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A Weyl-invariant rigid string
Volume 199, n u m b e r 2
PHYSICS LETTERS B
17 December 1987
A WEYL-INVARIANT RIGID STRING
U. L I N D S T R O M ~, M. ROCEK and P. VAN N I E U W E N H U I Z E N I n s t i t u t e f o r T h e o r e t i c a l Physics, S t a t e U n i v e r s i t y o f N e w Y o r k at S t o n y B r o o k , S t o n y B r o o k , N Y 1 1 7 9 4 - 3 8 4 0 ,
USA
Received 4 August 1987
We present a new WeyMnvariant string with a rigidity term. We give the first-order action and compute the critical dimension to be D = 14, using a particular background.
1. Introduction. The rigid string was introduced by Polyakov [ 1 ] as a candidate for a QCD string. Its action consists of the usual volume element plus a term quadratic in the extrinsic curvature K~b ~ IR=So f d2~x/~[1/Ro+½(yabK'~,,) 2] ,
(1.1)
where y refers to the induced metric on the world sheet ?~,b =OoXuOt, X I', y--dety~h,
/z=I,...,D,
y~'7~c=~ch ,
(1.2)
and -n~O,ObX , i J__(~ij n~nu -
','-{ni,}
i p n,,O~X =0
.
(1.3)
Here the n ~are the D - 2 unit normals to the world sheet and R 2 _ So~To, To being the string tension and So the rigidity parameter. Ro has the dimension of length while So is dimensionless. The dependence of the action on the normal fields n' is spurious, see (1.6). Classical aspects of this theory have recently been discussed in ref. [ 2] but its nonlinearity makes it difficult to quantize. The most promising approach seems to be a covariant path integral quantization similar to that described in refs. [3,4] for the N a m b u - G o t o string. In refs. [ 3,4] the first-order
form for the N a m b u - G o t o string, given in refs, [ 5,6 ], is used; however, no such formalism exists for the rigid string. In this note we propose an action which contains a rigidity term while maintaining the geometrical form of a volume element. In particular, the action: (i) contains (1.1) to lowest nontrivial order in So, (ii) is a straightforward generalization of the N a m b u - G o t o action in that it changes the measure to depend on a metric which is the sum of the induced metric and a term built from the extrinsic curvature, (iii) can be derived from a first-order action and (iv), in that form, has all the symmetries of (1.1) as well as a manifest Weyl invariance. In second-order form the action is I = To J dZ{x/~t(ya,, +Fab) ,
(1.4)
where F a b -=RoKacy 2 ~ cd Kdb ~ 2 i =RoKac7
cd
K ai b = R o2V a n i "Vbn
i
,
(1.5)
and K~b=VaObX u _- i
_
i
I~
K~b-nuK,b,
K a ib n u i p
,
p
i i KabKca=KabKca.
(1.6)
The covariant derivatives involve the Christoffel connection
Supported in part by NSF grant No. P H Y 85-07627. On leave of absence from ITP, University of Stockholm, S11346 Stockholm, Sweden. ~ We use euclidean signature in the world sheet.
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
{ab c} =- ½( Oby~. + O~Ybc - OcT~h) :OaObXltOc
Xp
.
(1.7) 219
Volume 199, number 2
PHYSICS LETTERS B
The expansion of the integrand in (1.4) follows from the expansion of the product of two e-tensors, and reads
I= To f d2~x/y(1 + yat'F~t,+ detyFah ) '/2 I D 2~,ab~,cdv/* To f d2¢,~[ '* x - - ~*'..o,V y l"~aclXbd
+o(R%)].
(1.8)
Here
17 December 1987
(1.4) shares with (1.12), immediately suggests how to find a first-order action with Weyl invariance.
