for which the SO (3, 2) content reproduces the sequence 10@35~84~... up to some level. If such an algebra were to exist, it would give a consistent coupling of afinite number of higher-spin gauge fields to conformal gravity.
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As far as the quantum aspects of these Chern-Simons theories are concerned, one might expect them to be finite because, as discussed by Witten [4 ] and Deser et al. [ 8 ] for pure gravity, each higher-spin field contributes only a finite number of degrees of freedom. In our case, however, there are an infinite number of such fields, so the total number of degrees of freedom is in fact infinite and the argument may not apply. Although a "theory of nothing" may be finite, it remains to be seen whether an "infinity of nothing" is finite too. We are grateful for conversations with M. Gtinaydin, E. Sezgin and K.S. Stelle. P.K.T. gratefully acknowledges the hospitality of the Center for Theoretical Physics at Texas A&M University.
References [ 1 ] R. Haag, J. Lopusanski and M.F. Sohnius, Nucl. Phys. B 88 (1975) 257. [2] E. Fradkin and M.A. Vasiliev, Ann. Phys. (NY) 177 (1987) 63; Phys. Len. B 189 (1987) 89; Nucl. Phys. B 291 (1987) 141. [3] M.P. Blencowe, Class. Quantum Grav. 6 (1989) 443. [4] A. Achficcaro and P.K. Townsend, Phys. Lett. B 180 (1986) 89; E. Witten, Nucl. Phys. B 311 (1988) 46. [ 5 ] P. van Nieuwenhuizen, Phys. Rev. D 32 ( 1985 ) 872; J.H. Horne and E. Witten, Phys. Rev. Lett. 62 (1989 ) 501. [ 6 ] E. Bergshoeff, M.P. Blencowe and K.S. SteUe, Area-preserving diffeomorphisms and higher-spin algebras, preprint Imperial/ TH-88-89/9. [ 7] M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Phys. Lett. B 69 (1977) 304. [ 8 ] S. Deser, J. McCarthy and Z. Yang, Polynomial formulations and renormalizability in quantum gravity, preprint BRX TH259 (1989).
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C O M F O R M A L H I G H E R S P I N I N (2+1) D I M E N S I O N S C.N. P O P E and P.K. T O W N S E N D Centerfor Theoretical Physics, Texas A&M University, College Station, TX 77843-4242, USA Received 10 April 1989
The conformal algebra in (2 + 1) dimensions allows an infinite-dimensional extension. We show that the Chern-Simons action for this algebra yields a generalisation of conformal gravity which contains an infinite tower of gauge fields of all integer spins greater than two, consistently coupled to conformal gravity. A similar extension of the N= 1 conformal superalgebra yields a conformally-invariant gauge theory of all integer and half-integer spins starting with spin-3.
The theorem of Haag, Lopusanski and Sohnius [ 1 ] effectively forbids the existence o f interacting, local, relativistic, four-dimensional field theories for massless particles of spin greater than two. The theorem assumes that the n u m b e r o f fields is finite. Thus it might be possible to circumvent it in a field theory with an infinite number o f fields. Most authors who have attempted to do this have reached the conclusion that a consistent massless "higher-spin" field theory would necessarily describe particles of arbitrarily-high spin. Fradkin and Vasiliev [ 2 ], in particular, have argued that a crucial ingredient for the consistency o f such a theory is the existence of an infinite-dimensional "gauge" group containing the spacetime symmetry group in question (the anti-de Sitter group in their case). Some progress has been made towards the construction, along these lines, of a gauge-invariant action, but the results are, as yet, inconclusive. There is, of course, no evidence for an infinite number of massless fields in nature, so if such a higher-spin field theory were to be constructed, we would wish to arrange for a spontaneous breakdown of the infinite-dimensional gauge invariance. The resulting field theory might then describe an infinite tower ofmassiveparticles of ever-increasing spin. This reminds us of string theory; indeed it is conceivable that string theory is a spontaneously-broken higherspin field theory. One motivation for the study o f the Permanent address: DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
