Physics Letters B 284 (1992) I 1-16 Norlh-Holland

PHYSICS LETTERS B

Matter coupling and spontaneous symmetry breaking in topological gravity T.T. B u r w i c k ~'-~ A.H. C h a m s e d d i n e Instttute ~)r lheorettcal Physics. L ntler~tt~ o/ Zurt~h. S~honberggasse 9. (H-8001 Zurt~h. Swttzerland

and K.A. M e i s s n e r lnstttute for Theorett~al Ph)'~t~. l~ ar~aw L'nt~erstt). ul Hoza 69. PL-00-681 14"ar~al~;Poland

Rccelxed 10 April 1992

Matter is coupled to three-&menslonalgravity such that the topological phase is allowed and the (anti-) de Sitter or Polncare symmetD' remains intact Spontaneous s~mmetry breaking to the Lorentz group occurs tf a scalar field is included This H~ggs field can then be used to couple matter so that the famdmr form of the matter couphng is estabhshed m the broken phase We also gl~, e the sypersymmetrlzat~onof this construction

I. I n t r o d u c t i o n

Many attempts have been made to formulate a q u a n t u m theory of four-&menstonal gravtty (see ref. [ 1 ] ). In this approach ~t ~s hoped that ifgravtty can be formulated as a renormahzable theory, then this would improve the prospects of untfymg gravtty wtth the other known interactions. The recent developments m three-&mens~onal gravtty provide an excellent testing ground for this program. There ~t was shown that by formulating three-dimensional gravity as a topological gauge theory of the groups SO( 1, 3 ), SO (2, 2 ) or ISO ( 1,2 ), the theory becomes fintte [ 2 ]. The mam &fficult~ m advancing this program ~s the couphng of matter A first difficulty of mtroducmg matter hes m the nature of topological gravtty. It allows for the unbroken phase ofgravtty where the dretbem ts degenerate: ~;,-'~ - 0 . In th~s topological phase, the notion of geometry loses ~ts meaning. Physics m the usual sense, t Supported b~r the Swiss National Sctence Foundation z ~ddressafter 1 May 1992 SLAC, Stanford Umvers~ty,Stanford, CA 94309, USA

where space-ttme xs equipped wtth dtstances, arises only away from this phase. Actually, m ref. [2] the partition function was seen to be d o m i n a t e d by geometrical universes, but the appearance of the topological phase was essentml for its derivation. Matter couphng, however, lS usually formulated by using the mverse drcibem e~ which would become singular in the topological phase In topological gravity th~s coupling has to be introduced such that only e~, ts used. Moreovcr, we want to reqmre the matter coupling to reproduce the famdmr form if restrtcted to lnverltble drelbems. A second difficulty stems from the fact that e~~, is part of the gauge field A and cannot be used by ttself wtthoul breaking the gauge m v a n a n c e . It ~s then suggestive to break the gauge symmetry to the Lorentz group SO( 1, 2 ) so that e~'~would correspond to the broken symmetry. To break the symmetry we employ some kind of Hlggs mechanism. However, writlng a usual Hlggs potential in the action requires a metric for the volume element. Again, since no dreibeln and therefore no metric g ; , ~ = e ~ , e ~ can be used w~thout breaking the tangent space symmetry "'by hand", the Htggs field potentml terms cannot be wrtt-

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ten in the usual way and an a l t e r n a u v e construction will be applied. It is surprising that despite o f these difficulues matter lnteraeUons can be introduced. In the followlng we will discuss the case o f a scalar field. The plan o f this paper is as follows. In section 2 we g~ve the coupling o f three-dlmensnonal topological gravity to matter In section 3 the construction IS generahzed to the supers3,mmetnc case. Some c o m m e n t s and the conclusnon are in section 4.

