Gauge symmetry in extended objects and the stability of compactified membranes

Nuclear Physics B356 ( 1991) 208-228 North-Holland

GAUGE SYMMETRY IN EXTENDED OBJECTS AND THE STABILITY OF COMPACTIFIED MEMBRANES Kazuo FUJIKAWA

Uji Research Center, Yukawa b~stitute for Theoretical Physics, Kyoto Unit'ersity, Uji 611, Japan Jisuke KUBO

College of Liberal Arts, Kanazawa Unil'ersity, Kanazawa 920, Japan Received 28 December 1990

A large number of gauge degrees of freedom inherent in relativistic extended objects often cause dynamical instability. This problem is studied by taking bosonic membranes and balls as examples. The bosonic membrane with two spatial coordinates compactified is examined in detail, and it is pointed out that the compactification produces a stabilizing potential barrier for the variables orthogonal to the compactified spaces. By gauge transforming away the variables in the compactified spaces, we suggest a possible confinement of unstable modes in compactified spaces.

1. Introduction

According to the general ideas of Dirac [ 1], Nambu [2] and Goto [3], relativistic extended objects are described by a reparametrization invariant lagrangian [4,5]. This reparametrization invariance, in the case of higher-dimensional extended objects such as membranes [5-8] and balls [3], strongly restricts the possible form of the action and often leads to dynamical instability. For example, a relativistic ball in 4-dimensional space-time can have a meaning only as a massless scalar point particle [9]. The relativistic supermembrane may not exist at all, since it may become an infinitely stretched one-dimensional string-like object [10]. We here analyse the effects of a large number of gauge degrees of freedom inherent in such higher-dimensional extended objects. By understanding the effects of gauge freedom better, we may, for example, find a possible mechanism for stabilizing the supermembrane with a part of space coordinates compactified [11]. To utilize the gauge freedom maximally, we formulate a bosonic membrane with compactified space coordinates in terms of the light-cone gauge [12], synchronous gauge [13] and covariant gauge fixings [14, 15]. We show that the synchronous and covariant gauge formulations cover the configurations which cannot be easily I)550-3213/91/$03.50 ai~ 1991 - Elsevier Science Publishers B.V. (North.Holhmd)

K Fujikawa, J. Kubo / Extended objects

209

recognized in the light-cone gauge, and we suggest a possible confinement mechanism of unstable modes in compactified spaces. These new configurations could be essential to analyse the stability of compactified supermembranes [11].

2. Light-cone and synchronous gauge formulations 2.1. LIGHT-CONEGAUGE FORMULATION The reparametrization invariant bosonic p-brane (p-dimensional extended object) in D-dimensional space-time is described by

s =

v/- det O,,X t' ObXt~ d p+'or,

(l)

with ora, X~(or), or°-z,

a = 0, 1 , 2 , . . . , p , /~ = 0 , 1 , . . . , D -

0 < o r k~
1,

fork=l,...,p.

(2)

The action (1) is invariant under p + 1 reparametrization gauge transformations

~ -~,~,- = f . ( ~ ) ,

(3)

and correspondingly one obtains p + 1 first-class constraints [1] rho = P~P~ + tr,2 det(OkX ~ OtX~,) = O, 4'k = ak XZ Pg = 0

(4)

with k, l = 1, 2,..., p, and vanishing hamiltonian, ~ = 0. If one goes to the light-cone frame by choosing the gauge conditions

Xo --

X o +X o-! V~

- z - X + ( z , or~ ) - z = 0 ,

x , - e+ (~,' ~k) -p+ (~) - 0 ,

(5)

one obtains the light-cone hamiltonian (density) from eqs. (4) 1 h = 2p+ [p2 + K2 det(OkXO, X)] '

(6)

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K. Fujikawa, J. Kubo / E.~tended objects

where X=(XI, XZ,...,xD-2); p+ in eqs. (5) in fact becomes ~--independent as a result of the dynamics. We still have p - 1 gauge degrees of freedom left in eq. (6). In the case of a relativistic ball ( p = 3) in D = 4 dimensional space-time, the potential part in eq. (6)vanishes det(0 k X OtX) = det(0 k X " ) d e t ( 0 I X ' ' ) = 0,