2. First-order formulation. It is interesting to note that whereas it seems very hard to find a gravitational first-order action for (1.1) ~2, introducing independent gravitational fields in (1.1) gives a firstorder action for (1.4). The first-order action is (suppressing the coupling constants) I 1 = f d2{ ½x/ggab(yab +F~b) ,
detr F~h -= 7
(2.1)
( 1.9)
(with the antisymmetrization having strength one), and we will return to the higher-order terms. The first two terms in the expansion can be seen to correspond to (1.1) if we use (1.6) and
ya[cydlt'KUacK~#,=R ( 2 ) ( ~ )
,
(1.10)
and observe that the two-dimensional curvature scalar density x/Y R(2)(Y) is a total divergence. Thus, our action differs from Polyakov's action in the first two terms by only a total derivative, but our action contains an infinite series in extrinsic curvatures. Note that (1.4) is but one example of a generalization of the ordinary second-order action for the bosonic string. (The invariance under scaling, Xi~2X ', which was used in ref. [ 1 ] to single out the second term in (1.1) is already broken by the higherorder K-terms in (1.8).) If one views the rigid string as an effective theory rather than a fundamental one, it is natural to consider generalizations of ( 1.4) where the F-term is only the first in a series of higher derivative terms modifying the induced metric. We choose to study (1.4) for reasons of simplicity and because it has a geometric interpretation. We were further led to our model by analogy to the Born-Infeld action [7], which originally read f d4x[ det(t/~,,-£J~;) ] ,1/4
(1.11)
where Fab is as in (1.5) but with
K~b = V ~( F)OaX/'
(2.2)
i.e., we have introduced both an independent metric and an independent connection. The introduction of an independent connection is convenient when deriving field equations, e.g., but not necessary (the 1.5 order formalism [ 8 ], it does not mean that one goes on shell). The dynamics is independent of the order in which g and F are solved. The action (2.1) has the usual Weyl invariance
gat,--+Qg~t, ,
(2.3)
along with invariance under diffeomorphisms. Furthermore the F-term is invariant under scalings X i ~ k X i, although, unlike in the action (1.1), this does not make it unique in first-order formalism. The same property is shared by, e.g., xfggabV 2X/ZV aOb X/*, where indices are raised with the induced metric in the covariant d'alembertian. If we do not use an independent F , it can be shown that the most general first-order X-translation invariant action of the type (2.1), with at most four derivatives, is
/=f
d2{
x/ggab[yah+OCFab+flR(Z)(y)yat,],
(2.4)
with c~ and ¢/some constants. This can be shown using that V ayec= 0 implies
but which is equivalent to f d4x[det(~/ah + £ b ) ] ,/2 = J d4[det(t/~_f~,,)]
1/2 ,
(1.12)
with f,~, the electromagnetic field strength. For us f~,--,F,~,, the square of the extrinsic curvatures. This geometric volume element form, which our action 220
~2 As observed in ref. [2], one can trivially obtain a first-order action for (1.1) by taking the usual first-order action for the Nambu-Goto term while keeping the rigidity term completely in terms of the induced metric. There are other similar modifications possible, but we do not find them very satisfactory, since they lack the geometric interpretation as a volume element. As a curiosity we mention that there is also the possibility of a first-order action with only the connection and X as independent fields.
Volume 199, number 2
PHYSICS LETTERS B
O(oX ~ Vc Ob) X ~ = 0 ,
(2.5)
and differentiating (2.5) to obtain various relations between terms with two X-fields and four derivatives. The case a = 0 gives an action which, after elimination of g, gives the usual first-order action for the N a m b u - G o t o action with the Euler term R (2) We choose f l = 0 , i.e., the specific form in (1.4) and (2.1), because it allows a convenient first-order treatment of the connection. Varying the two gravitational fields g and F in (2.1) we find 6g~-
~I , j g~-- g, l,g c d (Tc~l+ F,.j) + x/g(7,1, +F,f,) = 0 ,
(2.6a) 6 F ~ - 2xfg g~t'ycdV ~OcX~O,.X ¢' = 0.
(2.6b)
Dividing by .,/g, taking the determinant and finally the square root of both sides we find from (2.6a) xfgg~'(y~,+F,b)=2x/det(y~,,+F,~,)
.
(2.7)
Peeling off x/g g"~'?2~d and using (1.2) and (1.7) we obtain from (2.6b) F ,l,, = { ab c} =O~ObX~' OcX ~' .
(2.8)
Together (2.7) and (2.8) give (1.4) when plugged back into (2.1). Observe that the first-order form for the connection works quite differently here than in the usual gravitational first-order formulations. Normally the connection is introduced via the curvature scalar, and its variation results in the metricity condition. See ref. [9] for such a treatment of the ordinary bosonic string. Here it is introduced via the scalar action and its variation yields the connection of the induced metric. In complete analogy to the usual string action, if we make a gauge choice in (2.1), such as conformal gauge, (2.6) will have to be supplemented to the equations of motion as a (generalized) Virasoro constraint. 3. Field equations and geometry. The field equations for the action (1.4) can be conveniently derived from the first-order action (2.1) in the conformal gauge
x ~ g " " = d ~/' .
(3.1)
When F satisfies its own field equations, its varia-
17 December 1987
tions need not be considered. Variation with respect to X ~' then gives (~,J,[O,Ot,Xl,+ 0~(V,Kf, " ,7, + F , Id, K j c , ~,..(, ± w , v ,w, J , ~I., r~ vl,~l JJ
=0,
(3.2)
which again has to be supplemented by (2.6) (with ga~--" 3 ah) .
