A geometrical approach to string field theory

Nuclear Phs ~tes B284 (1987) 334-348 North-Holland Amsterdam

A GEOMETRICAL

APPROACH

TO STRING

FIELD THEORY

*

Korkut BARDAKCI* * La)~tenee Berl~ele~ Laborator) and Depatmlent o/Ph~lLs, ~ rare*sin ./ ( ahfolma BerLelc) C a h # . m a o4"20 L ~4

Recep~ed 15 September 1986

Starting from an extended ,~ersxonof reparametnzat~onm',ar~am.e,,a~.,_onqruct a lagranglan denslt~ for the interacting open bosomc string ~&eexpand the lagrangmn dcn~lt,~m pm~ers of th~ fields and ,,hov, that, the free part at least is sahsfactorx

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

O n e of the o u t s t a n d i n g problems of string theory is to fmd a coxartant, second q u a n u z e d reformulation, preferably based on a geometrical m v a n a n c e principle Recently, there has been a good deal of progress m this field [1 7] The m a m tool c o m m o n l y m use is the BRST approach [8, 9] to string q u a n u z a u o n Even the string m t e r a c n o n can now be incorporated [7, 10] m this l r a m e ~ o r k Despite ~ts obwou~ success, the standard approach, at least m th~s author's o p i n i o n , lacks a clear geometric f o u n d a u o n A c a n & d a t e for thts missing geometrtc principle was proposed in an earher paper [11] In this approach, the standard r e p a r a m e t r i z a t i o n mvarmnce of the string ~ promoted to a gauge m x a n a n c e by m a k i n g the group parameters d e p e n d e n t on the strmg coordmates This appears to be a r e a s o n a b l e principle on g e o m e m c grounds, ~mce, m a second quanttzed theory', it allows for i n d e p e n d e n t reparametrizattons of different strings The dtfferentlal geometry associated with this gauge p r m o p l e was developed m ref [11], ho~ever. the p r o b l e m of writing down a string lagrangtan based on these tdeas was largely left open In this paper, we report partml progress along these hnes The free part of the l a g r a n g l a n presented in th~s paper ts at least clas~lcall,v sat~sfactor'v A ntce feature of this a p p r o a c h is that the interaction ~s umquely determined b~ the gauge principle However, a full analysis of the mteractmg theory appears rather difficult * This work supported m part bv the National Science Foundatmn under grant PH~t85-15857 and in part b5 the Darector, Office of High Energy and Nuclear Physics, Dl~tston of Itlgh Energ\ Ph,~su.sol the US Department of Energy under contract DE-AC03-76SEO0098 ** Address after September 1, 1986 CERN CH-1211 (,enexa 23 S,altzerland 0550-3213/87/$03 50' Elsexmr Scmnce Pubhsher~ B \ (North-Holland Phy~ms Pubhshmg Dlvlsaon)

K Bardal~u / Strmgfleld theory

335

and will not be attempted in this paper Another important ingredient not included by the present treatment is the antlcommutmg coordinates of the BRST formulatlon It ~s known [1, 2] that full gauge fixing must introduce either the antl-commutmg coordinates or some equivalent comphcatlon Here, we should point out that the string action does not exist without gauge fixing, the actxon is gwen by

A = fDX

(X),

(1 1)

