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H-invariance, imbedding ghost, and strings with extrinsic curvature
Volume 202, number 2
PHYSICS LETTERS B
3 March 1988
H - I N V A R I A N C E , I M B E D D I N G G H O S T , A N D S T R I N G S W I T H E X T R I N S I C CURVATURE K.S. V I S W A N A T H A N and Xiaoan Z H O U
Department of Physics, Simon Fraser University, Burnaby, B.C., Canada VSA 1S6 Received 8 September 1987
The extrinsic curvature term in the theory of smooth strings is shown to possess an additional invariance, called H-invariance, on stationary world-sheet surfaces. H-variation is a normal variation in which the normal is in the direction of the mean curvature vector. The additional Faddeev-Popov ghost arising from this invariance is determined and its effect on the Liouville modes is discussed. It is shown that the conformal dimension of this ghost field is ½and that the quantum smooth strings are stable for d<25.
Recently, Kleinert [ 1 ] has pointed out that the theory of strings with extrinsic curvature (smooth strings) as formulated by Polyakov [2] may contain additional negative n o r m states due to the higher derivative term in the action. I f this happens then it will be difficult to interpret the quantum theory o f smooth strings. This argument is based on our experience with higher derivative gravity theory [ 3 ]. It is the purpose o f this article to show that the smooth string theory does not contain additional negative n o r m states. Unlike the case o f quadratic gravity, smooth string theory is a theory o f an immersed two-dimensional world sheet in a d-dimensional space-time. This immersion makes it possible to remove the ghost states from the physical spectrum. A key observation made here is the invariance o f the extrinsic curvature term in the action under normal variations o f the world sheet Xu(a) in the direction o f the mean curvature vector provided that the R i e m a n n two-surfaces of genus p o f the world sheet are H-stationary. This property has been known in the mathematics literature [4,5 ]. In the q u a n t u m theory of strings, one considers path integrals over all immersion o f the world sheet in a d-dimensional (euclidean) space and over all metrics o f a given topology (genus p). H-invariance o f the action introduces an additional Faddeev - P o p o v ( F - P ) determinant in the path integral. We call the associated ghost an imbedding ghost and it is a scalar grassmannian. The imbedding ghost makes a contribution to the Liouville modes and the confor-
mal anomaly. It is known that the Liouville action arises in the q u a n t u m theory o f strings [ 6 ] as well as in the theory o f smooth strings [ 7-10 ]. The coefficient of the Liouville kinetic term for smooth strings was evaluated in refs. [ 7,8 ] and was found to be proportional to ( 2 6 - d ) / 4 8 n and hence it was concluded that the q u a n t u m theory is stable for d~< 26. Pisarski [7] has considered the critical case d = 2 6 carefully and he concludes that smooth strings are stable perturbatively for d < 26. However, in both refs. [7,8] it was assumed that the smooth string theory and the N a m b u - G o t o theory have the same n u m b e r o f ghost fields. Incorporation o f the imbedding ghost changes the coefficient o f the Liouville kinetic term to ( 2 5 - d ) / 4 8 n and hence we conclude that smooth string theory is stable for d < 25. We now turn to the details. The smooth string action contains, besides the N - G term, a term that depends on the second fundamental form o f the world sheet XU(a) immersed in a d-dimensional manifold, 1 Sextrinsi c -- 2~
0
f d2a xfg (h~a)(h~b) M
(a,b=l,2;r=3
.... , d ) ,
(1)
where ao is a dimensionless coupling constant. The h ~b are components of the second fundamental form defined by ¢
r
OaObX= l"abOcX + h aber ,
(2a) 217
Volume 202, number 2
PHYSICS LETTERS B
e;e~=tLs,
(2b,c)
er'OaX=O.
er (r = 3 ..... d) are d - 2 orthonormal normal vectors to the world sheet X ( a ) and Feb are the connection coefficients determined by the induced metric. gab=OaX'ObX. The Weingarten formula [4] takes the form Oa er = - h a rb O b X - } ' ~ rsa e s •
(3)
Raising and lowering of indices are accomplished with gab and g~b, respectively. An equivalent form for eq. (l) is given by
,J
Sexlrinsic--2ao
d2°" ~/~ (AX)2 '
(4)
ab
er,
(7)
where the e~ are defined in eqs. ( 2 a ) - ( 2 c ) . The submanifold M is called stationary if 5 .f o~n d r = 0
(8)
M
for all normal variations of M. In (8) 5 =0/0tlt=o,
(9)
a is the mean curvature a = (l/n) (H.H)1/2 = (1/n)[(hra)2] ~/2
d v = x / ~ d t r l Ado'2 ^... ^ dtr, .
where A X ~ = (1/X/~)( Oaw/g gabObX) .