The model described by (2.1) has two geometries associated with it: There is the geometry described by the induced metric Tab and corresponding Christoffel symbols (cf. (1.2), (1.7)); however, this metric is not directly related to the conformal structure. As can be seen from (2.6), (2.7), the first-order metric g~b corresponds to 7~h+F,~. This metric defines the second geometry. The Christoffel connection for this geometry can be expressed in terms of quantities defined in the first geometry: {abc}),+F=(rdc+FJ
ab ~,
+ ~(2V (~F~,~c-V ~F~h)
(3.3)
where the covariant derivative is with respect to the connection (1.7). In the conformal gauge (3.1), the g~b field equation (2.6a) imposes an orthonormal gauge condition not on the induced metric g~b but rather on the metric (7~b+F~b) of the second geometry. It is also this metric that gives the volume element in the second-order action (1.4). 4. Critical dimension. Since the action (2.1) has the classical Weyl invariance (2.3), we can ask if there is a critical dimension where this invariance persists in the quantized theory. Two issues are at stake: do the higher derivatives in the model change the anomaly or is the Liouville form preserved, and, assuming the latter, what is the critical dimension. We have investigated the critical dimension using Fujikawa's approach [ 10]. As in the usual bosonic string, the divergent terms do not cancel by themselves and we discard them. We then consider the remaining finite, propagating, terms and find that they cancel in the critical dimension D = 14. In Fujikawa's approach one begins by choosing a coordinate invariant measure in the path integral, e.g., for scalar fields X, ~X., X - (g)1/42. The anomaly is then the regulated jacobian for the change of
221
Volume 199, number2
PHYSICSLETTERSB
variables ) ~ X. More explicitly, the generating functional is
17 December 1987 1
lim e x p ( ! dt tr f d 2 { l n p f ~d2k e x p ( - lk.~) • M.o~ o
J" ~ ) ? e x p ( - t X(gX )
=I
etL ]
X exp( - p, (gTp,/M 2) exp (ik. {) ) ,
J"
•
(4.
The only non-WeyMnvariant factor is the (infinitedimensional) jacobian determinant, which can be written as det ~-~ = e x p ( ¼ t r l n g ) .
(4.2)
To evaluate (4.2) one regulates the trace with the field operator for JT. One then has to evaluate [ 10 ]
l
lira e x p ( ( D - 2 ) f d t t r 0 xf
f d2{ lap
d2k exp(B)(1 + (l//.t) {A2 + ½[A2,B]
(4.3)
M~oo
1 2 +-g(A , B- BA, 2 +A1BA~)}) ) ,
where we have chosen the conformal gauge , , ~ = p and defined p , - p ( ' - 1)/2 For our model, clearly the gauge fixing term and the ghost field operator are unmodified as compared to the usual case, and hence the ghost contribution to the anomaly remains - 26. However, the field operator for the X ~ fields is nonlinear, and we choose to expand around a specific background where the calculation is simple: X*' = ~ " ~ + q " .
(4.8)
where B_= --p-l(k2)2,
A I - 4i5p- l/2k2 k'Op-l/2 , A2--p-'/Z[2k2D+4(k'O)2-Rff2k2]p -'/2 .
(4.9)
Dropping the divergent term, and using symmetric integration to eliminate (k.0) 2 in favour of ~k ' 2[~, (4.8) becomes, after some algebra,
(4.4)
We also go to conformal gauge g,,b=p&,b. The expansion of the action to second order in q is easy since V ~0t,X~ is linear in q. The contributions from the connection conspire to give a simple field operator Cu. = - [] [auL + auv.(1 - S o [] )1,
(4.5)
where a~L and a~,~ are longitudinal and transverse projection operators respectively: T L ~l,~=6~,~-6~.