where the lagrangmn density is functionally integrated over the string coordinates X " ( o ) In our treatment, the lagrangian density ~s invariant under the loop dependent reparametrizatlons of the stnng coordinates (see eq (2 8)). Thas leads to the divergence of the integral for the action m a manner famlhar from standard gauge theories, what ~s new here is that it is the action m contrast to the path integral, that is divergent. Therefore, what we present here is the lagranglan density and not the action, to construct the action, one must first gauge fix completely through the standard Faddeev-Popov procedure, winch will presumably introduce the antIcommuting coordinates absent m the treatment gwen here We felt, however, that the construcUon of the lagrangian density is of sufficaent interest in its own right to be presented separately We hope to return to the ~mportant problem of gauge fixing in the future Although this paper ~s based on the development given in ref [11], the present treatment differs from the prevmus one m several ~mportant respects For this reason, and for the convenience of the reader unfamiliar with ref [11], we shall present a self-contained treatment However, we will also point out the differences between the old and the new treatments One fundamental difference is the following in ref [11], only the reparametrlzauon of the parameter o that labels the string coordinate was considered However, we know that this is not the full story, and that separate reparametrlzatlons of the left- and rlght-mowng components of the string must be considered To formulate this larger mvanance requirement, one has first to define a string field which is a functmn of the right and left movers, instead of the coordinate X ( o ) Thxs presents a problem, since right (or left) movers do not form a commuting set of varmbles and therefore cannot be treated as ordinary coordinates It is possible that the idea of non-commutative geometry [7] is relevant here, however, we bypass this problem by replacing the non-commutative coordinates by commuting coordinates by means of a coherent state representation In the next section, it will be shown that the string dynamics simplifies considerably when expressed m terms of the new coordinates, and there is a parallel slmphflcanon m the dffferenUal geometry appropriate to the problem This differential geometry xs developed m sects 2 and 3 The fundamental dynamical quantity turns out to be the torsion tensor budt out of the vlelbems There is only a single non-vanishing scalar that can be constructed from the torsion tensor, and that xs our

336

K Bardal~¢l / Strmgjteld theor~

c a n & d a t e for the lagranglan density This umqueness is a veD race feature of the a p p r o a c h presented here In sect 4, the lagrangmn ~s expanded to quadratic order and is shown to be satisfactory classically The last section summarizes our conclusions and lists some of the open problems

2. Differential geometry in the loop space In this section, the coordinate that specifies the location of the string points will be denoted by Y"(o), instead of the more customary X~(o) This is because we wish to reserve the latter symbol for commuting coordinates to be defined later We will also frequently suppress the Lorentz index /x to slmphfy writing The right and left movers are defined by

Kr 1(o) = Y ' ( o ) ~ l

(2 1)

where the prime denotes differentiation with respect to o The K ' s satisfy the following c o m m u t a t i o n relations

[ K ~ ( o ) , K~(o')] = - [ K ~ ( o ) , K ( ( o ' ) ] = 2 t 3 ' ( 0 ' - o)3'", [K~(o), K((o')] = 0

(2 2)

One can then expand the right and left movers into the Fourier modes and define two c o m m u t i n g sets of string creation and annihilation operators This defines a F o c k space, which corresponds to the products of Hermlte functions ol various orders in the variable Y(o) The string field, which as a functional of Y, can then be e x p a n d e d into a series of Hermite functions, or alternatlvel), into a series of states of the F o c k space defined above We find it convement to write this expansion in a c o m p a c t form in terms of coherent states

O: f DXrf DXlO(Xr, X,)exp{f do [/k'r(O)~r(O

) q- ]'X'l(O ) Sl(O )] llO ) (2 3)

In this equation, ~ is the string field expanded in terms of string states described earlier The state 10) on the right-hand side is the vacuum of this Fock space, and Kr l act as operators on this vacuum Xr~ are functions of o, and they are the c o m m u t i n g coordinates mentioned earlier In effect, we have traded the n o n - c o m m u t i n g operators Kr, l(O), which could not be used as coordinates, for the c o m m u t ing variables Xr, l(o ), which can therefore serve as the new coordinates As a result, the old string field (~ is now replaced by the nev~ field ,~ Also, the original

K Bardakcl / Strlngfieldtheor~

337

coordinate Y In the end gets replaced by two sets of coordinates X r and X). This means that we can now study lnvarmnce under separate reparametrlzatlons of the right and left coordinates Invarlance under reparametrlzatlons of the original coordinate Y corresponds to invariance under the s~multaneous reparametrlzatmns of X r and X 1, and IS a much less powerful requirement Next we study the action of the mflnltesimal generators of the right and left reparametrIzatlons (conformal transformations) on the state defined by eq (2 3) The generators are the well-known Vlrasoro operators (2 4) which, when apphed to ~, yield the following equation

o)a;= f DXrf DX, [Lr,,(o),( Xr, Jl) ] ×exp{fdo'[Kr(o')Xr(a')+Kl(a')Xl(O')])[O

),

(25a)

where, Lr,,(o ) =

Xr'l(O ) (~Xr, l(O ) .