(5)
In this form the extrinsic curvature term (4) has explicit reparametrization invariance. We now discuss its invariance under H-variations. Although it is known to the mathematicians, it has not so far been exploited in the physics of smooth strings. We give a brief discussion of it here and refer the reader to the book of C h e n [ 4 ] for details. Let M be a compact n-dimensional submanifold (with or without boundary) of a euclidean R ~ (n = 2 for the string theory). Let X denote the position vector of M in R e. Then X = X ( a l , ..., a~), where cr~, ..., ~, are local coordinates of M. Consider the following variation of X: X ' ( a l ..... a , , t)
(10)
.... , a , ) ( ( c r l , . . . , c r , ) ,
where ~(tr~ .... , a , ) is a differentiable function and t and infinitesimal parameter. ~(trl, ..., a , ) is a unit normal vector field of M in R d. We require that both q) and 0 ~ vanish identically on the boundary 0M of M, For a submanifold without boundary, there are no requirements on ~ . The variation in (6) is called a normal variation and some of its consequences are discussed by Kleinert [ 1 ]. If ~ is a unit normal vector field which is in the direction of the mean curvature vector H, then the variation (7) is called an H-variation of M in R d. The mean curvature vector is defined by
(11)
M is called H-stationary if (8) is true for all H-variations of M. We sketch here the proof but leave out the details (see ref. [4]). Now O~X, ..., O~X, e,+~, ..., ed define the natural orientation of R d. Let us choose e,+L=(. From (6) and (9) it follows that 8X=CI)en+l
(12)
.
The change in the induced metric is 5gab=-2fbh~,~l, 8~=
8gab=2cbh~+l ~b ,
- qb(tr h~+l)~fg.
(13a,b) (13c)
Using (12), (13) and the formula of Gauss (eq. (2a)) and Weingarten (eq. (3)), a long calculation yields 8 a 2 = (2/n 2) X [(tr h"+~)(Aq~+ q~llA~+~ II2-12q~)],
(6)
218
H = h a rb g
and
M
= X ( a ~ ..... a , ) + t q ~ ( a l
3 March 1988
(14)
provided that the variation is an H-variation; viz. the variation in which ~ is parallel to the mean curvature vector. In eq. (14) we have used the following notations: IIAn+~ 112= (ha~+l b ) ( h ~ + l 12 =lrn+ l,tlrnt+l
a) ,
(15)
(16a)
where l~+ l,~ = er0te,+ l = -- l~,+ ~
(16b)
For arbitrary normal variations there are terms proportional to (tr hr) (r 4: n + 1 ) which do not vanish. Then we find for the variation of the action for extrinsic curvature:
Volume 202, number 2
PHYSICS LETTERS B
3 March 1988
t rh~3)---*'~3)~ab,,ab ~ =v.n M
+~jJA.+l [I2 o e " - l - n ~ o e "+' ] d r .
(17)
(Remember that for smooth strings, n = 2 and f o ? dv is essentially S~xtri,~ic.) Integrating the first term in (17) by parts and imposing the vanishing boundary conditions on • and its gradient, we find
(21)
With this gauge fixing choice, the F - P determinant is given by
Av_v[x,g] = d e t ( - A -
{IA3 {{2 + l 2) .
(22)
Thus the action for extrinsic curvature is H-invariant if and only if the mean curvature a satisfies ( for n = 2)
IJA3 II2 depends on X, while l 2 (see eq. (16)) depends on the local coordinates a. The F - P determinant in eq. (22) arising from H-invariance plays a crucial role in cancelling the longitudinal (unphysical) modes in the mass spectrum of smooth strings if we replace l 2 by a suitable constant and ignore the coupling term IIA 3 ]]2 [ 9,10 ]. Eq. ( 22) can be expressed in terms of a ghost e and an anti-ghost ~ field. Since this determinant is over scalar functions, the ghost and antighost are scalar grassmannian fields. We call these the imbedding ghost fields. We write
A a - - a ( l 2 + 2 a 2 -- IIA3 I]2) = 0 ,
det( - A -
M M -hOe "+' + [IA~+l I[2) d r .