latter is that it might shed light on a possible "unbroken phase" o f string theory. It was recently suggested by Blencowe [ 3] that a spacetime of (2 + 1 ) dimensions, rather than (3 + 1 ) dimensions, could be a useful arena for the study o f consistent higher-spin field theories. His starting point was an infinite-dimensional algebra extending the three-dimensional anti-de Sitter algebra. The action is just the Chern-Simons term for this algebra and it contains the Chern-Simons term for the anti-de Sitter algebra, which was shown in ref. [4] to be equivalent to the usual Einstein-Hilbert action for (2 + 1 ) gravity with a cosmological constant. It also contains an infinite sequence o f higher-spin fields. Their equations o f motion are consistent since they are equivalent to the vanishing of the field strengths of the higher-spin gauge algebra. In this letter, we report on some investigations into conformally-invariant consistent higher-spin field theories in (2 + 1 ) dimensions. The action for (2 + 1 ) conformal gravity is the Chern-Simons term for the conformal algebra Sp (4, R) _~SO (3, 2) [ 5 ]. The original higher-spin algebra of ref. [ 2 ] was in fact an infinite-dimensional extension o f SO (3, 2), viewed in that context as the anti-de Sitter algebra of four dimensions. The same infinite-dimensional algebra can, however, be considered to be an extension of SO(3, 2 ) viewed instead as the conformal algebra o f three dimensions. The main point of this paper is to show that the (2 + 1 ) C h e r n - S i m o n s action for this infinite-dimensional algebra generalises the results o f 245
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ref. [ 5 ] to a conformally-invariant action of ever-increasing spins 2, 3, 4, .... The field equations for the higher-spin fields involve higher-spin generalisations of the Cotton tensor of the graviton field equation. The supersymmetric extension is obtained by consideration of the infinite-dimensional extension of OSp(1 [4). Unlike the suggestions for higher-spin field theories in four dimensions, there is no necessity in three dimensions to introduce an infinite-dimensional extension of the spacetime symmetry algebra; a finitedimensional extension can also be consistent. One simply takes the Chern-Simons action for a n y YangMills group containing the spacetime symmetry group as a subgroup. For example, in ref. [6] it was observed that the group SU(N, N) × S U ( N , N), which contains the anti-de Sitter group SO (2, 2) ~ SU ( 1, 1 ) × S U ( 1, 1 ), yields a consistent theory for spins 2, 3 .... , N. For our purposes a useful toy example of this type is obtained from the Chern-Simons action for SO(4, 2) Yang-Mills. The adjoint of this group decomposes into the 10 (adjoint) and 5 (vector) representations of the conformal group SO (3, 2). As we shall now show, this model describes the interaction of a pure spin-2, but non-gauge, field with conformal gravity. We choose the generators of the conformal group SO(3, 2) robe (Ma, Pa, Ka, D), where a = 0 , 1, 2, and the extra generators of the group SO(4, 2) to be (Ja, U, V). Given a real 2 × 2 representation ~'afor the 2 + 1 Dirac matrices, a convenient representation for SO (4, 2) is provided by
Mo=½
D=½"~®0"3 @~
f
R a ( M ) =doga+ ½eaocogb ^ ¢0c + ~abc(2e b ^ f c _ ½sb ^ s c) , R ( D ) = d b + 2ea ^ f b r l a b - - 2 U A V, Ra(p)=dea+e%~eb ^oS-ea R a ( K ) = d f a + (.abcO,)b^ f c + f a
Au,
^ b+s a ^ v,
R a ( J ) =dsa+eabcO)b A s C + 2ea ^ v - - 2 f a ^ U, R(U)=du+b^ R(V) =dv-b
u - - e a Asbrlab,
(4)
^ v+ f ~ ^ Sbtlab ,
where qab is the ( 2 + 1 ) Minkowski metric with "mostly + " signature. These two-forms in fact coincide with the curvatures of conformal gravity in (3 + 1 ) dimensions [ 7 ], but the interpretation is of course quite different here. One can see on inspection that the equations Ra,(U)=0, RJ(P)=0, Rj(M)=O and e U ~ R j (J) = 0 can be solved algebraically forfu a, ogua, stu~ 1and vu, where su, =suae~a. Note that we are interpreting eu ~ as the dreibein, and that we are assuming it to be invertible. The solutions are Stuv ] = - O[uu~ ] -btuUul , wua= -- ( e~aeub -- ½eSe,b )e~pa( Opeob+ bpe b + ups~b) ,
1 crp -- ~1 [SuAS Av--SuvS--~guv(SpaS --S 2 ) ] , 1 t' 1 v . = - ~e a~t~s.l a - ~1 u bc09l~b S ul c + !zfu ~ -- ~-zfu~u~.