(2 5b)

In this equation the ( + ) sign attached t o / . lmphes that the equation 1s vahd only for that positive Fourier modes of L ( o ) . winch anminlate the vacuum state on the right-hand side of (2 5a) This restriction is not a serious problem, since ultimately we will have to restrict the general conformal transformations to tins special class m any case At this point, we would hke to make two comments All we have done so far is make a change of coordinates The conformal generators, gtven by eq (2 5b), satisfy the same standard algebra as before Since the whole dynamics is ultimately expressible m terms of the conformal (Vxrasoro) generators alone, one can, if one wishes, skip the detailed construction given above Also, one should keep m mind that the restriction to L.'s with posmve n means that we are left with only half of the algebra from the beginning From now one, we will forget about ~ and t h e / . ' s , and work exclusively with q~ and the L's given by (2 5b) An additional advantage of the transforming from quantities with tilde to quantities without tilde is the following The L's have quadratic derivatives m their definition and are nasty objects to deal with, for example Leibmz's rule breaks down and they cannot readily be identified with generators of dlffeomorphlsms (12) In contrast, the L's generate dlffeomorphlsms, they satisfy Lelbnlz's rule. and readily fit in a differential geometric framework Our next tast is to construct objects that transform covanantly under the transformations generated by (2 5b) The tools needed to solve tins problem were

338

K Bwda&z / Strmg[wldtheor)

developed in ref [11], and we will briefly review parts of that paper for the sake of a complete and self-contained treatment Also, for slmphcity, we will restrict ourselves to the open string In tins case, the left and right movers can be identified

xr(o ) = x (o) =- x ( o ),

(26)

and so we can avoid the comphcation of dealing with two sets of coordinates We hasten to add, however, that there does not appear to be any fundamental obstacle to extending the present treatment to the closed string We will start with a few prehmmary definitions The parameter ~ wdl vary from 0 to 1 Occasionally, we will instead use the complex variable z = exp[27r,o],

(2 7)

N o w let us consider the reparametrlzatlons

Y(o') = x(o), where of the global useful

(2 s)

o ' = g(o, X) We note that the reparametrlzatlon function g is a functional transformed coordinate X, as well as being a function of the parameter o A reparametrlzatlon group is thus promoted into a gauge transtormatlon It is also to consider the infinitesimal reparametrizatIons o ' = o + f ( o , X),

(2 9)

where f is an infinitesimal function periodic in o wlth unit period It can therefore be expanded into the standard Fourier series We will also need the tuo special subgroups generated by the positive and negative frequency parts of ]

f(* )(o) = Y'~ J,f ± )exp[2~r,no]

(2

10)

tt-- 1

Under the group defined by eq (2 8), scalar fields transform as

UO(X)U t = 0 ( X )

(2 11)

Next, vectors and tensors have to be defined The reparametrlzatIon group naturally admits two kinds of vectors The first kind, which will be called conformal vectors, carry o as the index, and transform in the following fashion

u0'"'(o, x ) u

1=

Y)] "o'"'(o', x ) ,

(2 12)

where, again ~' = g(o, X) Two conformal vectors 0~ and 02 can be contracted Into

K Bardal, cl / Strtngfield theom

339

a scalar S in a group m v a r l a n t w a y if and only if their weights a d d up to one

s=

f

do~(ll-")(o,

v,.,~,(o, x)

(2 13)

l. lv 2

T h e s e c o n d k i n d are the usual c o n t r a v a r l a n t a n d c o v a r l a n t vectors of n e m a n n l a n g e o m e t r y T h e y carry a Lorentz index ~ in a d d i t i o n to o, a n d they t r a n s f o r m In the standard fashion

vv.o(x)u

¢~X ~°

-~ = f do' - -

8~.'o'

v"'°'(~),

8Xp,' o'

uG(x)u-l= f do' 8X,~W,.o.(~) H e r e X ~'° =- X~'(o) - X . ( o ) the t r a n s f o r m a t i o n matrix

8x,~

_

(2 14)

F o r future use, we also write d o w n the exphclt form of

G.8( 0 ' - g(o. ; ) ) +

;;(~(o. ;))8g(o. ;)

2.,o,

(2 15)

8X~,' ."