2. Matter coupling to three-dimensional topological gravit)

F r o m the work o f W m e n it is now estabhshed that three-dimensional q u a n t u m gravity becomes a finite theory when formulated as a gauge theory o f G = S O ( I . 3), S O ( 2 . 2 ) or I S O ( I , 2) d e p e n d i n g on the sign o f the cosmologncal constant [2]. The gauge m v a n a n t action Ls o f the C h e r n - S l m o n s type Sg=4k~ f
(2.1)

where 4 is an S O ( 2 , 2) gauge field (the other two cases are recovered by Wick rotation or an I n o n u Wlgner group contraction ) 4 = ~.4 ~./,u,

l = a , 3, a = O , 1 , 2 ,

and the quadratic form is defined b~ ( J I/~Jc 1~) = ~ It~ , .

(2 2 )

The connection w~th gravity is m a d e through the identification A ''~ - e",

.4 . t , _ ~o ''~' .

difficulty of introducing non-trivml matter B.~ "nontrivial" we mean couplings which, in the non-topological phase, reduce to the familiar interactions. The familiar fbrm of the bosonlc matter coupling requires the inverse dreLbem e~'. This. however, is singular m the topological phase where c'"~,= 0 . Moreover, ~,¢1,, is part o f the gauge field and cannot be used by itself without breaking the s~ m m e t ~ . At the quantum level. the acUon (2.4) generates divergent one-loop dingrams that are cancelled b.~ ghost diagrams arising from Lorentz- and translatlon-m~arlance. Since we do not want to lose these ghost diagrams, we should try to break the gauge spontaneously. Let us try to couple a scalar field I1 Without using a metric, the onl.,, coupling that could be introduced would be to multipl~ the acuon (2 1 ) by factors of I1. This, cert a m b , does not give interesting physics. Attempts have been made to introduce an a d d m o n a l antls~mmetric tensor [3] or fields living m representations o f only the Lorentz group SO( 1,2 ) [4]. These. however, have a trivial physical content. Here, instead. we take a different strategy. We consider a field H ' and identlf~ I I ~ = H [5]. We will see that when expanding a r o u n d a flat background and using a linear a p p r o x i m a t i o n , the II J coupling will reproduce the thmlhar 0"110,I1 after eliminating the II a by its equation o f mouon. Slncc the H ' will takc a non-zero ~acuum expectation value ( V E V ) which breaks the s y m m e t ~ to the Lorentz group, we call it a Higgs field This Hlggs field can then be used to couple other m a t t e r fields. Since no metric is at our disposal to write volume elements, the o n b Higgs terms that can be written (aparl from possible factors I t ~H~ multiplying them ) are

( 2.3 )

In terms o f e and the spin connection (o the action (2.1) takes the form

"%=-

Sg=kg f ~,,t,,e"(R t ' ' - ~et'e ' ) ,

where

(2.4)

where R ~"= d ( o h' + ~ o ~ . At the classical level, when e~, is restricted to the subspace o f lnvertlblc fields, the action (2.4) ~s cqmvalent to the E i n s t e l n - H l l b e r t action. However, this cqulvalencc breaks down at the q u a n t u m level, where the q u a n t u m theory o f (2.4) is finite. The main disadvantage in th~s formulation is the 12

18 June 1992

f

,t~< i, t l ~(IzDHI~F ' " + 2 D I I I ~ D H ~ D I I I') •

(25)

DI, f t ~= 0,, I t 1+..1/," It1~ . F ~U=dA zu+ ! . 4 ~ .

With the Hlggs terms given m (2 5 ) we may now ask for a possible VEV m the translatlonar)' direction I1 J= ( H

~) + I ~ ~

( 2 6)

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where

( H a) = 0 ,

( n 3 ) -= ( I I ) .