(4.6)
Thus the computation for the longitudinal modes is unmodified, and gives a contribution to the anomaly of + 2. The remaining contribution to the integral in (4.3) is
222
where we evaluate the spatial part of the trace in a plane wave basis. Pulling the plane waves through, rescaling k by l~-MSff '/2 and using the BakerCampbell-Hausdorff theorem, we are finally left with
+~[B,[B, A2]] +la2-a-![A2~,.s,,B] 2
I
lim exp(!dttrln,oexp(-pt~OpJM2)),
L a b ~i,~=~b~u~,
(4.7)
exp(-(D-2)
f d2~ lnp([Z lnp+ 3Rff2) ) (4.10)
The Ro term is all that remains of the background dependence and it is non-covariant because of our particular choice (4.4). Its contribution to the anomaly should be considered together with all the other background field contributions in a general background (which we have not calculated). The first trerm is twice the usual contribution to the anomaly (of. ref. [ 11 ]), and hence adding up all our anomaly coefficients and requiring that they cancel we find --26+2+2(DcRIv--2) =0, i.e., DCRIT= 14. We end this section with some comments on the computation of the critical dimension. Defining a = in p, we have focused our attention on the usual trDcr terms or, covariantly, x/g RE] -~R. (Note that the anomaly E]tr = x/gR is local.) There may be other
Volume 199, number 2
PHYSICS LETTERS B
propagating terms possible, such as a[2ay"~F,b, etc., and they should vanish, too (we expect that d i m e n sional analysis, covariance and locality o f the a n o m aly will rule t h e m out). F u r t h e r m o r e there are the usual n o n p r o p a g a t i n g divergent terms in our m o d e l p r o p o r t i o n a l to ~ / g = e x p ( a ) as well as new terms p r o p o r t i o n a l to e x p ( l a ) ; covariantly these would read (gT)j/4 o f [g det (),,~,+Fab)] t/4, etc. These terms are usually d i s c a r d e d by referring to a renormalization prescription, but the e x p ( a ) terms are known to cancel for the spinning string, a n d it would be gratifying if both types o f terms were to cancel in a s u p e r s y m m e t r i c version o f our m o d e l as well. Even if all terms with propagating ~ were to cancel, the algebraic d e p e n d e n c e on a o f the effective action might still be such that no classical m i n i m u m exists. It would be interesting to do a full one-loop background field c o m p u t a t i o n for general X.", and check that all the described properties hold. Such a comp u t a t i o n is all the m o r e desirable since the c o m p u tation we have p e r f o r m e d does not d i s c r i m i n a t e between P o l y a k o v ' s (see footnote 2) a n d our model.
5. Conclusions. In this p a p e r we have presented a new string; it is Weyl invariant yet it has a rigidity term and is geometric. To lowest o r d e r in the rigidity constant it is the same as Polyakov's rigid string. In second-order form the action is an area, but this area is not in terms o f the i n d u c e d metric, but rather in terms o f the i n d u c e d metric plus extrinsic curvature terms. It a d m i t s a first-order form, both with respect to the metric and with respect to the connection, in which the rigidity term has an extra constant scale invariance. W h e n analyzed using F u j i k a w a ' s m e t h o d in a special background, the Liouville m o d e is found to be non-propagating in D - - 1 4 .
17 December 1987
A n u m b e r o f open p r o b l e m s remain. The m o d e l has higher derivatives on the world sheet a n d is therefore p r e s u m a b l y non-unitary when regarded as a D = 2 field theory. It is unclear to us what implications this has for spacetime unitarity, a n d this is clearly the most i m p o r t a n t p r o b l e m to address. Secondly, an interesting question is whether this m o d e l a d m i t s (local) supersymmetrization. The answer is only partially known for Polyakov's rigid string [ 12], and m a y be easier to obtain in our model. Finally, a technical p r o b l e m remains: to u n d e r s t a n d the field d e p e n d e n t contributions to the a n o m a l y and, in particular, to d e t e r m i n e the fl-function for So. We would like to thank Marc G r i s a r u for discussions about the anomaly.
References [ 1] A.M. Polyakov, Nucl. Phys. B 268 (1986) 406. [2] T.L. Curtright, G.I. Ghandour and C.K. Zachos, Phys. Rev. D34(1986) 3811. [3] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [41 A.M. Polyakov, Phys. Len. B 103 (1981) 211. [ 5 ] L. Brink, P. di Vecchia and P. Howe, Phys. Len. B 65 (1976) 471. [6] S, Deser and B. Zumino, Phys. Len. B 65 (1976) 369. [7] M. Born and L. Infetd, Proc. R. Soc. A 144 (1939) 425. [8] P.K. Townsend and P. van Nieuwenhuizen, Phys. Lett. B 67 (1977) 439. [9] U. Lindstr6m and M. Ro~ek, Class. Quant. Grav. 4 (1987) L79. [ 10] K. Fujikawa, Phys. Rev. I,ett. 42 (1979) 1195: Phys. Rev. D 21 (1980) 2848; 22 (1980) 1499; Nucl. Phys. B 226 (1983) 437. [ 11 ] A. Eastaugh, L. Mezincescu, E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. Lett. 57 (1986) 29. [12] T,L. Curtright and P. van Nieuwenhuizen, Superstrings, preprint ITP-SB-87-20.