T h e s e d e f i n i t i o n s easily generahze to multi-index tensors W e also need to d e f m e the c o v a r l a n t derivative of a c o n f o r m a l vector of weight n b y

D.,,O (n) (~', X ) -

8~,(")(.~. x ) 8 X ~°

co~,o , O ~ ( " ) ( T, X ) - n O~( ~o.(,,,) ep(")( T, X ) ,

(2 16) where % . , , ( X ) is a (spin) c o n n e c t i o n field with suitable t r a n s f o r m a t i o n p r o p e r t i e s T h e c o v a r i a n t derivative is defined to t r a n s f o r m like a covarlant vector with respect to the indices /~ a n d o, a n d like a c o n f o r m a l vector of weight n with respect to ~F i n a l l y , we i n t r o d u c e c o n t r a v a r i a n t a n d covariant vlelbelns e ~" ~ ' ( X ) and e.~ ~,( X ), w h i c h t r a n s f o r m as follows In the indices /~ a n d o, they t r a n s f o r m like cont r a v a r l a n t a n d covarlant vectors, as their n a m e s indicate In the index ~-. the c o v a r l a n t vlelbeln transforms as a c o n f o r m a l vector of weight n a n d the cont r a v a r l a n t vlelbeln as a c o n f o r m a l vector of weight 1 - n The index a, x~hlch IS a L o r e n t z i n d e x j u s t like/x, is a s p e c t a t o r u n d e r these t r a n s f o r m a t i o n s and does n o t t r a n s f o r m at all The two vielbexns are not i n d e p e n d e n t a n d satisfy the usual inverse relation f d're ,uo' a'r% ° , ~ , = 8 , ~ , 8 ( o -

o')

(2 17)

340

K Bardah~t / StrmgfieMtheot~

T h e c o n f o r m a l weight n is so far a r b i t r a r y However. the sum of the weights ol the two vlelbelns must a d d up to one m order for eq (2 17) to be r e p a r a m e t r l z a t l o n lnvarlant A n interesting question is whether the t r a n s f o r m a t i o n s on /,o and the ones on r c o u l d be d e c o u p l e d and thereby a larger lnvariance group could be i n t r o d u c e d A t this stage of the game, such an enlargement is still possible However, later on. we shall need the fact that X~'(r) is a conformal vector of weight one in r, which is true o n l y if X - - * X as r ~ g ( r ) Therefore, it a p p e a r s that the t ~ o t r a n s f o r m a t i o n s c a n n o t be d e c o u p l e d after all

3. The torsion and the lagrangian densil3 T h e s t a n d a r d expression of differential geometry for torsion T = d e - we

(3 1)

a p p l i e s w i t h o u t m o d i f i c a t i o n also m our case Writing it out in lull, we have

T . . ~,~,, ,~,,.,,

8 X ~"

wuo

,,OT,,(e~,, ., , , . , )

- n 3 . , , ( ~ % o .,)e.,.,,,,,,T,,

(ttoe*/~'cr')

(32a)

T h e torsion is an a n t l s y m m e t r l c covarlant tensor in the indices /.to a n d /.t'o', a n d it is a c o n f o r m a l vector of weight n in the index r " It IS con',enlent to convert all the indices into c o n f o r m a l indices by defining

T2..,..

=

ff d o d o '

e ~ ~'e ~'0' "'
(32b)

N o w that we have a tensor with simple t r a n s f o r m a t i o n properties (eq (2 12)), we c a n go b a c k to the d y n a m i c a l p r o b l e m that m o t i v a t e d the m a t h e m a t i c a l d e v e l o p m e n t , n a m e l y , to construct a lagranglan density that transforms like a scalar u n d e r the r e p a r a m e t r l z a t i o n group gwen by eq (2 8) There are some further restrictions o n the l a g r a n g i a n density for it to r e p r o d u c e the k n o w n free string d y n a m i c s The m o s t i m p o r t a n t of these is the requirement that the equations of m o t i o n should be e x p r e s s i b l e in terms of the c o n f o r m a l o p e r a t o r s L ( o ) of eq (2 5b) F u r t h e r m o r e , the L ' s a p p e a r linearly in these equations, so that the lagranglan density should be h n e a r in these o p e r a t o r s The torsion as a good starting p o i n t for the lagranglan, since it has nice t r a n s f o r m a t i o n properties and it is h n e a r in the derivatives with respect to X All we have to do is to convert those derivatives into the L o p e r a t o r s b y m u l t i p l y i n g eq (3 2b) by the tangent vectors X'~ a n d X2,,,, and contract over the L o r e n t z indices c~ and a ' W e r e m i n d the reader that X . ' is a c o n f o r m a l vector