(5) i ~ V = ~ ® ® a _ ®a2
(1)
where a+ =a~ +ia2, and 0.1, 0"2, 0"3 denote the Pauli matrices. We introduce the one-form Yang-Mills SO (4, 2 ) potential A=ogaM~+bD+e~Pa+f~K~+SaJ~+uU+vV,
and the Chern-Simons action 246
^b-sa
fu~ = - ½[Rl,,(o9) - ¼R(og)gu,]
,
i Ja = ~ Y~®0"3®0"2, i
(3)
t r ( A d A + ~ A 3) .
The field equations are simply the vanishing of the SO (4, 2 ) field strengths, which are
Pa=½ 7~®0"+®~, Ka = ½ ~'a®0"- ®~,
U = ~ "~®0"+ ® 0 " 2 ,
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(2)
where s = s~g. f = f u~ and g.~ = e.ae~a is the three-dimensional metric. Here R.~(og) = eo~aRu°~(o9). where R ~ ( o g ) - 28fuO9~+ ¢%cO9u%9~c. The remaining independent gauge fields are therefore e. a, bg. S(u~) and uu. Their gauge transformations follow from 6A=dA+[A,A]
,
(6)
where we expand the SO (4, 2)-valued parameter A as
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A =Aa( M)Ma + A ( D )D + Aa( p)P: +A:(K)Ka+A:(J)Ja+A(U)U+A(V)V.
(7)
Substituting (2) and (7) into (6), we can read off the component gauge transformation rules. In particular, one finds that
6(K)bu=2Au(K), 6(j)uu=-A~,(J), 6
(8)
where Au = A aeu:. Thus the gauge parameters A ~(K), Aa(J) a n d A ( V ) can be used to set
b~=uu=s=O ,
(9)
where we recall that s=sUu. Hence, the only new independent field introduced by the extension of SO (3, 2) to SO(4, 2) is a symmetric traceless tensor 1 s~ ~ = s(u~)- ~sgu~. In this gauge, (5) reduces to
stu~j = 0 ,
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is the Cotton tensor density. It is symmetric, traceless, covariantly conserved and conformally covariant. Eqs. (14) follow from varying the only remaining independent fields, namely gu, and gu~, in the action
I=
f
2 a ~aF2 ~F zp~ d 3 x ~.uup (F o uzO~F3.p~+ ~F
_ 2gu~V~gp~).
(16)
This action is invariant under the conformal transformation gu,~t2Zgu,, gu~--,t2gu, for an arbitrary function g2, as guaranteed by its construction. Note that the symmetric tensor fields gu~ is not a gauge field; its gauge transformation involved the parameter Aa (J), which has been used to set u = 0. Since gu~ is traceless, it can be represented by a totally-symmetric four-index multi-spinor of SL (2, • ), g,~py6,for which the linearised action can be written as
v~=-¼V~/,
1 a w a=__ (e,, eub-- ~e~, e,,b)Eppa Opeob ,
L,, = -- ½(Ru,, - ¼Rgm,) - ½( ~ j ~ ~ - ~ g ~ ) ,
(lO)
where V~ is the usual covariant derivative defined with the Levi-Civita connection F%p = ½gUO( O,g~z+ 0pg,~- 0~g~a) and Ru,-RZup~ is the usual (symmetric) Ricci tensor. The Riemann tensor is given by
RPauv=OuFP~a-OvFPu~+FP~F'~va-FPvaF'~uo.
( 11 )
In three dimensions, it can be expressed in terms of the Ricci tensor, the Ricci scalar R=RUu and g - det(gu~ ) as follows: R % , , = I g l - 'e%aeu~( R ~ - ½Rg ~) •
(12)
Taking into account Bianchi identities on the curvatures (4), the only remaining equations of motion are
e~°~Rp/(K)e~=O,
eUP~Rpaa(J)e~=O.