(7)

where k,I = 1,2,3, and m = 1,2; the matrix in the right-hand side of eq. (7) contains at least one vanishing column. This means that we have no stabilizing force for the stretching (or decaying) ball. After removing the remaining two degrees of freedom in X by two ( = p - 1) remaining gauge degrees of freedom, one finds that a massless scalar point particle described by the center of mass coordinates is the only meaningful (singular) solution for a relativistic ball in D = 4 dimensions. The ball in D - 4 thus becomes a purely topological object [9]. A further comment on this problem will be given later. A more fundamental result, which is also related to the behavior of the potential, has been stated by de Wit et ai. in 1988. They conclude that the sttpennembrane ( p = 2 ) in D = 11, for example, is dynamically unstable [10]. Moreover they suggest, without detailed analyses, that this instability will not be influenced by compactification unless all the space coordinates are compactified. In other words, the supermembrane will not exist quantum mechanically. An intuitive picture behind their argument is illustrated by using the (closed) bosonic membrane ( p - 2) in (1) as follows [10]: In the light-cone gauge (5), the hamiltonian for a bosonic membrane is given by

h = 2p+ [pZ + r2 det(OkXOtX)] m

(8)

where k, ! = l, 2, and X = (X I, X 2,..., x l ~ - 2) in D-dimensional space-time. The potential part in eq. (8) is rewritten as

'(ektokx'"O,X")2>/0

det(OkXOtX ) =_~

(9)

with an anti-symmetric symbol e kt and m, n = 1,2,..., D - 2. From this expression, one can see that the potential vanishes either for a tr2-independent configuration

(10) or for a o-~-independent configuration X = X ( r , tr-').

(11)

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At the semi-classical level, the string-like configurations (10) and (11) can thus stretch indefinitely without any potential barrier. The quantum mechanical fluctuation could stabilize the above singular configurations. In the case of supermembrahe however, the above authors [10] argued that the zero-point energies cancel and the above unstable configurations can indeed lead to an infinitely stretched string-like configuration. They showed this result by using an SU(N) matrix model regularization in the light-cone gauge for a finite N, and extended the result for N = 00 by assuming the existence of a well-defined limit. On the other hand, the analysis of a closed supermembrane with toms compactifled space coordinates by Duff et al. [11] shows that a quadratic approximation to the supermembrane exhibits a well-defined harmonic oscillator-type behavior and thus suggests a stable supermembrane. This apparently stable behavior of a compactified supermembrane could of course be destabilized if one incorporates the effects of the full potential. Since the issue of stability is very fundamental, we investigate a possible picture behind the above apparently stable compactified supermembrane. If one can establish a reasonable physical picture for the stabilizing mechanism, the apparently stable behavior in ref. [11] may have a more fundamental meaning. For this purpose, we consider a torus (Sm × S ~) compactified configuration [11] 1

Xl(,r,0. k) = a0. t + x '(~') + ~x-xY ' ( r , 0 . k ) ,

X2(r,0. k)

1

=b0.2 + x 2 ( r ) + .~_x_xy2(,r, 0.1,), 1

Xt'(-r,0. k) = x # ( r ) + ~x--xYt'(z,0. k),

(12)

for g = 3 , . . . , D - 2. For more general solutions, one may multiply the a and b terms in eqs. (12) by integers. We note that all YU(r, 0.k), # = 1,..., D - 2, are periodic in 0.1 and 0.2, Y/z(T, 0.1 + 1,O "2) = Ytt( T, 0.1, 0.2 + 1) = Yg(T,0.1,0.2) •

(13)

The theory in (8) is then specified by 1 h ~.

2p+

[ p2 + K2 det(0 h X OiX)],

dp - OI( 02XP ) - 02( OIXP ) =bOiP2-aO2p

1

I + -~K [O,(O,.YP)-O,.(O,YP)]

=0,

(14)

K. Fujikawa, J. Kubo / Extended objects

212 and

aV~p, =

- f (a,Yp) d2(r,

by'p2 = -

f (a2rP) d2cr,

(15)

where & stands for the so-called "area-preserving symmetry" [16], and pt and P2 stand for the momenta circulating inside the compactified spaces,

Pl = 2~'ll/a,

P2 =

2~'!2/b

(16)

with integral it and 12. The momenta P = ( P I , P2,...,Po_2) in the right-hand sides of eqs. (15) do not contain the trk-independent center-of-mass momenta due to the periodicity of Y= ( y t , y2,...,yl>-2) described in eq. (13). The potential part (9) in eq. (14) becomes

Itc2lEklj r 2 det(akXa/X ) = _v ~, k X m atX" =-~l,e Ok" t'tr

)2 + ((Vr-da +O,Y')c~2YN-O2Y'atY N

,

)2 (17)

where M, N = 3,..., D - 2 . We note that each term in eq. (17) is separately positive-semi-definite classically. To search for a potential valley, we assume YN= yN('r, trl),

N=3,...,D-2,

(18)

i.e. yN is o-2-independent, for which the first term in eq. (17)vanishes. In this case, the remaining terms in eq. (17) become (a2Y

2 + (V~h + a2Y2)21 (o~,YN)2 + ((¢-K-a + c~,V')(C--Kb + ~2Y2) -~2 Y' a,Y2) 2 (19)

and the second constraint in eq. (15)

V~bP2 = - f (c~2Y' P, +/~2Y2 P2) d2tr.