K Bardal~t /

341

StrlngJwldtheorv

of weight one, and so this operauon preserves the tensor character of the torsion Accordingly, we define a contracted torsion tensor by "'"-X'

T ~ ' I TIt - -

ar

(3 3)

X'o d ' r l -Ta.t ".¢. .'' '*'

Before proceeding with the computation of the contracted torsmn, let us point out an important simplification It was noticed in reference [11] that the vlelbems can be simplified considerably by imposing certain condmons on them, which are conSlstent with their transformation properties Here, we need only a weaker version of those constraints

X reg~,,,,,~r __

X '~oe~o, aT -- X 'areo ~,

X ~ . e o, r ,

-

(3 4)

-

where the two reduced vmlbems e °," and e o ~ satisfy an inverse relationship

f dre

(35)

"e o, , = 8 ( a - o ' )

One can easily show that the constraints imposed by (3 4) are consistent with the transformation laws (2 14), by malong use of the exphclt form of the transformation matrix given by eq (2 15) The key point is the appearance of the tangent vector m the second term on the nght-hand side of (2 15) After these prehmlnarles, eq (3 3) can be worked out explicitly as follows Start with eq (3 2a), then perform the contractions indicated by eqs (3 2b) and (3 3), and finally substitute the constraints given by eq. (3 4) in the resulting equation This yields the following expression T , "rt

It

l " , "r ~

T~,,,<,=X~,,,,,,F.~,,

t

"r .gt

(3 6a)

+X~,,~,,T;,,,

where F'<-8(r'-'r 1- r~

--

''

e ¢'''

-

doe

%,¢.

-(r,-+r')

(3 6b)

n

and

T~,;~'= 8( r ' - r") 0<(e'"") - f do { 8 ' ( r ' - r") e°' ~% ,,,- n S ( r ' - r")

xeo,'a,,,(,,,o + fdofdo'{e

° "e°"~"L,,(eo,,,,,)}-(r,~,r

a,

')

"r ~p

}

(36c)

342

Ix B a r d a A ~ t

/ gttm~jteldtheot~

In this expression,

8 L =_ L ( o ) = •¥ tL,, ,

8X~,

,

Eq (3 6) is the starting point for the construction of the lagranglan densit) As expected, it is linear in L ' s and contains no other dynamical operators Also, we note that F and T are separately tensors, since they appear in the expression for torsion multiplied by two linearly independent vectors X~',¢, and ¥,~,,~, We therefore finally end up with two conformal tensors F and T, which are Lorentz scalars Our dynamical variables are also Lorentz scalars, namely, e ° ~ and ¢0 ~ In lew of this, it is clear that our starting point was redundant For example, instead of defining transformation laws with vectors with Lorentz radices as in eq (2 14), we could have started with the transformation laws of contracted vectors directly For example, the transformation law for the contracted vlelbem can easdv be deduced from its definition and from eqs (2 14) and (2 15)

ue" ~(x)u 1=~.(¢. Y)[.,'(o. 2)] 1 ×le°'"(x)+fdo"L...(g(o. V))e"'"(X)}.