(18)
(19)
where A3 denotes the Weingarten map with respect to the unit vector in the direction of H. It is shown in ref. [ 5 ] that there exist H-invariant immersions of M of arbitrary genus p in R a. Chen [4] shows, for example, that a two-toms in R 3 is H-stationary if and only if a = x/-2b, where a and b are radii of the circles generating the torus. The classical action for strings with extrinsic curvature is taken in the first-order form [ 2 ]
S[x, gl=~' f ~/~d2a+~o f d2a [x/g(AX~)2 + 2 ~b(O~X~ObX~ -gab)] •
(20)
This action is manifestly reparametrization invariant. It is also H-invariant (on-shell) provided that the immersions M described by X ( a ) are H-stationary surfaces and ~ is chosen to be zero at the classical level. The quantum theory involves path integrals over all metrics of given topology (i.e. ofgenus-p surfaces) and over all immersion of the world sheet X(a) in d dimensions, including those that are H-stationary. We assume that the space of H-stationary surfaces o f g e n u s p is dense [4]. Then it is neccessary to use the F - P procedure to determine the Haar measure with respect to H-variations in addition to the usual F - P determinants associated with the reparametrization invariance. The H-gauge fixing condition is taken in the form
IIA3 II 2 + l 2)
det( - TA+ TI 2 + TI]A3 IJ2),
(23)
where the string tension T is introduced on dimensional grounds, det( - TA+ TI 2 + T[IA3 II2)
f Tfd2a f =jrdeJ tdJ expkj X e ( - - A + I 2 + IrA3 II 2 ) e ) •
(24)
It is seen from (24) that the term #llA3 JJ2e represents the coupling of the imbedding ghosts with X(a) through the fourth derivatives of X. #12e represents a mass-like term for the ghost although l 2 depends on local coordinates. Both these coupling terms do not, however, contribute to the conformal anomaly. Let us recall that the F - P determinant for the reparametrization invariance in the conformal gauge gob= e¢,~ob is given by o¢[g] = (det V-- )(det V :) = f t d c ] [de] [dbl [db] ×exp(-f
/- (b~-V ~c~+ c . c . ) ) -d2a 5 7 x/g
(25)
Here V z acts from rank-one tensors to rank-two tensors and V e is its complex conjugate. It has been re219
Volume 202, number 2
PHYSICS LETTERS B
marked by Polyakov [2], F~Srster [ 11 ], David [ 12], and Pisarski [ 7 ] that the LiouviUe action arises in the quantization of smooth strings. We now turn to the calculation of the Liouville action. We follow the procedure outlined in Friedan's [ 13 ] lecture notes. The stress-energy tensor is defined by 0 ~ = O L + OX~b'~'~'~ ,
(26)
where Og~ = - ( 4rc/x/~)~S/rg ~ ,
(27)
3 March 1988
singular part at short distances of ( e ( z ) £ ( w ) ) . We find ( e(z)~( w) ) ~ - ~ l n l z - w l .
(34)
Substituting (32a), (32b) and (34) into (31 ), we find ( Ox;_b'<'e'~OX'wb'C'e'e)c"- ½ ( 2 5 - d ) / ( z - w )
4.
(35)
Comparing (29) and (35) we find that 2 ~ 2 5 - d . Thus the effective Liouville kinetic term has the form d2°"
I
S(g, 0, = ( ~ 2 4 d ) ~ -~nv/-g (:g
ab
Oa~bO ) .
(36'
and
where we have set the string tension T = 1. Note that the ghost "mass" term and the extrinsic curvature term are not conformal invariants [ 6 ] and hence have no contribution to the Liouville mode kinetic energy. If we look only at the leading singularity, then (see ref. [13] for details)
The coefficient of the Liouville term is thus proportional to 25 - d, indicating that the smooth strings are stable for d ~ 25. It burns out that for d = 25 a more careful study of the Liouville action is needed and Pisarski [ 7 ] has shown that at the critical dimension the smooth string may in fact be perturbatively unstable. Pisarski [ 7] demonstrated that smooth strings are stable for d < 2 6 but he did not consider the imbedding ghost. Applying Pisarski's argument and using eq. (36) above, we conclude that the quantum theory of smooth strings is well defined for d < 25. It is interesting to compare the above result with the one in conformal anomaly [ 14] calculations. From the action (22), we see that both e(z, ~,) and ~(z, Z) have the same conformal dimension, that is,
( Ox~bx.e,e OX,~x,e,g) = - 2 / 2 ( z - w) 4 .
J=l -J,
ox,z Z bx,~,e_ -- V ~XV ~ X - 2b~VzC z - (V ~b~)c ~ -V~V~e.