223
PHYSICS LETTERS B
17 December 1987
A WEYL-INVARIANT RIGID STRING
U. L I N D S T R O M ~, M. ROCEK and P. VAN N I E U W E N H U I Z E N I n s t i t u t e f o r T h e o r e t i c a l Physics, S t a t e U n i v e r s i t y o f N e w Y o r k at S t o n y B r o o k , S t o n y B r o o k , N Y 1 1 7 9 4 - 3 8 4 0 ,
USA
Received 4 August 1987
We present a new WeyMnvariant string with a rigidity term. We give the first-order action and compute the critical dimension to be D = 14, using a particular background.
1. Introduction. The rigid string was introduced by Polyakov [ 1 ] as a candidate for a QCD string. Its action consists of the usual volume element plus a term quadratic in the extrinsic curvature K~b ~ IR=So f d2~x/~[1/Ro+½(yabK'~,,) 2] ,
(1.1)
where y refers to the induced metric on the world sheet ?~,b =OoXuOt, X I', y--dety~h,
/z=I,...,D,
y~'7~c=~ch ,
(1.2)
and -n~O,ObX , i J__(~ij n~nu -
','-{ni,}
i p n,,O~X =0
.
(1.3)
Here the n ~are the D - 2 unit normals to the world sheet and R 2 _ So~To, To being the string tension and So the rigidity parameter. Ro has the dimension of length while So is dimensionless. The dependence of the action on the normal fields n' is spurious, see (1.6). Classical aspects of this theory have recently been discussed in ref. [ 2] but its nonlinearity makes it difficult to quantize. The most promising approach seems to be a covariant path integral quantization similar to that described in refs. [3,4] for the N a m b u - G o t o string. In refs. [ 3,4] the first-order
form for the N a m b u - G o t o string, given in refs, [ 5,6 ], is used; however, no such formalism exists for the rigid string. In this note we propose an action which contains a rigidity term while maintaining the geometrical form of a volume element. In particular, the action: (i) contains (1.1) to lowest nontrivial order in So, (ii) is a straightforward generalization of the N a m b u - G o t o action in that it changes the measure to depend on a metric which is the sum of the induced metric and a term built from the extrinsic curvature, (iii) can be derived from a first-order action and (iv), in that form, has all the symmetries of (1.1) as well as a manifest Weyl invariance. In second-order form the action is I = To J dZ{x/~t(ya,, +Fab) ,
(1.4)
where F a b -=RoKacy 2 ~ cd Kdb ~ 2 i =RoKac7
cd
K ai b = R o2V a n i "Vbn
i
,
(1.5)
and K~b=VaObX u _- i
_
i
I~
K~b-nuK,b,
K a ib n u i p
,
p
i i KabKca=KabKca.
(1.6)
The covariant derivatives involve the Christoffel connection
Supported in part by NSF grant No. P H Y 85-07627. On leave of absence from ITP, University of Stockholm, S11346 Stockholm, Sweden. ~ We use euclidean signature in the world sheet.
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
{ab c} =- ½( Oby~. + O~Ybc - OcT~h) :OaObXltOc
Xp
.
(1.7) 219
Volume 199, number 2
PHYSICS LETTERS B
The expansion of the integrand in (1.4) follows from the expansion of the product of two e-tensors, and reads
I= To f d2~x/y(1 + yat'F~t,+ detyFah ) '/2 I D 2~,ab~,cdv/* To f d2¢,~[ '* x - - ~*'..o,V y l"~aclXbd
+o(R%)].
(1.8)
Here
17 December 1987
(1.4) shares with (1.12), immediately suggests how to find a first-order action with Weyl invariance.