(2.7)

Actually, the argument should have been reversed: It is the direction of the non-zero ( H e) that dccldcs which p a n of the gauge fields in (2.3) separates to be Identified with the drelbeln. To look for non-zero ( I I ) we have to consider the part of the action (2.5) given by

S'h = f c,,b, [ ItH2e"R t" + e"ebe ' ( - ltH 2 +,~II 4 ) ] . (2.8) The prime indicates that in (2.5) we set H " = 0 which is sufficient for obtaining a VEV in the translation direction. With ( 1/ 3! ) Ca,,e"e~'e ' = d 3x~,,@, the last two terms in (2.8) are seen to be the usual scalar potential. It is a well-known feature (and problem!) that the VEV of a Hlggs field changes the cosmological constant. For convenience, we may assume that we tuned the coupling constants such that the effective cosmological constant vanishes. Then we may go to the fiat background: ( e ,a, ) _- d , ,a ,

(tu~/') = 0 .

(2.9)

For such a background, the first term in (2.8) will not contribute. With 2 > 0 . the Hlggs potential in (2.8) is then minimized by

( H)

(2.10)

=v@/2.~.

if i t > 0 , otherwise the VEV will vanish. The VEV (2. ! 0 ) breaks the tangent space symmetr3' to the Lorentz group SO( 1, 2) that leaves (2.7) in,,arlant. Plugging (2.10) back into (2.8), we find the total action to be

the linear approximation the terms of (2.5) that are quadratic in II ~are

2~ ~ d3x(2tt"d~O,,H-3H2-tt~H,)

.

(2.13)

Eliminating the tt" by its equation of motion from (2.13), this turns into 21~ f d 3 ¥(O"II 0 , H - 3 H 2) .

(2.14)

In (2.14) we recognize the usual kinetic term of the Hlggs field around the flat background (2.9). For a general gravitational background and including also higher than quadratic terms in (2.13), the elimination of H ~ by equations of motion becomes a formidable task and will not be attacked here. We take (2.14) as sufficient in determining the structure of the H ~ sector. Alternatively. for analyzing the system (2.5), the gauge condition H o = 0 could be imposed and a kinetic term for the Hlggs f i e l d / / w o u l d be generated by a Wcyl scaling that absorbs the H 2 m the first term of (2.8). Having Included the Hlggs field H 4, we are now able to couple other matter fields. The simplest matter interaction to construct is that of a scalar multipier. Let X ~be a scalar muhlplet in the fundamental representation of SO(2, 2) with the identifications X " = n", A'~= 0. One possible action that reproduces the familiar form at the classical level is

Sm= km f ~ am DII fDHt~DII c (Xt~D,J( t Ht ) . (2.15 ) If we expand the Hlggs field around the broken phase (2.7), (2.10 ) the matter action (2.15 ) takes the form Sm

/1-"

18 June 1992

= --

/.'~ jf cl. . .3-.. ..

LltVI.~ l

a~ll~t .,,, ~.-,,~..,

" d (OvO--evTZd)

Sg+Sh = f E,,,,[(k, + -~)e"R'" + O ( I 1 "~)

- ( , k , + --~.)e"e"e'] +O(14') .

(2.11)

Except for HlggS quantum fluctuations, this is of the same form as the gravity action (2.4) but with a different cosmological constant. We find this effective cosmological constant to be cancelled if

1 1 2 = - ~k~ )t •

,

(2.16)

where h~ =km/t-~/422. The action (2.16) is just the first-order formulation of a scalar field action. To see this, assume the non-topological phase where e~~, is lnvertlble, and substitute the equation of motion of zc, = ½e~U0,,0+ O(/-/4 )

(2.17)

(2 12) into the action (2.16) to get

This allows to use the flat background (2.9), and in 13

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Sm-

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~ ~ d33:ee~:e""O,,O0.O+O(ll ').

(2.18)

Thus (2.15) reproduces the canonical form at the classical level. In the spontaneously broken phase, the total actmn, which Is the sum of (2.11 ) and (2.16 ), has onl.~ the S O ( 1 . 2 ) Lorentz symmet~'.