(13)
These are equivalent to
(where ~u~= ~#~u# for any spinor ~#). This contrasts with the action for the graviton field h~,~-gu,,-~1~,~, for which, in two-component spinor notation, the linearised action is
f d3x h~flYaO,~'(D6~6f+O~Orq)h,an6,
,
Of Og
Of Og ,
(14)
{f, g}_=
(15)
w h e r e f a n d g are any pair of functions of q" and p,. The algebra of SO ( 3, 2) can now be realised in terms of Poisson brackets for the generators
where
CU~= euPOVp(R° u ~R6~)
( 18 )
where [] - 0"0u= - 10~P0~# is the d'alembertian. This action has the gauge invariance 6h(~pya= 0~zClya) (P). In general, the linearised gauge-invariant irreducible spin-s action (s>_- 1 ) will involve ( 2 s - 1 ) derivatives on a totally-symmetric 2s-index multispinor potential. Such fields will play a role in the infinite-dimensional extension of the conformal algebra to which we now turn our attention. We shall introduce this infinite-dimensional algebra through a realisation in terms of the coordinates (q~, p#) of a four-dimensional phase space, which we take to be two commuting two-component spinors of SL (2, R ). The Poisson bracket relation for this phase space is defined to be
C up+ e~(~Vpg~)~g~ + C°(ug~)~V~%= 0 , ~_P~('UVpSV)cr:O
17
3x
a
0q ~ Op,~
Op,~ Oq'~
(19)
247
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M ' ~ = q'~pp - ½(i~'~lypy , D=qYp,~,
P~'P=q~qP,
Ko~=p,~pp,
(20)
corresponding to the generators M,, D, P~ and K, that we introduced earlier. An obvious infinite-dimensional extension of SO (3, 2) can now be obtained by considering the Poisson-bracket algebra of arbitrary functions on phase space, which is the algebra of symplectic diffeomorphisms of phase space. We shall be interested in the infinite-dimensional subalgebra for which the generators are the even functions on phase space. (By an even function, we mean one for which f ( q , p) = f ( - q, - p ) . ) We denote these generators by • (a, b) '~t. . . . . fll...flb=q°"'"q"aPa, ""PBb,
(21)
where a + b is taken to be even. They can be organised into the subspaces G ( r ) = {qb(a, b ) [ a + b = r ) ,
(22)
in terms of which the infinite-dimensional algebra is G(2)~G(4)~G(6)~
....
(23)
the general structure of this algebra is {G(r), G ( s ) } = G ( r + s - 2 ) ,
(24)
which shows that whilst the SO ( 3, 2 ) generators G (2) alone form a subalgebra, the inclusion of any of the higher-order generators in (23) then requires the inclusion of all of them. With respect to the SO (3, 2) subgroup, the representations in the sequence (23) are 1 0 ~ 3 5 ~ 8 4 ~ .... We shall now show that the Chern-Simons action for this infinite-dimensional algebra is equivalent to an action for an infinite sequence of gauge fields describing irreducible spins 2, 3, 4 ..... We write the gauge field A as A=
~
~ ( a, b) °'~..... a,....a~,
{a, bla+ b even}
× A (a, b) a ' a " ....... .
(25)
The equations of motion, following from the ChernSimons action, are the vanishing of the complete set of field strengths. For the purposes of determining the spin content it is sufficient to examine the linearised equations, for which we may expand the dreibein e~,a about the flat background ~ . Thus the gauge poten248
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tial one-form A (2, 0),,p-e~p is assigned a preferred role in the theory in that those terms in a given field strength of the form e ^ A (a, b) will contribute to the linearised equations. Specifically, using the Poissonbracket relation {PY6, q)(a, b) ....... a,...ab} =2b~l~ l qO(a+ 1, b - 1 )a) ....... a2...ab),
(26)
we find that the linearised field strengths are given by R ~ ( a, b ) u, a,...a~ ....... = 20 o,A ( a, b ) ~j a'p~
.......