(20)

The vanishing of the potential for the components YN which are orthogonal to the

K. Fujikawa,Z Kubo / Extendedobjects

213

compactified spaces then requires

OzYI= lf-Kbk O2Y2=O,

(21)

on the classical level. But the last relation in eq. (21)cannot be realized, since

f (v b + 02y2)d20 r= l f K b - - 0

(22)

means a de-compactification of the compactified space X2. Also the condition (20) cannot be satisfied by eq. (21), since P2 in the right-hand side of eq. (20) does not contain the zero mode and a non-vanishing P2 in the left-hand side is expected to be required in general for the stability of the compactified solution (12). We thus conclude that the potential valley for the components YN in the non-compactified spaces disappears as long as the compactified spaces in solution (12) are not de-compactified*. As for an analysis of slightly more general configurations, for which the first term in the potential (17) vanishes, see appendix A.

2.2. SYNCHRONOUS GAUGE FORMULATION

The physical picture involved in the case of the compactified solution (12) becomes more transparent if one uses the so-called synchronous gauge which preserves manifest rotation invariance. This gauge is specified by X°=r

(or Y ° = 0 ) ,

y l = y2=0

(23)

with y l and y2 defined in eqs. (12). The hamiltonian is then given by h = f dZo'{P 2 + P2! + P] + (Kab) 2 + r [ a Z ( a 2 Y ) 2 + b2(OiY) 2] + det(Ot, YOIY)} !/2 (24) with afrP I = -OiYP ,

bY'P2 = -02YP.

(25)

The zero frequency components of eq. (25) give alf-Kp, = - f O,YP d2o " ,

bl/r~p2 = - f 02YP d2o• ,

(26)

* The stability of compactified spaces appears to be implicitly assumed in ref. [10], since the authors there state that the supermembrane is unstable unless all the space coordinates are compactified.

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K. Fujikawa, J. Kubo / E.rtended objects

with quantized Pi = 2wli/a and P2 = 2rd2/b as before. The variables P and Y in eq. (24) now include the ( D - l)th components, y=(y3,y4,...,yO-t),

J' = ( P , , P.,,..., e o - , ) ,

(27)

where 1 X ~, -xV~(~.) + .__~yw(7., o'*), YK

/z = 3 , . . . , D - 1.

(28)

See appendix B for a derivation of eq. (24). The "potential terms" in eq. (24)

(abK)'-+

+ b-'(02¥) 2] +

(29)

clearly show that the compactification (12) generates a potential barrier for the orthogonal components. Although the hamiltonian (24) contains a square root and is not easily analysed, the string-like configuration Y = Y(r, o"~) gives a behavior which is no worse than a corresponding compactified string [ 17] in the synchronous gauge. In fact, if one assumes p~ ~ 0 and P2 #= 0 in eqs. (26), which are expected to be required to stabilize the compactified solution, the tr2-independent Y is not allowed; this situation is analogous to that in connection with the constraint (20) in the light-cone gauge. We note that the configuration (23) cannot easily be recognized in the light-cone gauge, since two of the three gauge degrees of freedom are used in the light-cone gauge to define the infinite momentum frame in the non-compactified space-time, as seen in eqs. (5). The space compactification thus tends to stabilize the membrane on the semiclassical level not only by modifying the potential as in (29) but also by imposing the extra constraints (26).

3. Covariant gauge formulation

In sect. 2 we found that the spatial compactification (12) generates a stabilizing potential barrier for orthogonal non-compactified variables on the semi-classical level. This phenomenon may be termed "confinement of unstable modes" in compactified spaces. If this behavior should be maintained in fully quantized theory, one would obtain a mechanism to save the unstable supermembrane [10] and also the decaying ball in D = 4 dimensions [9] by starting, for example from D = 7 with three space coordinates compactified. In this connection, it may be worth mentioning that the number of unstable modes in p-branes generally agrees with the number of (spatial) gauge degrees of freedom.