(38a)

where o'=g(o,

X),

~-' = g(~-, 2 )

(3 8b)

Also, we could have replaced eq (2 16) by the contracted equation obtained by multiplying both sides by 3~p,o ' We would then encounter some complications due to the fact that the Lo's, unlike ordinary derivatives, do not commute In ref [2], mathematical techniques ,~ ere developed for overcoming precisely such problems in the context of free strings Nevertheless, we found ~t easier to start with a redundant formulation and latter project the Lorentz indices along the tangential direction lor tv, o r e a s o n ' ~

(a) One can make use of the standard formulas in differential geometry, in particular, eq (3 1) at the beginning The projection along tangential directions (eq (3 3)) involves only trivial algebra In thts fashion, the problem of non-commutatlvity IS entirely bypassed The only non-trivial step is the consistenc;y of the constraints (3 4) with reparametrlzation invarlance (b) The interaction terms were automatically taken into account without an) special effort Examining the two tensors F and T in eq (3 6), it is clear that F carries no dynamics, since no L ' s appear in its definition To simplify matters, '~e

K Bardaku / Strmgfieldtheorv

343

set

F;."= 0,

(3 9a)

winch imphes that ~0..=8(0-

.)

(3 9b)

In tins fashion, we are left with only the e ' s as dynamical variables Also, from eq (3 9b), ~t is easy to show that the curvature tensor vanishes Tins can be established by c o m p u t i n g the c o m m u t a t o r of two covarlant derivatives As a result, torsion is the only tensor available for the construcnon of the lagrangIan W e now face the task of constructing the lagrangian density from the torsion tensor It is clear that we must somehow contract the indices "r, T' and ~'" and end up with a conformal scalar After taking into account the ant~symmetry in ~" and "r'. the only posslblhty turns out to be the following Set ~' = r " (or ~- = -c"), and then integrate over both ~" and r ' Reparametrlzatlon lnvarlance demands that the c o n f o r m a l weight on each integrated variable be one The conformal weight of T is easily calculated to be 2 - n in the upper indices and to be n - 1 in the lower one It then follows that we have to set n = 1 For convemence, we d~splay the torsion tensor with n = 1 and ,0 gwen by eq (3 9b)

T£,, -'

<'

f

da[O

(e °,)e°*"

+ffdodo'[e°"e°'"Lo(eo,.,,)l-('~T

eo

,,,1 ')

(310)

T h e scalar torsion is then given by

r-- f f

(311)

W h a t we need, however, is the scalar density, since ultimately T has to be functionally integrated over X As in general relativity, we then introduce the determinant D = det(e o ~).

(3 12)

and take the lagranglan density to be ~9¢= const

DT

(3 13)

There 1s, however, a serious problem with tins choice If, for example, we c o m p u t e the free p a r t of tins lagrangian along the lines described in the next section, all the terms containing the L operators turn out to be total derivatives, and integrate to

344

K Bardal~t / Stlmgpeld theom

zero in the action The root cause of the p r o b l e m can be traced back to the s y m m e t r y between r ' and ~-" m the d e f m m o n of T T h e l e ~s no way to overcome this p r o b l e m without giving up the full r e p a r a m e t r I z a t i o n mvarmnce The D~rac d e l t a f u n c t m n 6 ( ~ - ' - r " ) is the only c o n f o r m a l tensor available for contractions, a n d it is necessarily s y m m e m c m its arguments W h a t we need is an a n n s v m m e t r i c f u n c h o n in r ' and ~-" to save the action from vanishing, and there is no way to i n t r o d u c e such a function without s y m m e t r y b r e a k i n g All ~s not lost, however, if the s u b g r o u p s g e n e r a t e d separately by the positive and negative frequency m o d e s r e m a i n u n b r o k e n (see eq (2 10)) In the next sectmn, it will b e c o m e clear that, at least for the free string, this restricted i n v a n a n c e is all that is needed to e h m l n a t e ghosts It m a y also be helpful to notice that, m terms of the variable c of eq (2 7), the r e p a r a m e t n z a t l o n s are now being restricted to positive power series m : and _ i separately, rather than a Laurent series that includes b o t h posttlve a n d negative p o w e r s s i m u l t a n e o u s l y This should not be surprising, however, since m any c o v a r i a n t t r e a t m e n t of the string, such a restriction seems me,,ltable Having a c c e p t e d the r e d u c t i o n of the s y m m e t r y group, we replace eq (3 11) by

T= f f f drdr'dr"a< ) ( r ' , r " ) T ~ L

<,

(3 14a)

w h e r e the a n t l s y m m e t r i c delta function used to contract over the indices r ' and r " is given b y