2. First-order formulation. It is interesting to note that whereas it seems very hard to find a gravitational first-order action for (1.1) ~2, introducing independent gravitational fields in (1.1) gives a firstorder action for (1.4). The first-order action is (suppressing the coupling constants) I 1 = f d2{ ½x/ggab(yab +F~b) ,
detr F~h -= 7
(2.1)
( 1.9)
(with the antisymmetrization having strength one), and we will return to the higher-order terms. The first two terms in the expansion can be seen to correspond to (1.1) if we use (1.6) and
ya[cydlt'KUacK~#,=R ( 2 ) ( ~ )
,
(1.10)
and observe that the two-dimensional curvature scalar density x/Y R(2)(Y) is a total divergence. Thus, our action differs from Polyakov's action in the first two terms by only a total derivative, but our action contains an infinite series in extrinsic curvatures. Note that (1.4) is but one example of a generalization of the ordinary second-order action for the bosonic string. (The invariance under scaling, Xi~2X ', which was used in ref. [ 1 ] to single out the second term in (1.1) is already broken by the higherorder K-terms in (1.8).) If one views the rigid string as an effective theory rather than a fundamental one, it is natural to consider generalizations of ( 1.4) where the F-term is only the first in a series of higher derivative terms modifying the induced metric. We choose to study (1.4) for reasons of simplicity and because it has a geometric interpretation. We were further led to our model by analogy to the Born-Infeld action [7], which originally read f d4x[ det(t/~,,-£J~;) ] ,1/4
(1.11)
where Fab is as in (1.5) but with
K~b = V ~( F)OaX/'
(2.2)
i.e., we have introduced both an independent metric and an independent connection. The introduction of an independent connection is convenient when deriving field equations, e.g., but not necessary (the 1.5 order formalism [ 8 ], it does not mean that one goes on shell). The dynamics is independent of the order in which g and F are solved. The action (2.1) has the usual Weyl invariance
gat,--+Qg~t, ,
(2.3)
along with invariance under diffeomorphisms. Furthermore the F-term is invariant under scalings X i ~ k X i, although, unlike in the action (1.1), this does not make it unique in first-order formalism. The same property is shared by, e.g., xfggabV 2X/ZV aOb X/*, where indices are raised with the induced metric in the covariant d'alembertian. If we do not use an independent F , it can be shown that the most general first-order X-translation invariant action of the type (2.1), with at most four derivatives, is
/=f
d2{
x/ggab[yah+OCFab+flR(Z)(y)yat,],
(2.4)
with c~ and ¢/some constants. This can be shown using that V ayec= 0 implies
but which is equivalent to f d4x[det(~/ah + £ b ) ] ,/2 = J d4[det(t/~_f~,,)]
1/2 ,
(1.12)
with f,~, the electromagnetic field strength. For us f~,--,F,~,, the square of the extrinsic curvatures. This geometric volume element form, which our action 220
~2 As observed in ref. [2], one can trivially obtain a first-order action for (1.1) by taking the usual first-order action for the Nambu-Goto term while keeping the rigidity term completely in terms of the induced metric. There are other similar modifications possible, but we do not find them very satisfactory, since they lack the geometric interpretation as a volume element. As a curiosity we mention that there is also the possibility of a first-order action with only the connection and X as independent fields.
Volume 199, number 2
PHYSICS LETTERS B
O(oX ~ Vc Ob) X ~ = 0 ,
(2.5)
and differentiating (2.5) to obtain various relations between terms with two X-fields and four derivatives. The case a = 0 gives an action which, after elimination of g, gives the usual first-order action for the N a m b u - G o t o action with the Euler term R (2) We choose f l = 0 , i.e., the specific form in (1.4) and (2.1), because it allows a convenient first-order treatment of the connection. Varying the two gravitational fields g and F in (2.1) we find 6g~-
~I , j g~-- g, l,g c d (Tc~l+ F,.j) + x/g(7,1, +F,f,) = 0 ,
(2.6a) 6 F ~ - 2xfg g~t'ycdV ~OcX~O,.X ¢' = 0.
(2.6b)
Dividing by .,/g, taking the determinant and finally the square root of both sides we find from (2.6a) xfgg~'(y~,+F,b)=2x/det(y~,,+F,~,)
.
(2.7)
Peeling off x/g g"~'?2~d and using (1.2) and (1.7) we obtain from (2.6b) F ,l,, = { ab c} =O~ObX~' OcX ~' .
(2.8)
Together (2.7) and (2.8) give (1.4) when plugged back into (2.1). Observe that the first-order form for the connection works quite differently here than in the usual gravitational first-order formulations. Normally the connection is introduced via the curvature scalar, and its variation results in the metricity condition. See ref. [9] for such a treatment of the ordinary bosonic string. Here it is introduced via the scalar action and its variation yields the connection of the induced metric. In complete analogy to the usual string action, if we make a gauge choice in (2.1), such as conformal gauge, (2.6) will have to be supplemented to the equations of motion as a (generalized) Virasoro constraint. 3. Field equations and geometry. The field equations for the action (1.4) can be conveniently derived from the first-order action (2.1) in the conformal gauge
x ~ g " " = d ~/' .
(3.1)
When F satisfies its own field equations, its varia-
17 December 1987
tions need not be considered. Variation with respect to X ~' then gives (~,J,[O,Ot,Xl,+ 0~(V,Kf, " ,7, + F , Id, K j c , ~,..(, ± w , v ,w, J , ~I., r~ vl,~l JJ
=0,
(3.2)
which again has to be supplemented by (2.6) (with ga~--" 3 ah) .