K. Fujikawa, J. Kubo / Extended objects

215

3.1. CONFORMAL GAUGE-TYPE FORMULATION

In this section, we examine the physical picture suggested by the synchronous gauge in (29) from a viewpoint of covariant quantization. Namely, we examine the BRST cohomology in the membrane theory and see whether the variables in the gauge (23) are really cancelled by Faddeev-Popov ghosts in the physical sector. Our covariant treatment in the following as it stands is an approximate one (strong-tension limit), but we hope that the covariant formulation will eventually provide a more general and gauge independent basis for the analysis of membranes. Our covariant treatment should thus be regarded as a first attempt toward this direction. Besides, our covariant formulation gives a better understanding of the area preserving symmetry. We thus start with the action defined by [14] S = f d3cr[½00Xu OoX~, - ½det Gt, t + iC.o(OoC°-OkC t' ) + iC, OoC* ] ,

(30)

with

Gkt=Ot, X ~"OIX~, + iC--, OiC° + iC.lOkC °,

k , l = 1,2.

(31)

The variables C a, a = 0, 1,2, stand for the reparametrization ghosts, and Ca the corresponding anti-ghosts. This gauge is a generalization of the conformal gauge in string theory. Note that the action (30) is a polynomial up to quartic terms. The hamiltonian H =/~Y d2cr and the BRST charge Q for the action (30) is given by

Z-'f~= ~'[0o Xr' OoX,, + det G,! ] + iCoOt, C t'

(32)

and

Q- f d2 (C°(½ooX"OoX,+½detGkt)+CkOkX'OoX, -i~_,o(Ct'Ot, C° +C°a, Ct') -iC, t,CldlCt'),

(33)

which satisfy

[H,Q] = 0 .

Q2=0,

(34)

The BRST transformations are given by

[ Q , X ~"] = [Q,C a]

-iCaOa x~ ,

= --iCl'Ob C a ,

[ Q, Col - ~ + i O0(CaCa) ' -- r,,,,

k---

(35)

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K. Fujikawa, J. Kubo / Extended objects

with ~ from eq. (32), and the energy-momentum tensor Tok defined by Tok = OoXt" 0k Xt, + i O~(C.kCa) - iC'k OtCt + iC't OkCt + 2iCo OkCO.

(36)

We here note the existence of an exact relation in (30)

ao(~,~'~- ~,f',) =o,

(37)

[n,o,~'~-a~',] =o.

(38)

or

By noting the relation (34) and the Jacobi identity [H, {Q, o,C 2 -02¢~!} ] + cyclic permutations = 0 ,

(39)

In, L,~] =o,

(4o)

we find with

L,~={o,o,e~-o~e,} = O,Toz - OzTo, ,

(41)

where Tok is defined in eq. (36); LiE generates the so-called area preserving symmetry [16] G w - f d2°" w(°'l, °'2)LI2

= f d2tr(02 W Tol -- 01w To2 )

(42)

and (1 + i G w ) ~ ( X t ' , . . . ) =

g t ( X t ' ( t r ° , o '' + 0 2 w , o ' 2 - O , w ) , . . . ) ,

(43)

for a general functional gr. It is interesting that the symmetry generated by L l2, which is analogous to the Gauss-law operator in the A 0 = 0 gauge in ordinary gauge theory, appears in the present covariant quantization as an exact symmetry. By imposing the constraints

(~,~ - a ~ ' , ) ~ = o,

Gw~ = 0,

(44)

in the Schr6dinger picture, which are consistent with the hamiltonian constraint, one can eliminate certain non-oscillatory variables in our problem; see eq. (57) in subsect. 3.3 for the explicit identification of non-oscillatory variables.

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K. Fujikawa, J. Kubo / Extended objects

3.2. STRONG-TENSION EXPANSION

In the strong-tension limit, i.e. large K, one can incorporate the tension parameter K into the action (30) in a systematic way by [14]

1 KS XU + ~ Y "

1 _

1

- Ks'O'( X c") + S'2)(X~, Y ' , 1

+--SO)(X -~

)

~',,, C") 1

. y . C,,,C '~) + - S ( 4 ' ( Y ' , ~ ' , , , C " ) . c '

K

"

(45)

with X~ a classical solution. To be specific, we consider the toms (S t x S t) compactified solution [14] 1

X'=x"

1

+--P'~'+K -~K Y ' ( r ' ( r k ) '

/1, 4= 1,2,

X t = a ( r t + x t + l p t ~ - + 1 Y 1( ~-,O.k) K ~-K ' 1 2 X 2 __ bo.2 + x 2 + Lp2~. + ~ - r Y ( ~, trk) ' r

(46)

where the terms atr t and b t r 2 stand for the classical solutions. The momenta pl= 27rlt/a and p2= 2~r!2/b are quantized as before. In the following, we include the center of mass motion, x" + p ' r / x , into S (°) for simplicity. The quadratic term in eq. (45) is then given by

S (2) = f dStr{ ½(30Y" 30Y, - b 2 3,Y" O,Y. - a232Y" 32Y.) + ½(a 32Y' - b 3,y2) 2 +i~.o(Oo Co -- OkC k) + i~. k OoCk -- ib2C_., a,C ° - ia2C_.2a2C°) , (47) where # = 0 , . . . , D - 1. One may treat S t3) and S (4) a s perturbations. We perform the mode expansion of variables in eq. (47)

y/~ ___ E r

1

(v,,,..