+ 4 ( b + 1 )et~ty,~lA(a- 1, b + 1 )~1 ~a' a~(~2..... 6a-~ ). (27) Here, ca, a is just eu" with the Lorentz index replaced by the symmetric pair of spinor indices c~fl; i.e. e~,,a=eua(Ta)jEya. As explained previously for the SO (4, 2) model, the linearised equations of motion R ~ " ( a , b) = 0 will allow some oftheA ( a - 1, b + 1 ) to be algebraically solved for. To determine which these are, we need to identify those SL(2, ~) representations of a given field strength that occur in the nonderivative term in (27). To see how this works, we shall fist concentrate on the 35 representation of SO (3, 2) which corresponds to the G ( 4 ) generators qb(4, 0); (/)(3, 1 ); q~(2, 2); qo( 1, 3); qb(0, 4). These contain the SL(2, R) representations
5; 5~3; 5~3~1; 5~3; 5,
(28)
respectively. The corresponding gauge potentials, which carry an additional vector index, therefore contain the above representations multiplied by a 3 of SL (2, ~). Similarly, because we are in three dimensions, the antisymmetric-tensor field strengths also contain the above representations multiplied by the 3. Thus starting from the Rain(4, 0 ) = 0 field equation, which contains the 3 ® 5 = 7 @ 5 ~ 3 , we see that we can solve algebraically for the corresponding representations of A(3, 1), which contains the 3® ( 5 ~ 3) = 7 G 5 @ 3 ~ 5 ~ 3 ~ 1 . We are left with the 5~33G1 representations in A(3, 1 ). Continuing this process for the remainder of the field equations, we find that we are left with the following representations inA(4, 0);A(3, 1);A(2, 2);A(1, 3);A(0, 4) respectively: 7 ® 5 0 3 ; 5 G 3 ~ 1 ; 3;
.;
(29)
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The potentials A(a, b) are still subject to gauge transformations. In particular, because of (26), a gauge parameter A(a, b) in a given SL(2, •) representation will give rise to a shift symmetry of the corresponding SL(2, R) representation in (29) of the potential A(a+ 1, b - 1 ). Hence any representation in a member of the sequence (29) that appears also in the member to its immediate right in the sequence (28) corresponds to a field that can be gauged away. For our example, this leaves only one irreducible SL(2, R) representation, namely the 7 in A(4, 0) at the left-most end of the sequence (29). It follows from the scale invariance of the lagrangian that the equation of motion for this field is the vanishing of the corresponding 7 representation in the field strength R ~in(0, 4 ) for an auxiliary gauge field at the right-most end of the sequence. This auxiliary field is solved for in terms of a single derivative of a potential appearing in R'in( 1, 3). In turn, this field is auxiliary, and is solved for in terms of a single derivative of a potential appearing in R l~n(2, 2 ). This process is continued until the physical gauge field at the left-most end of the sequence is reached. Thus the auxiliary field at the right-most end of the sequence will be expressed in terms of four derivatives on the physical field. Since the curvature itself contains one derivative, it follows that the field equation for the physical spin-3 field is of fifth order in derivatives. This is precisely what is required of an irreducible spin-3 gauge field, i.e. one for which all the lower-spin components can be gauged away. In SL(2, R) spinor notation, the linearised irreducible spin-3 action for a six-index totallysymmetric multispinor h,a.., 6 takes the form
f d3x
(30,~, ~ ...0,~,~5h~l...~,,6
+ 10[] O~,,eL..OmB3hpl...,e3 ....... + 3 ( [] ) 23,, a,hp~,2...~6),
(30)
where ( [] )2 is the square of the d'alembertian. In summary, we have seen that the 35 representation of SO ( 3, 2 ), corresponding to the G (4) generators of the infinite-dimensional algebra, ultimately describes just a single irreducible spin-3 gauge field. Analogous arguments lead to the conclusion that the subsequent 84 representation, corresponding to the G ( 6 ) generators, produces just a seventh-order equation for an irreducible spin-4 gauge field, and so
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on. Thus the final physical spin-content of the infinite dimensional theory is simply the set of all integer-spin>~ 2 irreducible gauge fields, each occuring exactly once. In fact, the linearised action is I=l
d3x
2 ~2r+/~2s1"/a...... 2sll-"l~r~ ii ] s=2 r=0
× ~,,~...~,~,_~_~ ms-~-~ha,...~_~_,,~. >...~.,
(31) where the first summation is over all integer spins s>~ 2, and ( [ ] ) r denotes the rth power of the d'alembertian. The symbol C~ is the combination factorp!/ [q!(p-q)!]. The action 31) has the gauge invariances ~ha'"'az'= 0 (ma2a ~3...azO --~s)
(32)
for the set of 2s-index multispmor totally-symmetric spin-s gauge fields. Variation of (31 ) with respect to the spin-2 field h ....... yields the linearised Cotton tensor. Variation with respect to the other fields yields the linearisations of the higher-spin analogues of the Cotton tensor. Note that the sum over spins in (31 ) could be extended to include s = 1. This would amount to the addition of U ( 1 ) factor in the algebra, corresponding to an electromagnetic field that does not couple to any of the higher spins. (In three dimensions, it is not true that everything must couple to gravity. ) One might suppose that the supersymmetric extension of the above model is obtained simply by extending the generators qb(a, b) to include those for which a + b is odd. Let us call these the odd generators. The problem is that the Poisson bracket of two odd generators is antisymmetric, whereas for a superalgebra one needs a symmetric product for odd generators. One can circumvent this problem by introducing the product [ , ] + defined by
[f+,g+]--{f+,g+}, [f_,g_]+=-2f_g_,
[f+, g_ ]_ - {f+, g_ }, (33)
where f+ and g+ are even functions on phase space, f_ and g_ are odd functions, and {, } is the Poisson bracket defined in (19). (Recall that the coordinates q" a n d p , of phase space are commuting spinors. ) One can readily verify that this product satisfies the super Jacobi identities. With the product defined above, the algebra gen249
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erated by G ( 1 ) @ G (2) is the N = I superconformal algebra OSp(114). Its infinite-dimensional extension is simply {G(1)~G(2)}~{G(3)~G(4)}~{G(5)0)G(6)} ~3....