3. Topological supergravity and matter coupling Since SO(2, 2) ~ S O ( I , 2) x S O ( 1 , 2) and OSP (21 1 ) is the graded version of SO ( 1.2 ), the supersymmetnc analogue of the construcuon given m the prevmus section ~s achieved b~ gauging OSP(211)xOSP(211)[2,6]. We shall adopt the notauon ofref. [7] for the matrix representauon o f O S P ( 2 1 1 ). Let q). and qL, be the gaugc fields o f the two OSP (21 1 ) gauge groups transforming as

DG=dG + qS. G-G~, .

q~ --,12~q).12; -1 +.Q.d.Q ;- ' ,

(3.1)

where 12, and g22 are two elements of the two respecuve groups. These can be represented in the matrix form

(3.5)

In order to distinguish the group indices of the secondOSP(211)letusdenotcthemby&,fl...Thcn the matrix representation of G is c"=k(/s /Hf,

~"),

(3.6)

where both r/,, and ~,, arc Majorana splnors, and H,a, and 0 are real. It will also be necessary to define the eqmvalcnt representation 67 transforming as ,f22 67(2 i-~

(3.7)

and whosc matrix form is

67 = ( [ (:I I T~ -1] {3~'

""0K' )

(3.8)

_r~fl

We first write the pure supergravlty action [6] S,g

O,-,12, O, 12?' +,Qj d,Qi-1

18 June 1992

-g

[Str(qS~ dq~ + ~q) ~ ) - 1 -

,

whose component form is Ssg = ~ I

[ ( , l l a d . q"l _ l ~'a'"A'"|"1') a t, ,

+4q~D~ ~ - 1-,2] ,

o__(:

,32,

whcre

where D, = d + -1,.The action in ( 3.10 ) can be put into a more famlhar form by recxpresslng it in tcrms of

~u,, = ~.,a¢': •

( 3.3 )

c o " = ~ ( A T + 13),

It is also convenient to write

~'- = ~(~Ul + q/.,) •

A,/~= ~a( r.)~ ,

Then

where the z, are the SO (2, I ) generators

S,~ =k~g

1(010) -1

,

r,=~

10)

'

(3.11)

"" '

(3 12)

Introduce now the Hlggs field G transforming as G-'-(21G-f2y I

[e"(R,,

e"=~(,,17--l~),

+4~u_ (d+(o)~u+ +2q?+ e~,+ + 2q)_ e~,_ ] .

,

I

(3.4)

and the covarlant derivative of G. transforming as is defined by 14

(3.10)

[6]

el,,, = A : . .

ro=~

(3.9)

G.

where R , = d , ~ , - ~ , t , , o / ' o ) ' . Using oJ"=~'a"oJ~,, the bosonlc part agrees w~th (2.4). Apart from trace factors Str(G67), the most general expression for the Higgs interactions compatible with ( 3.9 ) and the dlagonahzauon in ( 3 11 ) is

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PHYSICS LETTERS B

t" & , = J {~/~[Str(G D"G(d~, + ~ ) )

-Str(f~ DG(dq~2 +

ground ( 3.17 ) any value o f the ¢ is allowed. The total acnon lakes a p a m c u l a r l y interesting form if we shift ¢p--.h+~p.

• _¢) ) ]

+ ~ 2 S t r ( G DG D G D G ) } .

(3.13)

Analogously to the bosomc case, we ma~ look for a non-zero VEV of

G=(G)+G.

(3.14)

0)

(3.15)

and h is m the umt & r e c n o n o f t l = h + H " t , The supergroup OSP(211 ) × O S P ( 2 1 1 ) has ten degrecs o f freedom. Out o f these, seven may be used to rotate ( G ) into the & r e c n o n given by (3.15 ). Therefore. a non-zero VEV (3.15 ) would leave onl~ three degrees o f freedom and we will see that these correspond to the Lorentz group. Since we have to vary the acnon in the direcnon given by (3.15 ) wc only need to look at the terms

f

S~g + S,h

t ( k,g +/~h - )

×[eUR,+40

(d+to)~.+20+e~+]