(28)
We see from (28) that the action for the imbedding ghost e(z, ~) contributes to the stress-energy tensor a term O$z = - (V 2e)(V z e ) ,
(29)
(30)
We can find the same singularity by a direct Feynman diagram calculation of the OPE (~x,b,,..~ ~~ox, b.c.~,e\ ~ 2 [O~O,,(X(z)X(w) ) ]2 ww /
(37)
which leads to J=½.
(38)
~2Z
+ 2[ O~Ow(a(z)e(w) ) ]2 +40~ ( b=cW)O~( c~bw~ )
The contribution of the extra ghost to the conformal anomaly is therefore [ 13 ]
+ [ 2 ( b = c w )OzO,, (c2bww) + z ~ w ]
Ae(m)=(--d~ + J--~)m3 +~m =~m 3+~m.
+O~(b:,cW)O,,(czb,~,,) .
(39)
(31)
( X~'(z)X,,(w) ) ~ - ½du~I n ( z - w ) ,
(32a)
As stated before, since the extrinsic curvature term is not conformal invariant, it has no contribution to the conformal anomaly. Therefore, the total anomaly term of the smooth strings is
( cZbww) ~ 1 / ( z - w ) ,
(32b)
A ( m ) ~---AN_G(m) + h e ( m ) ,
The Green functions ( X ( z ) X ( w ) ) and (cZbww) are given by [ 13 ]
where
while (~(z) e(w) ) satisfies ( l / 2 n ) ( - 0 z 0 e ) ( e ( z ) ~ ( w ) ) = d 2 ( z - w) .
(33)
As in (32a) and (32b), we need only to know the 220
(40)
AN_G(m) = ~ D ( m 3 - rn) + -~( m - 13m 3) +2otto,
(41)
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PHYSICS LETTERS B
a n d A e ( m ) is given b y ( 3 9 ) . F r o m ( 3 9 ) , ( 4 0 ) a n d ( 4 1 ) , it is seen that A(m) v a n i s h e s for d = 25 a n d a = -~. A l t h o u g h the a n o m a l y t e r m v a n i s h e s o n l y at the critical d i m e n s i o n d = 25, a c o n s i s t e n t q u a n t u m t h e o r y m a y exist for d < 25 prov i d e d the Liouville m o d e s are t r e a t e d as d y n a m i c a l degrees o f f r e e d o m . I n that case, a d d i t i o n a l a n o m a l y t e r m s c o r r e s p o n d i n g to Liouville m o d e s c a n t h e n be a d d e d to eq. ( 4 0 ) [ 15 ] so that the total a n o m a l y c a n b e m a d e to v a n i s h for a n y d < 25.
References [ 1] H. Kleinert, Phys. Lett. B 174 (1986) 335; Phys, Rev. Lett. 58 (1987) 1915. [2] A.M. Polyakov, Nucl. Phys. B 268 (1986) 406. [3] J. Julve and M. Torin, Nuovo Cimento 46B (1978) 137; E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 104 (1981) 377. [4] B.Y. Chen, Total mean curvature and submanifolds of finite type (World Scientific, Singapore, 1984 ). [ 5 ] T.J. Wilmore, Total curvature in riemannian geometry (Ellis Horword, England, 1982).
3 March 1988
[6] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [7] R.D. Pisarski, Fermilab preprint (1987); Phys. Rev. Lett. 58 (1987) 1300. [8] T. Matsuki and K.S. Viswanathan, Phys. Rev. D, to be published. [9] K.S. Viswanathan and Zhou Xiaoan, submitted to Phys. Rev. D. [ 10] K.S. Viswanathan and Zhou Xiaoan, submitted to Phys. Rev. D. [ 11 ] D. F6rster, Phys. Lett. A 114 (1986) I 15. [ 12] F. David, Europhys. Lett. 2 (1986) 577. [ 13 ] D. Friedan, Introduction to Polyakov's string theory, in: Recent advances in field theory and statistical mechanics, ed. R. Stora, Les Houches Summer School (1982). [ 14 ] J.H. Schwarz, Faddeev-Popov ghosts and BRS symmetry in string theories, CalTech preprint (1985 ). [ 15] J.L. Gervais and A. Neveu, Nucl. Phys. B 209 (1982) 125; B 224 (1983) 329; B 238 (1984) 125, 396; B 257 (1985) 59; T.L. Curtright and C.B. Thorn, Phys. Rev. Lett. 48 (1982) 1309; E. Braaten, T. Curtright and C. Thorn, Phys. Lett. B 118 (1982) 115; R. Marnelius, Nucl. Phys. B 211 (1983) 14; S. Hwang, Phys. Rev. D 28 (1983) 2614.