I

'o.,,,"u,,,. + v.* mr! e °

.,,,,,) ##

K Fujikawa, J. Kubo / Extended objects

218

with t~ ~ 1.2. and

(v' "v~) =--( - i ) E' (

1 (2 rr rob. 217"ha) [ Y.,,, e -''",','~ . u . . , - Y,,,~, e''"," . u*. ] V~to3/2

+

,

....... ( 2 . r n C a / b . -

2rrm b C ~ ) [ i r Z , . . -

}

Ym.]U,...

OJt t l l l

C ° = c° + - ~

( C , . . e ,o,.....' u,,., + C~,,,, e'" .....' u,,*,,),

C" = e ° + - ~

( e,,,,, ~-".,." ,,,,,,, + e~,,,, ~- - " ° ....." ,""* l t l l l ]

( C ' . C 2 ) =- ( c ' , c 2 ) -

~,(

"~

1 . . V ~ w , , , , ( 2 7 r m b 2 . 2 ~ r n a 2 ) [ C , . . e - ' ° ° .....~"u,,,,, + C,* e,.,,,,,,~"u,,*,,]

+ ab ( 27rn. - 27rm )t~,,,,,u,,,,,I ,

I

tOmn

(~,,~2) _(~, ~2)_ E'

1 --t eiW,,,,,ru .*m] v~-w,,,, (2"n'm,2rrn)[C-',,,.e-"°'""~'u,,,,, + Cm,

2 r r n ~ - , - 2rrm-Ok m n

,,,,urn,, , (48)

a

where we used the vector notation for (C 2, C 2), for example, and E' means the sum over m and n with m 2 + n 2 4= O. We also defined

Urn,, -- exp[21rimtr t + 2 r r i n t r 2 ] , to,,,,,- ¢(21rmb)2 + (2~rna) a .

(49)

The zero modes of ghosts and anti-ghosts in eqs. (48) are expressed by lower case letters without m and n indices. Some combinations of variables in eqs. (48) satisfy

219

K. Fujikawa, J. K u b o / E x t e n d e d objects

simple relations: .

bOtY l +a02 Y2

.

(Ym,, e-'',.,: u,,~. + Y,*. e''.,,: Umn).

0oC° =0,C* :i:

i ~ , , tOmn(Cmn e -"it°ran,'r ~!1111! ~r~

Ct*nn eta~m,,r U m n ) ,

OoC 0 = b 2 OIC ! + a 2 0 2 C 2

-

~/

E'

~,,,,,(c,,.,~-,°,,,:.,,.,-c,.,,e°,.,:.,~,,). . . . .

(50)

The commutation relations of the coefficients in eqs. (48) are given by

[ Y,.~,~ Y,n~',~'] = 61. m't~..'g "" 9

9

[ Ym,,, r,,,*.,,.l = [ ~,,,,, Zm',,'l = 8,,,o.~,,,,..

{f i n n ,

m¢

__ _

=3

Dt

8

111111' 1111' ~

{c°,G}=~.

(51)

The non-oscillatory variables satisfy the hermiticity relations "*

fin

n

=

-C?

-- nl,

-- n

9

11

g"* in. = ?-m.-,,,

Z*m .

= -C

=

-Z

- - n | • - - 11 ~

-m.-.,

(52)

The BRST charge Q from eq. (33) is expressed as Q = coil + ct"T,,t.

+i ~

+~

1 ~,

~m.(Y,:,*,C,,,,,

~

1

*

")

[abrto,,,,,-p'(27rmb) -p2(2zrna)][

+ 2 E' [2rtm~:tb2 + 2~rn~:a 2 - to,,,,,?o]C,.,,C,.. , *

r,,,,,c,,,,,+c,,,,,Y,,,,,] * * (53)

K. Fujikawa, J. Kubo

..0

/

Extended objects

with the hamiltonian

l {

H=-~r

,

(Pa)-+(P'

)2

+(p2)2+(abK

)2

+2KE c

a¢ --t + 2r E oJ,,,,,[t V~'~l,'~ + Y,,,,,Y,,,,, + C,,,,,C,,,,, + - I~ltl " Itlll

(54)

C D I I I C I I , I, ]

and the zero modes of T0k in eq. (36) are given in the vector notation by

(Tin,Toz) -/(To, dZ~r,To2 d2cr) = (ap' ' bp 2 ) -

Er

--t t -(2~r,n.2rrn)[V~*r t - n l n - i n~* l l + Y,~,,Y,,,,, + Cm.Cm, , + C,.,,C,.,,]

(55) The expressions for H in eq. (54) and Q in eq. (53) are valid in the present strong tension approximation, while To~ is "exact" in the sense that To~ in eq. (36) is quadratic in field variables. The non-oscillatory variables are suppressed in eqs.