(34)
where the bracketed terms are irreps of OSp (114). The Chern-Simons action for this infinite dimensional superalgebra yields, after elimination of all auxiliary gauge fields, in the manner described previously, an action for irreducible spins 3, 2, ~, 3, 7, etc., i.e. all integer and half-integer spins t> 3, each occurring once. This action can be consistently truncated to the N = 1 conformal supergravity action of van Nieuwenhuizen [5 ], by setting to zero all fields with spins greater than two. As we have mentioned, the infinite-dimensional extensions of SP (4, R) and OSp (114) considered here cannot be consistently truncated to any other finite-dimensional (super)algebra. However, this does not preclude the existence of a finite-dimensional algebra containing Sp(4,R) for which the Sp(4, ~) content reproduces that of th~ infinite-dimensional algebra up to some level. For example, as shown in ref. [ 6 ], the first N representations of the infinitedimensional extension of the anti-de Sitter algebra SO (2, 2) are reproduced by Chern-Simons action based on the finite-dimensional algebra SU(N, N) × SU(N, N), even though the latter is not a subalgebra of the infinite-dimensional algebra. Similarly, there may be a finite-dimensional algebra containing SO (3, 2) for which the SO (3, 2) content reproduces the sequence 10@35~84~... up to some level. If such an algebra were to exist, it would give a consistent coupling of afinite number of higher-spin gauge fields to conformal gravity.
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As far as the quantum aspects of these Chern-Simons theories are concerned, one might expect them to be finite because, as discussed by Witten [4 ] and Deser et al. [ 8 ] for pure gravity, each higher-spin field contributes only a finite number of degrees of freedom. In our case, however, there are an infinite number of such fields, so the total number of degrees of freedom is in fact infinite and the argument may not apply. Although a "theory of nothing" may be finite, it remains to be seen whether an "infinity of nothing" is finite too. We are grateful for conversations with M. Gtinaydin, E. Sezgin and K.S. Stelle. P.K.T. gratefully acknowledges the hospitality of the Center for Theoretical Physics at Texas A&M University.
References [ 1 ] R. Haag, J. Lopusanski and M.F. Sohnius, Nucl. Phys. B 88 (1975) 257. [2] E. Fradkin and M.A. Vasiliev, Ann. Phys. (NY) 177 (1987) 63; Phys. Len. B 189 (1987) 89; Nucl. Phys. B 291 (1987) 141. [3] M.P. Blencowe, Class. Quantum Grav. 6 (1989) 443. [4] A. Achficcaro and P.K. Townsend, Phys. Lett. B 180 (1986) 89; E. Witten, Nucl. Phys. B 311 (1988) 46. [ 5 ] P. van Nieuwenhuizen, Phys. Rev. D 32 ( 1985 ) 872; J.H. Horne and E. Witten, Phys. Rev. Lett. 62 (1989 ) 501. [ 6 ] E. Bergshoeff, M.P. Blencowe and K.S. SteUe, Area-preserving diffeomorphisms and higher-spin algebras, preprint Imperial/ TH-88-89/9. [ 7] M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Phys. Lett. B 69 (1977) 304. [ 8 ] S. Deser, J. McCarthy and Z. Yang, Polynomial formulations and renormalizability in quantum gravity, preprint BRX TH259 (1989).