[exp(2rrm(r'-r"))-exp(2~rm(r"-r'))]

8' ' ( r ' , r " ) = t ~

(3 14b)

11-- 1

This ~s the u m q u e function which Is both a n t l s y m m e t r i c in its arguments a n d is m v a r l a n t u n d e r the subgroups of r e p a r a m e t r i z a t i o n s discussed above The lagrangian density, with a convenient normalization, is defined as before ~ ' = ~,DT,

(3 15)

where D is gwen in eq (3 12) and T bv (3 14a) Because of the a n t l s y m m e t r y of 3 ( ) ( r ' , r " ) in the variables ~" and r " , the disaster mentxoned earlier is avoided, a n d the crucial terms in the action do not vanish

4. The free lagrangian densi~ A l t h o u g h we have a lagranglan density, we c a n n o t m a k e any sense of it yet, unless we can e x p a n d the vlelbeln a r o u n d some convenient function As m general relativity, such an expansion is necessary in order to solve eq (3 5) and to c o m p u t e the d e t e r m i n a n t (3 12) explicitly The most natural e x p a n s i o n is a r o u n d a delta

K Bardal,ct / Strmgfield theory

345

function. Setting e ° "=6(o-

(4 la)

r) +E °"

we have eo, , = 8 ( o

- T) - E*'°+

det( eo ,) = 1 -

fd~U,'+

fd~'E~,"E'"°+

•

(4 l b )

Such an expansion lmphes the spontaneous breakdown of the reparametrlzatlon lnvarlance One expects, however, that, as in general relatlwty or gauge theories, the global part of the lnvanance will remain intact This is not true in the present case, and even the global part of the group is completely (but, of course, only spontaneously) broken To see this, let us restrict the transformations given by (3 8a) to functions g independent of X, winch form the global group Ueo,.u-l=g,(.r)[g,(o)]

leg(O,, g(~)"

(4 2)

The delta function does not transform properly since g'(,)(g'(o))Is(g(o)

-

(4.3)

and therefore the global part of the group is also broken spontaneously This is good news, since an unbroken global symmetry would show up in the physical S matrix as degenerate multlplets and conserved quantum numbers, and we know that the bosonlc string has no such extra symmetries beyond the usual space-time symmetries It is of some interest to find the transformation properties of the field E under h n e a r gauge transformations generated by infinitesimal functions given by eq (2 9) They are given by E °''--* E . . . .

6 ( o - r ) f ' ( a , X ) + L , [ f ( o , X)]

(4 4)

The spontaneous breaking of the global symmetry is also clear from this transformation law We now proceed to expand the lagranglan density to quadratic order Along the way, we verify an important and highly non-trivial consistency check There are no linear terms m E m the action The lagranglan density does have some linear terms which, however, turn out to be total derivatives and vanish by parnal integration In displaying the quadratic terms, it ~s best to use the Fourier modes, which also make comparison with existing literature easier We define the Fourier modes by the

346

K Bardal,u / Strmg /teld theom

expansmn E o T= ~

~ E .....e x p [ 2 ~ r z ( n o + m r ) ] ,

(45a)

L~= ~ L.exp(2~rmo)

(4

5b)

O(2 The quadratic part of the lagrangxan density (315), after a straightforward calculation, is given by

,,~q~l ~ n =

~_~ { E .... ( L o - 1 ) E . . . . i

m

=

+E'°L

mE

0

.....

E "' LInE

n

"

i

-E

m, " ( L o - 1 ) E

.... + E

..... L,,E"'°-

E ..... L

,,E ' " °

+27rt(n + m ) E . . . . . E . . . . . . 2~';(2n + m) × ( E °+°' °E . . . .