(53)-(55). The Casimir energy E c in H given in eq. (54) is defined by

E¢ = ( D- 3)3'E' to,,,,,

(°_3)

(ab) 2 --

E

(a2m2+b2n2) - a / 2 + -

,(,

+

En

,}, (56)

in the #-function regulation [14].

3.3. BRST COHOMOLOGY

From the mode expansion in eqs. (48), we obtain

aO2Y I - b O i Y 2 = r ' ,o,,,,, ~ [irZ,,,,,a 2192CI - b 2 tglC2 -- ( - i )

O2C I -

OiC. 2 = i

%],,,,,, ,

Y'/abto,,,,,C.,,,,,u,,,,,,

r tomn ab ~mnUmn.

(57)

"

K. Fujikawa, J. Kubo / Extended objects

221

Using the commutation relations (51), the first constraint in (44) eliminates the (conjugate variable) £~m~-dependence of qt, the non-oscillatory ghost modes are eliminated altogether. Using the second constraint in (44), we can eliminate the non-oscillatory l~m.-dependence of 1/'; this is seen by noting G w • a02 YI-

b O l Y 2 --,

(aO~ + boe)w + terms homogeneous in Y' and y 2 ,

Gw" bO~Y ! +

a O 2 Y 2 -o

terms homogeneous in y! and y2,

(58)

under the action of Gw given by eq. (42). By choosing w suitably, one can thus eliminate the ~,,,-dependence of ~ , which is analogous to the elimination of the longitudinal mode of A# in QCD. As for the analysis of the oscillatory modes, it is convenient to use the conventional Fock space language. [To be precise, one may obtain a Faddeev-Popov determinant from an explicit elimination of ~,,,, in (58), which is analogous to the Coulomb gauge fixing, but this complication does not appear yet at the level of approximation we are working in.] If we impose the on-shell conditions [see eqs. (54) and (55)] m

HI ) = 0,

Tokl ) = 0,

k = 1,2,

(59)

on all the states, we may choose the (true)vacuum 10) such that ~',~10) = O,

c'~lO)* O,

(a =0,1,2),


(60)

for ghost and anti-ghost zero modes, to be consistent with QI0) = 0. As the lowest states which satisfy eqs. (59), we consider the set of states in the sector of vanishing ghost number {Y~*II,1),

Y~II,1),

c~¢~*,11,1)}

(61)

with

p.ll,l>=(2rr/a)[1,1>,

P211,1>=(2rr/b)ll,1).

(62)

We consider the states with p~ ~ 0 and P2 q: 0, since these conditions are generally required for the stability of compactified solutions (46). The on-shell condition HI ) = 0 then becomes [(p~,)2 + (abK)2 + (2,r/a)2 + (2zr/b)2 + 2KoJ,, + 2KEc]Y~*II, 1 ) = 0 and similarly for Yt*tI 1, 1) and c"Ct*ll 1, 1).

(63)

K. Fujikawa, J. Kubo / F.rtended objects

222

We now examine the BRST cohomology for the set of states (61). Using the BRST charge (53), we obtain QY~~I 1,1) = - i

QY~II,I)= ~

~/

loll

p'C*~,ll,l),

K

l(

1 )Cm*'ll'l)"

abrto,,

ab t°2'

(64)

On the other hand, we find QC~t]I 1) = i '

- - - p.g~*+ r,

~

abKtot~- ~

' (

Y~ [1 1)

°')1 ab

'

(65) "

From eqs. (64) and (65), we obtain

Q2~.t~,ll, l ) = ( t o , , / K ) [ ( p , ) 2 + ( a b r - t a , , / a b ) 2 ] C ~ , l l , 1 ) ,

(66)

which is nil-potent if

( pg)2 + (abK - toll/ab) 2 = O.

(67)

This condition coincides with the on-shell condition (63) if E c = -2~oit.

(68)

From the explicit forms of E c in eq. (56) and ~o,,,, in eq. (49) combined with a numerical calculation, condition (68) is satisfied for D Z 28.5

(69)

meaning, D ~ 28 dimensions! One can also confirm that c'~Cl*~]1, 1) are not in Ker Q. The BRST cohomology then becomes KerQ-(Yk*]I,1),

[ip, Y~* + ( a b u - t o , , / a b ) V , ~ ] ] l , 1 ) } ,

(70)

and I m O - ( [ i p , Y~* + (abK-to,t/ab)Y,*l]ll,l>

}.