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PHYSICS LETTERS B
3 March 1988
H - I N V A R I A N C E , I M B E D D I N G G H O S T , A N D S T R I N G S W I T H E X T R I N S I C CURVATURE K.S. V I S W A N A T H A N and Xiaoan Z H O U
Department of Physics, Simon Fraser University, Burnaby, B.C., Canada VSA 1S6 Received 8 September 1987
The extrinsic curvature term in the theory of smooth strings is shown to possess an additional invariance, called H-invariance, on stationary world-sheet surfaces. H-variation is a normal variation in which the normal is in the direction of the mean curvature vector. The additional Faddeev-Popov ghost arising from this invariance is determined and its effect on the Liouville modes is discussed. It is shown that the conformal dimension of this ghost field is ½and that the quantum smooth strings are stable for d<25.
Recently, Kleinert [ 1 ] has pointed out that the theory of strings with extrinsic curvature (smooth strings) as formulated by Polyakov [2] may contain additional negative n o r m states due to the higher derivative term in the action. I f this happens then it will be difficult to interpret the quantum theory o f smooth strings. This argument is based on our experience with higher derivative gravity theory [ 3 ]. It is the purpose o f this article to show that the smooth string theory does not contain additional negative n o r m states. Unlike the case o f quadratic gravity, smooth string theory is a theory o f an immersed two-dimensional world sheet in a d-dimensional space-time. This immersion makes it possible to remove the ghost states from the physical spectrum. A key observation made here is the invariance o f the extrinsic curvature term in the action under normal variations o f the world sheet Xu(a) in the direction o f the mean curvature vector provided that the R i e m a n n two-surfaces of genus p o f the world sheet are H-stationary. This property has been known in the mathematics literature [4,5 ]. In the q u a n t u m theory of strings, one considers path integrals over all immersion o f the world sheet in a d-dimensional (euclidean) space and over all metrics o f a given topology (genus p). H-invariance o f the action introduces an additional Faddeev - P o p o v ( F - P ) determinant in the path integral. We call the associated ghost an imbedding ghost and it is a scalar grassmannian. The imbedding ghost makes a contribution to the Liouville modes and the confor-
mal anomaly. It is known that the Liouville action arises in the q u a n t u m theory o f strings [ 6 ] as well as in the theory o f smooth strings [ 7-10 ]. The coefficient of the Liouville kinetic term for smooth strings was evaluated in refs. [ 7,8 ] and was found to be proportional to ( 2 6 - d ) / 4 8 n and hence it was concluded that the q u a n t u m theory is stable for d~< 26. Pisarski [7] has considered the critical case d = 2 6 carefully and he concludes that smooth strings are stable perturbatively for d < 26. However, in both refs. [7,8] it was assumed that the smooth string theory and the N a m b u - G o t o theory have the same n u m b e r o f ghost fields. Incorporation o f the imbedding ghost changes the coefficient o f the Liouville kinetic term to ( 2 5 - d ) / 4 8 n and hence we conclude that smooth string theory is stable for d < 25. We now turn to the details. The smooth string action contains, besides the N - G term, a term that depends on the second fundamental form o f the world sheet XU(a) immersed in a d-dimensional manifold, 1 Sextrinsi c -- 2~
0
f d2a xfg (h~a)(h~b) M
(a,b=l,2;r=3
.... , d ) ,
(1)
where ao is a dimensionless coupling constant. The h ~b are components of the second fundamental form defined by ¢
r
OaObX= l"abOcX + h aber ,
(2a) 217
Volume 202, number 2
PHYSICS LETTERS B
e;e~=tLs,
(2b,c)
er'OaX=O.
er (r = 3 ..... d) are d - 2 orthonormal normal vectors to the world sheet X ( a ) and Feb are the connection coefficients determined by the induced metric. gab=OaX'ObX. The Weingarten formula [4] takes the form Oa er = - h a rb O b X - } ' ~ rsa e s •
(3)
Raising and lowering of indices are accomplished with gab and g~b, respectively. An equivalent form for eq. (l) is given by
,J
Sexlrinsic--2ao
d2°" ~/~ (AX)2 '
(4)
ab
er,
(7)
where the e~ are defined in eqs. ( 2 a ) - ( 2 c ) . The submanifold M is called stationary if 5 .f o~n d r = 0
(8)
M
for all normal variations of M. In (8) 5 =0/0tlt=o,
(9)
a is the mean curvature a = (l/n) (H.H)1/2 = (1/n)[(hra)2] ~/2
d v = x / ~ d t r l Ado'2 ^... ^ dtr, .
where A X ~ = (1/X/~)( Oaw/g gabObX) .