The model described by (2.1) has two geometries associated with it: There is the geometry described by the induced metric Tab and corresponding Christoffel symbols (cf. (1.2), (1.7)); however, this metric is not directly related to the conformal structure. As can be seen from (2.6), (2.7), the first-order metric g~b corresponds to 7~h+F,~. This metric defines the second geometry. The Christoffel connection for this geometry can be expressed in terms of quantities defined in the first geometry: {abc}),+F=(rdc+FJ
ab ~,
+ ~(2V (~F~,~c-V ~F~h)
(3.3)
where the covariant derivative is with respect to the connection (1.7). In the conformal gauge (3.1), the g~b field equation (2.6a) imposes an orthonormal gauge condition not on the induced metric g~b but rather on the metric (7~b+F~b) of the second geometry. It is also this metric that gives the volume element in the second-order action (1.4). 4. Critical dimension. Since the action (2.1) has the classical Weyl invariance (2.3), we can ask if there is a critical dimension where this invariance persists in the quantized theory. Two issues are at stake: do the higher derivatives in the model change the anomaly or is the Liouville form preserved, and, assuming the latter, what is the critical dimension. We have investigated the critical dimension using Fujikawa's approach [ 10]. As in the usual bosonic string, the divergent terms do not cancel by themselves and we discard them. We then consider the remaining finite, propagating, terms and find that they cancel in the critical dimension D = 14. In Fujikawa's approach one begins by choosing a coordinate invariant measure in the path integral, e.g., for scalar fields X, ~X., X - (g)1/42. The anomaly is then the regulated jacobian for the change of
221
Volume 199, number2
PHYSICSLETTERSB
variables ) ~ X. More explicitly, the generating functional is
17 December 1987 1
lim e x p ( ! dt tr f d 2 { l n p f ~d2k e x p ( - lk.~) • M.o~ o
J" ~ ) ? e x p ( - t X(gX )
=I
etL ]
X exp( - p, (gTp,/M 2) exp (ik. {) ) ,
J"
•
(4.
The only non-WeyMnvariant factor is the (infinitedimensional) jacobian determinant, which can be written as det ~-~ = e x p ( ¼ t r l n g ) .
(4.2)
To evaluate (4.2) one regulates the trace with the field operator for JT. One then has to evaluate [ 10 ]
l
lira e x p ( ( D - 2 ) f d t t r 0 xf
f d2{ lap
d2k exp(B)(1 + (l//.t) {A2 + ½[A2,B]
(4.3)
M~oo
1 2 +-g(A , B- BA, 2 +A1BA~)}) ) ,
where we have chosen the conformal gauge , , ~ = p and defined p , - p ( ' - 1)/2 For our model, clearly the gauge fixing term and the ghost field operator are unmodified as compared to the usual case, and hence the ghost contribution to the anomaly remains - 26. However, the field operator for the X ~ fields is nonlinear, and we choose to expand around a specific background where the calculation is simple: X*' = ~ " ~ + q " .
(4.8)
where B_= --p-l(k2)2,
A I - 4i5p- l/2k2 k'Op-l/2 , A2--p-'/Z[2k2D+4(k'O)2-Rff2k2]p -'/2 .
(4.9)
Dropping the divergent term, and using symmetric integration to eliminate (k.0) 2 in favour of ~k ' 2[~, (4.8) becomes, after some algebra,
(4.4)
We also go to conformal gauge g,,b=p&,b. The expansion of the action to second order in q is easy since V ~0t,X~ is linear in q. The contributions from the connection conspire to give a simple field operator Cu. = - [] [auL + auv.(1 - S o [] )1,
(4.5)
where a~L and a~,~ are longitudinal and transverse projection operators respectively: T L ~l,~=6~,~-6~.