"°

5h39)+4h2~2-h~ 3) ]0+ e~+

+ [/~ ( 3h-" + ¢p-~+ 2h~p)

- ~).(2ha+5h~+4h2¢2+h¢ ~)]0- e~_ +½2(hd~p-qdh)(

h~

- - r p -~) ~ _, _ ~ + }

(3 16)

where the prime m & c a t e s that we set Higgs components orthogonal to ( 3.15 ) to zero. L k c the bosonlc case, we may for convenience assume that the coefficients k , r # and 2 are tuned such that the effective cosmological constant vamshes. Thcn we may go to a flat background.

(e,,)-J,,, __

a

a

(co,,)=0,

(~'_+)=0.

(3.17)

With 2 > 0 . the potennal m (3.16) will then be minimized b~ x ~/22

Except for Hlggs q u a n t u m flueluanons and wtth zero q, the terms appearing m the acnon (3.20) are o f the same form as the original supergravlty a c n o n (3.12 ), but with a different cosmological constant. We find the cosmological constant to be cancelled ff

,u2= - ~k,~;, .

(3.21)

This will then allow for the flat background (3.17). Notice, that m this background the q u a d r a n c H~ggs terms m (3.13) are

+½0r"O,,~l-~r"O,,~-~(~tl+~)].

+ [/~(3h 2 +
(3.20)

tl J d ~x[ll"J~ Oj,h - 3h 2- ~II"11,

{lth-e R,-~%~,,e"e~'e'(llhE-2h ~)

+ 2/t (h-" + q - ' ) 0 _ ( d + (n)p'+

(h)=

Then the sum o f ( 3.9 ) and (3.13 ) becomes

+ 0 ( ~ , (7)}.

(~P)

a

(3.19)

+ (k,~ + 3#h 2_ 32114 ) ( _ gI ¢,~,, e a ~~b ~~t + 20_ e~'_ )

where

S~,=

18 June 1992

(3.22)

After rescahng 11<-,211 ", the h and H" terms are o f lhc same form as m (2.13 ), and the ~1, ¢ terms arc o f the Dirac type The presence o f the H~ggs field G does now allow to couple another matter field X which ~s a multlplet transforming hke G The m a m x representanon o f X is given by

,v_(~(o+s)6g +Tr"(r°)':~ -\

2"

;..-z,,)

(3.23)

>

A. matter m t e r a c n o n which reproduces the bosonlc matter interaenons (2.16) is

S,m=k,m f Slr(D-GDGGX)Str(GDX).

(3.24)

This can be seen by using the VEV ( 3.15 ), ( 3.18 ). Then (3.24) reduces to

S~,,, = -k;m f (~,~,,e"#'n' + 4 0 _ t,,~_ r t " - 4 0 _ e).)

(3.18)

f i l l > 0 . F o r / ~ < 0 the VEV o f h would be zero The field ~ ~s not driven to a certain value: m the back-

× (d(~--eand -- 2 0 - 2) + O (tp, (~) ,

( 3.25 )

where we used the shift (3.19 ) and did not write the 15

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~pcontributions. They always appear w~th gravltmOS and will not influence the bosomc part. Although th~s action has thc correct bosomc interactions for n'~ and O, however, s and ~ decouples, and ,:, does not acqmrc a propagator.