(71)

K. Fujikawa, J. Kubo / Extended objects

223

This gives then

KerQ/ImQ={Y~*ll,1)

k=3,...,D-

1}

(72)

in the vanishing ghost-number sector. Here we assumed that p~, in eq. (67) is time-like, and we choose p,, = ( p o , O , O , . . . , o ) ,

(73)

which is consistent with the large r expansion. In the present analysis of the lowest states (61) in the strong-tension limit, the BRST cohomology is consistent with the synchronous gauge (23) and it gives essentially a rotation invariant version of the light-cone gauge analysis in ref. [11].

3.4. STABILITY OF T H E G R O U N D STATE CONFIGURATION

It may be instructive to recall the case of compactified strings [17], for which the quadratic approximation is exact. One can formally arrive at the compactified string by letting b - , 0 and r - - , o0 with ~ - b r kept fixed in our formula. The conditions (67) and (68) then become (p~)2 + ( a ~ - 27r/a) 2 = 0 ,

(74)

and KEc=

--2rOJll

=

-4rrff.

(75)

The condition (75) with a string Casimir energy is satisfied only for D = 26, which determines the critical dimension. Since eq. (74) is exact, one may examine the stationary confguration which is obtained at

a~ - 2 rr/a = 0.

(76)

In this case, the corresponding string states Y/~*I1) and Y~I 1) become massless, and

QY~*I1) ~p"C*~ ll),

QY~*I1)= 0 .

(77)

This means that the compactified component Y~*I1) becomes physical, ai~d two light-cone components of Y/'*I1) become unphysical, to be consistent with a massless vector particle represented by Y~*I1). Unfortunately, we cannot investigate the stationary configuration in our treatment of bosonic membranes. The stationary configuration means that the first two terms in the expansion of the action (45) are comparable. This in turn suggests that all the four terms in eq. (45) become comparable, and our approximation scheme

K. Fujikawa, J. Kubo / Extended objects

224

breaks down. Nevertheless it is interesting to examine the ground state mass

f(a,b,K)

= (abK) 2 + (21r/a) 2 + ( 2 z r / b ) 2 + 2rto,, + 2KE c ,

(78)

which is obtained from eq. (63). By using the explicit form of E c in eq. (56), one can confirm that f(a,b,r) is unbounded from below in the domain ~ > a >I 0 and > b >t 0 for D > 3, and that it has no stable ground state. (In ref. [14], the renormalization of K was suggested to avoid this problem, but no well-defined procedure of renormalization is known at this moment.) If this behavior suggested by f ( a , b, r ) persists in the exact treatment, it means that the Casimir energy de-stabilizes the compactified bosonic membrane although the classical potential itself indicates a nice behavior. In passing, it should be noted that Ec is positive for any finite mode cut-off, but it becomes negative when one sums all the terms in eq. (56). In the case of a compactified supermembrane,the authors in ref. [11] obtain a vanishing Casimir energy, E~ = 0. It is expected that this vanishing Casimir energy persists as long as the supersymmetry is preserved. If this is the case, we expect that the ground state configuration of a compactified supermembrane is stable in the 1/K expansion scheme because of the global behavior in eq. (19). For E¢ = 0 however, all the states become massive in compactified membranes with P l #: 0 and P2 ~ 0, and there are no direct applications to unification schemes [13]. For massive (super)membranes, the configuration in (23) is expected to be generally realized because of rotation invariance, although the simple Fock space construction such as in (72) is not valid beyond the large K expansion. One of the applications of our formulation of bosonic membranes, which avoids the issue of ground state stability, is to consider a fixed-end membrane for example, in the lattice theoretical setting [14]. In this case, one interprets the classical solution (46) as specifying a square-shaped classical membrane with all the ends kept fixed. One may then expand all Y~' in terms of sine modes with Pt = P 2 = 0 in eq. (46) which preserve the boundary conditions. One can still ask an interesting question whether narrow string-like configurations or "spikes" develop in the orthogonal directions from this membrane, whose boundaries are kept fixed in the two-dimensional subspace. The appearance of uncancel!ed non-oscillatory modes signals such instability. The instability of supermembranes noted in ref. [10] may be investigated in this manner also. Our formulation of bosonic membranes in this paper is applicable to this class of problems as long as r is kept large.