(4.6)
Thus the computation for the longitudinal modes is unmodified, and gives a contribution to the anomaly of + 2. The remaining contribution to the integral in (4.3) is
222
where we evaluate the spatial part of the trace in a plane wave basis. Pulling the plane waves through, rescaling k by l~-MSff '/2 and using the BakerCampbell-Hausdorff theorem, we are finally left with
+~[B,[B, A2]] +la2-a-![A2~,.s,,B] 2
I
lim exp(!dttrln,oexp(-pt~OpJM2)),
L a b ~i,~=~b~u~,
(4.7)
exp(-(D-2)
f d2~ lnp([Z lnp+ 3Rff2) ) (4.10)
The Ro term is all that remains of the background dependence and it is non-covariant because of our particular choice (4.4). Its contribution to the anomaly should be considered together with all the other background field contributions in a general background (which we have not calculated). The first trerm is twice the usual contribution to the anomaly (of. ref. [ 11 ]), and hence adding up all our anomaly coefficients and requiring that they cancel we find --26+2+2(DcRIv--2) =0, i.e., DCRIT= 14. We end this section with some comments on the computation of the critical dimension. Defining a = in p, we have focused our attention on the usual trDcr terms or, covariantly, x/g RE] -~R. (Note that the anomaly E]tr = x/gR is local.) There may be other
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PHYSICS LETTERS B
propagating terms possible, such as a[2ay"~F,b, etc., and they should vanish, too (we expect that d i m e n sional analysis, covariance and locality o f the a n o m aly will rule t h e m out). F u r t h e r m o r e there are the usual n o n p r o p a g a t i n g divergent terms in our m o d e l p r o p o r t i o n a l to ~ / g = e x p ( a ) as well as new terms p r o p o r t i o n a l to e x p ( l a ) ; covariantly these would read (gT)j/4 o f [g det (),,~,+Fab)] t/4, etc. These terms are usually d i s c a r d e d by referring to a renormalization prescription, but the e x p ( a ) terms are known to cancel for the spinning string, a n d it would be gratifying if both types o f terms were to cancel in a s u p e r s y m m e t r i c version o f our m o d e l as well. Even if all terms with propagating ~ were to cancel, the algebraic d e p e n d e n c e on a o f the effective action might still be such that no classical m i n i m u m exists. It would be interesting to do a full one-loop background field c o m p u t a t i o n for general X.", and check that all the described properties hold. Such a comp u t a t i o n is all the m o r e desirable since the c o m p u tation we have p e r f o r m e d does not d i s c r i m i n a t e between P o l y a k o v ' s (see footnote 2) a n d our model.
5. Conclusions. In this p a p e r we have presented a new string; it is Weyl invariant yet it has a rigidity term and is geometric. To lowest o r d e r in the rigidity constant it is the same as Polyakov's rigid string. In second-order form the action is an area, but this area is not in terms o f the i n d u c e d metric, but rather in terms o f the i n d u c e d metric plus extrinsic curvature terms. It a d m i t s a first-order form, both with respect to the metric and with respect to the connection, in which the rigidity term has an extra constant scale invariance. W h e n analyzed using F u j i k a w a ' s m e t h o d in a special background, the Liouville m o d e is found to be non-propagating in D - - 1 4 .
17 December 1987
A n u m b e r o f open p r o b l e m s remain. The m o d e l has higher derivatives on the world sheet a n d is therefore p r e s u m a b l y non-unitary when regarded as a D = 2 field theory. It is unclear to us what implications this has for spacetime unitarity, a n d this is clearly the most i m p o r t a n t p r o b l e m to address. Secondly, an interesting question is whether this m o d e l a d m i t s (local) supersymmetrization. The answer is only partially known for Polyakov's rigid string [ 12], and m a y be easier to obtain in our model. Finally, a technical p r o b l e m remains: to u n d e r s t a n d the field d e p e n d e n t contributions to the a n o m a l y and, in particular, to d e t e r m i n e the fl-function for So. We would like to thank Marc G r i s a r u for discussions about the anomaly.
References [ 1] A.M. Polyakov, Nucl. Phys. B 268 (1986) 406. [2] T.L. Curtright, G.I. Ghandour and C.K. Zachos, Phys. Rev. D34(1986) 3811. [3] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [41 A.M. Polyakov, Phys. Len. B 103 (1981) 211. [ 5 ] L. Brink, P. di Vecchia and P. Howe, Phys. Len. B 65 (1976) 471. [6] S, Deser and B. Zumino, Phys. Len. B 65 (1976) 369. [7] M. Born and L. Infetd, Proc. R. Soc. A 144 (1939) 425. [8] P.K. Townsend and P. van Nieuwenhuizen, Phys. Lett. B 67 (1977) 439. [9] U. Lindstr6m and M. Ro~ek, Class. Quant. Grav. 4 (1987) L79. [ 10] K. Fujikawa, Phys. Rev. I,ett. 42 (1979) 1195: Phys. Rev. D 21 (1980) 2848; 22 (1980) 1499; Nucl. Phys. B 226 (1983) 437. [ 11 ] A. Eastaugh, L. Mezincescu, E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. Lett. 57 (1986) 29. [12] T,L. Curtright and P. van Nieuwenhuizen, Superstrings, preprint ITP-SB-87-20.
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