4. Conclusions and comments Wc have constructed matter mteract~ons coupled to grav~t~ m a topological wa~. The dretbem separates from the other Pomcare or (ant~-) de Sitter gauge fields only by spontaneous s y m m e t ~ breaking. The matter couphng was introduced wtthout using thc reverse drelbem, thereby allowing for the unbroken phase of gravity. This became possible by including a Hlggs field and usmg a first order formahsm. Restricting to the mx ert~ble drelbems, the matter couphng takes the famdmr form if the equations of motion arc used. We workcd out the three-&menslonal case, but the generahzatlon to topological gravtt~ m htgher & m e n s l o n s [8] is straightforward. We also presented the supersymmetrtc analogue of th~s construction. Future work should examine the q u a n t u m theory of the proposed matter interaction m a perturbat~vc setlmg Smce we included matter only b3 spontaneous symmetry breaking, wc can immcdmtcly dcduce that the pure gravity sector rcmams fimtc. Smcc threc-dtmenslonal gra~ ity ts a specafic example of a C h e r n - S i m o n s theory, the pcrturbatwe analys~s may be pcrformcd along the hnes of ref. [ 9 ]. For the case of pure gravtt.~ perturbation could be performed in thc unbroken background ¢ , , - 0 Having mcludcd matter interactions, the fields used will obtain propagators onl~ ~f one expands around some non-zero background. For the case of pure gravity th~s expansion and the perturbatlve analys~s was performed m ref. [10]. With mattcr man~ new verttces and dingrams arise. Apart from qucsttons about non-zero backgrounds an,, q u a n t u m analysis of a topologtcal lhcorx rcqmres the mtroductlon of a background metric to fix the gauge and derive propagators For pure gravity, the resulting q u a n t u m theor} remains m d e p c n d c n t of this background metric [2,1 1] in general, hov~ever, ~t ~s not guaranteed that a theor~

16

18 June 1992

which ~s metric independent at the classtcal level would remain so at the q u a n t u m le,,el [ 12 ]. It is then of interest to study whether the property of metric mdependencc is lost m the presence of matter. We want to emphasize that for proving a possible metrtc dependence ,t ~s not enough to find &vergences that can only be cancelled by using the background metrtc The SltUatton may be compared to Yang-Mdls theory m the axml gauge nU,4~,=0 where n ~' plays the role of the background metric. There, at the one-loop Icvcl counterterms havc been found that were dependent on n ~' [13 ]. Later, the s~tuat~on was re-invest~gated by using BRST methods and tt became posstble to control this gauge dependence [ ! 4 ]. A BRST analysis along the hnes of ref. [ 14 ] should also be apphed to study a posstble background dependence of topologtcal gravlt~ m the presence of matter. This will then dec~de whether topological grav~ty keeps all of its nice features after matter ~s coupled.

References [I]P van N,euwenhulzen, Ph3,s. Rep 68 (1981) 191, and references thereto [2]E Wltlen, Nucl Ph2,s B311 (1988)96, B323 (1989)113 [3] J Gegenberg,G Kunstatter and H P Lelvo, Phys Left B 252 (1990) 381 [4] S Carhp and J Gegenberg,Ph'.s Rex' D 44 ( 1991 ) 424 [ 5 ] A H Chamseddme.Ann Phys 113 ( 1978) 219, Nucl Phxs B 131 (1977)494. H G Pagcls. Ph,~s Re', D27 (1983) 2299 [ 6 ] -X Achucarroand P K Townsend,Phys Left B 180 (1986) 89 [7] A H Chamscddmc. k SalamandT Strathdec. Nucl Phys B 136 (1978) 248 [8]~H ('hamseddme. Nucl Ph',s B~46(1990)213 [9] L Alvarcz Gaume. J Labastlda and A V Ramallo, Nutl Ph)s B334 (1990) 103. E Guadagmn~,M Marlelhnt and M Mmtche',, Ph'.s Left B 334 (1989) 111, C P Martin. Ph~,,s Lctt B 241 (1990)513 [ 101S Dcscr. J McCarlh~and Z Yang.Ph3,s Len B 222 (1989) 61 [ I I ] D Ra~andl Smger. Ad', Math 7(1971)145, Ann Math 98 (1973) 154 [12]M BlauandG Thompson, Ph~s Left B255 (1991) 535 [131DM ('appcrandG Lelbbrandt, Phys Re', D25 (1982) 1002, t 009 [ 14] P Galgg. O Ptguct, '~ Rcbhahn and M Schweda, Ph3,s Lett B 175 (1986) 53