4. Conclusion

We studied the effects of a large number of gauge degrees of freedom inherent in the bosonic balls and membranes. From the study of synchronous and covariant gauge conditions, we are led to the analysis of a possible confinement of unstable

K. Fujikawa, 1. Kubo / Extended objects

225

modes in compactified spaces. This picture was examined in the 1/K expansion scheme of bosonic membranes. The BRST cohomology supports such a picture in the above approximation scheme, but at the same time the Casimir energy indicates a de-stabilization of the compactified ground state configuration. On the basis of a global consideration of classical potential and the vanishing Casimir energy, we also suggested a stable compactified supermembrane in the l / x expansion scheme. If our picture of confining unstable modes in compactified spaces is supported by further studies, the Dirac-Nambu-Goto action and its supersymmetrization will have many interesting applications. Our present work was mainly motivated by the interesting work in ref. [10], but the basic issue whether the seemingly stabilizing behavior of the classical potential (19) in compactified (super-)membranes is still de-stabilized in the non-perturbative matrix model regularization remains to be investigated. Our study may suggest that rather detailed analyses in the matrix model regularization such as the restoration of rotation invariance and the convergence properties at N - , ~ are required before the instability of compactified supermembrane suggested in ref. [10] is finally confirmed. The basic question is how close one can approach the configuration (21) by satisfying the constraints (13) and (20). A related interesting problem is a matrix regularization of the cot,ariant action (30); a preliminary investigation however indicates certain complications in the treatment of ghost variables. As for the stability of extended objects in general, it may be interesting to study a possible connection of the present problem with Derrick's theorem [18] in soliton theory, for example. We thank T. Inami, K. Kikkawa, M. Sasaki, S. Sawada and P. van Nieuwenhuizen for stimulating comments.

Appendix A The first term in the potential (17) in the light-cone gauge vanishes for the general string-like configurations

yN(r, or),

(A.1)

where or = a o r i + f l o r 2 ,

(A.2)

with mutually co-prime integers a and /3; the periodicity as in eq. (13)can be ensured by Y~(~',or+ 1)= YN(r, or). [For this general case, however, the basic domain of Ork in (2) covers the (degenerate) membrane configuration more than

K. Fujikawa, Z Kubo / Extended objects

226

once]. The potential for YN(r, a ) then becomes

+

+

(A.3)

-

which vanishes for

p(Vt~a +O,Y')=aO2Y ' , a( f-~b + O2Y Z) = ~3O,Y 2 .

(A.4)

This condition (A.4) cannot be consistent with (13).

Appendix B

The synchronous gauge (23) is specified by xo = X°(~', ak)

- r - 0,

x, =x'(~,,, ~) -[x'(~-) + a,~'] =0, (B.1)

X2 =X2('r, tr k) - [ x 2 ( ' r ) + bo"] = 0 , in the path integral formula of the evolution operator [19]

f ~ x ~ ~,P~,a(,b,)a(Xo)det[ 4~,,xb]exp[i f P~,,V~

d3tr],

(B.2)

with the constraints 6a in eqs. (4). We have

det[ 4,,,, Xb ] = det

2P0 2Pa l 2P2} ] 0 62( t r - tr' ) 0

0

(B.3)

b

for each time (slice) r. This determinant factor when combined with gives

fP,,X"d3,~=fPXd~,~+f[p,(~)~'(~ .) +p~(~)~(~)-h]

~(t~a)~(Xb)

dr,

(B.4)

!~ Fujikawa, Z Kubo / Extended objects

227

with

h=f

d20"(P 2 + P21 + p2

+ (abK)2+ K[a2(O2y)2 + b2(o,r)

l + i kl

m o,r n

where Y-{YN(r, trk);m = 3 ~ D - 1} are defined in eq. (28), and

P' =

1

aV~ O,YP,

P2 =

1

b¢-K02YP"

(B.6)

The remaining path integral measure is given by

(8.7) where we treat the zero-frequency components of P~ and /'2 in eq. (B.6)separately. The zero-frequency components, p~ and P2, are quantized with the phase factor in eq. (B.4)

exPlif(p,Yc' +p2~2)d~-] = explif(p, dx n + p 2 d x 2 ) ] ,

(B.S)

and the periodicity under x t -~ x I + a ,

x 2 --* x 2 + b,

(B.9)

resulting in

Pl = 27rll/a,

P2 = 2¢r12/b,

(B.10)

with integral l~ and 12. One may thus define the path integral (B.2) in terms of transverse variables only by

S~.~X~P~(aC-Kp, + S c ~ i Y p d 2 0 " ) ~ ( b C - K p 2 + S c ~ 2 Y p d 2 0 " ) e x p [ i f ( P X - h ) d 3 0 " ] ,

(B.11) with the quantized p~ and P2 as in eq. (B.10); the integers l~ and determined by the state to which (B.I 1) is applied.

!2 are

•

228

K. Fujikawa, J. Kubo / Ertended objects

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