(5)
In this form the extrinsic curvature term (4) has explicit reparametrization invariance. We now discuss its invariance under H-variations. Although it is known to the mathematicians, it has not so far been exploited in the physics of smooth strings. We give a brief discussion of it here and refer the reader to the book of C h e n [ 4 ] for details. Let M be a compact n-dimensional submanifold (with or without boundary) of a euclidean R ~ (n = 2 for the string theory). Let X denote the position vector of M in R e. Then X = X ( a l , ..., a~), where cr~, ..., ~, are local coordinates of M. Consider the following variation of X: X ' ( a l ..... a , , t)
(10)
.... , a , ) ( ( c r l , . . . , c r , ) ,
where ~(tr~ .... , a , ) is a differentiable function and t and infinitesimal parameter. ~(trl, ..., a , ) is a unit normal vector field of M in R d. We require that both q) and 0 ~ vanish identically on the boundary 0M of M, For a submanifold without boundary, there are no requirements on ~ . The variation in (6) is called a normal variation and some of its consequences are discussed by Kleinert [ 1 ]. If ~ is a unit normal vector field which is in the direction of the mean curvature vector H, then the variation (7) is called an H-variation of M in R d. The mean curvature vector is defined by
(11)
M is called H-stationary if (8) is true for all H-variations of M. We sketch here the proof but leave out the details (see ref. [4]). Now O~X, ..., O~X, e,+~, ..., ed define the natural orientation of R d. Let us choose e,+L=(. From (6) and (9) it follows that 8X=CI)en+l
(12)
.
The change in the induced metric is 5gab=-2fbh~,~l, 8~=
8gab=2cbh~+l ~b ,
- qb(tr h~+l)~fg.
(13a,b) (13c)
Using (12), (13) and the formula of Gauss (eq. (2a)) and Weingarten (eq. (3)), a long calculation yields 8 a 2 = (2/n 2) X [(tr h"+~)(Aq~+ q~llA~+~ II2-12q~)],
(6)
218
H = h a rb g
and
M
= X ( a ~ ..... a , ) + t q ~ ( a l
3 March 1988
(14)
provided that the variation is an H-variation; viz. the variation in which ~ is parallel to the mean curvature vector. In eq. (14) we have used the following notations: IIAn+~ 112= (ha~+l b ) ( h ~ + l 12 =lrn+ l,tlrnt+l
a) ,
(15)
(16a)
where l~+ l,~ = er0te,+ l = -- l~,+ ~
(16b)
For arbitrary normal variations there are terms proportional to (tr hr) (r 4: n + 1 ) which do not vanish. Then we find for the variation of the action for extrinsic curvature:
Volume 202, number 2
PHYSICS LETTERS B
3 March 1988
t rh~3)---*'~3)~ab,,ab ~ =v.n M
+~jJA.+l [I2 o e " - l - n ~ o e "+' ] d r .
(17)
(Remember that for smooth strings, n = 2 and f o ? dv is essentially S~xtri,~ic.) Integrating the first term in (17) by parts and imposing the vanishing boundary conditions on • and its gradient, we find
(21)
With this gauge fixing choice, the F - P determinant is given by
Av_v[x,g] = d e t ( - A -
{IA3 {{2 + l 2) .
(22)
Thus the action for extrinsic curvature is H-invariant if and only if the mean curvature a satisfies ( for n = 2)
IJA3 II2 depends on X, while l 2 (see eq. (16)) depends on the local coordinates a. The F - P determinant in eq. (22) arising from H-invariance plays a crucial role in cancelling the longitudinal (unphysical) modes in the mass spectrum of smooth strings if we replace l 2 by a suitable constant and ignore the coupling term IIA 3 ]]2 [ 9,10 ]. Eq. ( 22) can be expressed in terms of a ghost e and an anti-ghost ~ field. Since this determinant is over scalar functions, the ghost and antighost are scalar grassmannian fields. We call these the imbedding ghost fields. We write
A a - - a ( l 2 + 2 a 2 -- IIA3 I]2) = 0 ,
det( - A -
M M -hOe "+' + [IA~+l I[2) d r .
(18)
(19)
where A3 denotes the Weingarten map with respect to the unit vector in the direction of H. It is shown in ref. [ 5 ] that there exist H-invariant immersions of M of arbitrary genus p in R a. Chen [4] shows, for example, that a two-toms in R 3 is H-stationary if and only if a = x/-2b, where a and b are radii of the circles generating the torus. The classical action for strings with extrinsic curvature is taken in the first-order form [ 2 ]
S[x, gl=~' f ~/~d2a+~o f d2a [x/g(AX~)2 + 2 ~b(O~X~ObX~ -gab)] •
(20)
This action is manifestly reparametrization invariant. It is also H-invariant (on-shell) provided that the immersions M described by X ( a ) are H-stationary surfaces and ~ is chosen to be zero at the classical level. The quantum theory involves path integrals over all metrics of given topology (i.e. ofgenus-p surfaces) and over all immersion of the world sheet X(a) in d dimensions, including those that are H-stationary. We assume that the space of H-stationary surfaces o f g e n u s p is dense [4]. Then it is neccessary to use the F - P procedure to determine the Haar measure with respect to H-variations in addition to the usual F - P determinants associated with the reparametrization invariance. The H-gauge fixing condition is taken in the form
IIA3 II 2 + l 2)
det( - TA+ TI 2 + TI]A3 IJ2),
(23)
where the string tension T is introduced on dimensional grounds, det( - TA+ TI 2 + T[IA3 II2)
f Tfd2a f =jrdeJ tdJ expkj X e ( - - A + I 2 + IrA3 II 2 ) e ) •
(24)
It is seen from (24) that the term #llA3 JJ2e represents the coupling of the imbedding ghosts with X(a) through the fourth derivatives of X. #12e represents a mass-like term for the ghost although l 2 depends on local coordinates. Both these coupling terms do not, however, contribute to the conformal anomaly. Let us recall that the F - P determinant for the reparametrization invariance in the conformal gauge gob= e¢,~ob is given by o¢[g] = (det V-- )(det V :) = f t d c ] [de] [dbl [db] ×exp(-f
/- (b~-V ~c~+ c . c . ) ) -d2a 5 7 x/g
(25)
Here V z acts from rank-one tensors to rank-two tensors and V e is its complex conjugate. It has been re219
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PHYSICS LETTERS B
marked by Polyakov [2], F~Srster [ 11 ], David [ 12], and Pisarski [ 7 ] that the LiouviUe action arises in the quantization of smooth strings. We now turn to the calculation of the Liouville action. We follow the procedure outlined in Friedan's [ 13 ] lecture notes. The stress-energy tensor is defined by 0 ~ = O L + OX~b'~'~'~ ,
(26)
where Og~ = - ( 4rc/x/~)~S/rg ~ ,
(27)
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singular part at short distances of ( e ( z ) £ ( w ) ) . We find ( e(z)~( w) ) ~ - ~ l n l z - w l .
(34)
Substituting (32a), (32b) and (34) into (31 ), we find ( Ox;_b'<'e'~OX'wb'C'e'e)c"- ½ ( 2 5 - d ) / ( z - w )
4.
(35)
Comparing (29) and (35) we find that 2 ~ 2 5 - d . Thus the effective Liouville kinetic term has the form d2°"
I
S(g, 0, = ( ~ 2 4 d ) ~ -~nv/-g (:g
ab
Oa~bO ) .
(36'
and
where we have set the string tension T = 1. Note that the ghost "mass" term and the extrinsic curvature term are not conformal invariants [ 6 ] and hence have no contribution to the Liouville mode kinetic energy. If we look only at the leading singularity, then (see ref. [13] for details)
The coefficient of the Liouville term is thus proportional to 25 - d, indicating that the smooth strings are stable for d ~ 25. It burns out that for d = 25 a more careful study of the Liouville action is needed and Pisarski [ 7 ] has shown that at the critical dimension the smooth string may in fact be perturbatively unstable. Pisarski [ 7] demonstrated that smooth strings are stable for d < 2 6 but he did not consider the imbedding ghost. Applying Pisarski's argument and using eq. (36) above, we conclude that the quantum theory of smooth strings is well defined for d < 25. It is interesting to compare the above result with the one in conformal anomaly [ 14] calculations. From the action (22), we see that both e(z, ~,) and ~(z, Z) have the same conformal dimension, that is,
( Ox~bx.e,e OX,~x,e,g) = - 2 / 2 ( z - w) 4 .
J=l -J,
ox,z Z bx,~,e_ -- V ~XV ~ X - 2b~VzC z - (V ~b~)c ~ -V~V~e.
(28)
We see from (28) that the action for the imbedding ghost e(z, ~) contributes to the stress-energy tensor a term O$z = - (V 2e)(V z e ) ,
(29)
(30)
We can find the same singularity by a direct Feynman diagram calculation of the OPE (~x,b,,..~ ~~ox, b.c.~,e\ ~ 2 [O~O,,(X(z)X(w) ) ]2 ww /
(37)
which leads to J=½.
(38)
~2Z
+ 2[ O~Ow(a(z)e(w) ) ]2 +40~ ( b=cW)O~( c~bw~ )
The contribution of the extra ghost to the conformal anomaly is therefore [ 13 ]
+ [ 2 ( b = c w )OzO,, (c2bww) + z ~ w ]
Ae(m)=(--d~ + J--~)m3 +~m =~m 3+~m.
+O~(b:,cW)O,,(czb,~,,) .
(39)
(31)
( X~'(z)X,,(w) ) ~ - ½du~I n ( z - w ) ,
(32a)
As stated before, since the extrinsic curvature term is not conformal invariant, it has no contribution to the conformal anomaly. Therefore, the total anomaly term of the smooth strings is
( cZbww) ~ 1 / ( z - w ) ,
(32b)
A ( m ) ~---AN_G(m) + h e ( m ) ,
The Green functions ( X ( z ) X ( w ) ) and (cZbww) are given by [ 13 ]
where
while (~(z) e(w) ) satisfies ( l / 2 n ) ( - 0 z 0 e ) ( e ( z ) ~ ( w ) ) = d 2 ( z - w) .
(33)
As in (32a) and (32b), we need only to know the 220
(40)
AN_G(m) = ~ D ( m 3 - rn) + -~( m - 13m 3) +2otto,
(41)
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a n d A e ( m ) is given b y ( 3 9 ) . F r o m ( 3 9 ) , ( 4 0 ) a n d ( 4 1 ) , it is seen that A(m) v a n i s h e s for d = 25 a n d a = -~. A l t h o u g h the a n o m a l y t e r m v a n i s h e s o n l y at the critical d i m e n s i o n d = 25, a c o n s i s t e n t q u a n t u m t h e o r y m a y exist for d < 25 prov i d e d the Liouville m o d e s are t r e a t e d as d y n a m i c a l degrees o f f r e e d o m . I n that case, a d d i t i o n a l a n o m a l y t e r m s c o r r e s p o n d i n g to Liouville m o d e s c a n t h e n be a d d e d to eq. ( 4 0 ) [ 15 ] so that the total a n o m a l y c a n b e m a d e to v a n i s h for a n y d < 25.
References [ 1] H. Kleinert, Phys. Lett. B 174 (1986) 335; Phys, Rev. Lett. 58 (1987) 1915. [2] A.M. Polyakov, Nucl. Phys. B 268 (1986) 406. [3] J. Julve and M. Torin, Nuovo Cimento 46B (1978) 137; E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 104 (1981) 377. [4] B.Y. Chen, Total mean curvature and submanifolds of finite type (World Scientific, Singapore, 1984 ). [ 5 ] T.J. Wilmore, Total curvature in riemannian geometry (Ellis Horword, England, 1982).
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[6] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [7] R.D. Pisarski, Fermilab preprint (1987); Phys. Rev. Lett. 58 (1987) 1300. [8] T. Matsuki and K.S. Viswanathan, Phys. Rev. D, to be published. [9] K.S. Viswanathan and Zhou Xiaoan, submitted to Phys. Rev. D. [ 10] K.S. Viswanathan and Zhou Xiaoan, submitted to Phys. Rev. D. [ 11 ] D. F6rster, Phys. Lett. A 114 (1986) I 15. [ 12] F. David, Europhys. Lett. 2 (1986) 577. [ 13 ] D. Friedan, Introduction to Polyakov's string theory, in: Recent advances in field theory and statistical mechanics, ed. R. Stora, Les Houches Summer School (1982). [ 14 ] J.H. Schwarz, Faddeev-Popov ghosts and BRS symmetry in string theories, CalTech preprint (1985 ). [ 15] J.L. Gervais and A. Neveu, Nucl. Phys. B 209 (1982) 125; B 224 (1983) 329; B 238 (1984) 125, 396; B 257 (1985) 59; T.L. Curtright and C.B. Thorn, Phys. Rev. Lett. 48 (1982) 1309; E. Braaten, T. Curtright and C. Thorn, Phys. Lett. B 118 (1982) 115; R. Marnelius, Nucl. Phys. B 211 (1983) 14; S. Hwang, Phys. Rev. D 28 (1983) 2614.
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