+ E ..... "E ..... t} + E 4~,,E" "E ""

~t: 1 (46) where all the L's act to the right In writing down thx~ expression, we have made the replacement Lo-, Lo

1

(4 7)

where the new L o is a well-defined (normal ordered) operator It is knoun that a careful treatment with a cutoff justifies thl~ replacement [13] The lagranglan of (46) contains a large number of Stueckelberg fields, ab ,~ell as the physical string field [1,2] The physical string field '/" ~s eas) to identify, it is given by

n=

1

Under the hnear gauge transformations ol (4 4). it transform~ as ,,f,,( X ) ,

(49a)

= Y'. /,,exp(2~rmo)

(4 9b)

't'--, 't" + ~

L

t?= 1 where f(o)

K Bardal~et / Strmgfield theor)

347

Imposing the physical gauge L f l ' -- 0

for n > 1

(4 10)

decouples ,/, from all the other (Stueckelberg) fields We note that expressions containing factors of the form E "' ", whose indices sum to zero, appear only in the first three terms m eq (4 6) The rest of the expression does not depend on field components of the form E ", ~ at all It is therefore legitimate to define '/" as an mdependent field through eq. (4 8), since the field components that appear in its definition do not appear in the rest of the lagranglan Another curious feature of (4 6) is the absence of terms of the form E ° ' ' This is presumably a reflection of the gauge invanance, since these fields can be gauged away through the transformauon (4 4) It IS of some interest to compare eq (4 6) with some of the free string lagrang~ans that have appeared in literature Apart from some notational differences, (4 6) is essentially the lagrangian of Neveu et al [14] and is also equivalent to a truncated version of Banks and Peskm [2] In comparing, one should keep in mind the reality condition

(E" m ) * = E - "

"

(411)

which follows from the reality of the vielbeln We end this section with a word of caution about the vahdlty of naive manipulations we have been using throughout this paper Although it may appear that the lagrangaan (4 6) is invarmnt under the gauge transformations (4 4) m all space-time dimensions, a more careful treatment of operators [2,14] shows that there are, in fact, some anomalies which only cancel in the crmcal dimension 26

5. Conclusions and future directions In this paper, we have shown that coordinate dependent reparametrizatlon mvariance can be used to construct a lagranglan density, whose free part, at least, is satisfactory Given the starting point, the construction is essentially unique It is also encouraging that some minor miracles, like the disappearance of the potential hnear terms and the breaking of the global group, happen along the way Much remains yet to be done As explained in the Introduction, a complete gauge fixing is necessary to go from the lagranglan density to the action Also, the interaction terms have to be investigated to see whether the usual overlap integral emerges in some way or other We remind the reader that in our approach, the interacting lagrangian is completely fixed, and the right answer must automatically emerge for the program to succeed In investigating the full lagranglan, the most serious problem one encounters is the identification of the physical string field and the choice of the physical gauge

348

K Bardahtz / String held theot~

conditions to all orders This means finding the generahzations of eqs (4 8) and (4 10) to all orders so that the string field decouples from the Stueckelberg fields We hope to return to these problems m the future

I would hke to thank Herbert Neuberger for helpful discussions References [1] W Siegel Ph~s Lett 151B (1985)391,396 W Siegel and B Zwleba~h, Nucl Phys B263 (1986) 105 [2] T Banks and M E Peskm, Nucl Phys B264 (1986) 513, Y Banks, M E Pe~km, C R Prextschopf D Fnedan and E Martmcc Nucl Phw B274(1986)71 [3] M Kaku, Ph~s Lett 162B (1985) 97, Nucl Ph~s B267 (1985) 125 [4] A Nc~eu H Nlcolal and P West, Ph~s Lett 167B (1986)~(17 Y Kazama, A Neveu, H Nlcolm and P West Nud Phw 278 (1986) 833 H Arat~n and A H Zlmmerman, Phys Lett 166B (1986) 13/), 168B (1086) 75 [5] N Ohta Phs~ Re~ Lett 56 (1986) 44(t, S P de Alwls and N Ohta, Texas preprlnts (1986) [6] P Ramond, Florida preprmt (1985), D Pfeffer, P R a m o n d a n d V G J Rodgers, Nucl Ph3s B276(1986)131 [7] E Wltten, Nud Ph~s B268 (1986)253, Nucl Phvs B266 (1986)245 [8] M Kato and K Ogawa, Nucl Phys B212 (1983) 443 [9] W Siegel, Phvs Lett 149B (1984) 157, 162 [1(1] J L~kken and S Rabv, Los Alamos preprmt 11986) [ll] K Bardakcl Nucl Phys B271 (1986)561 [12] K Bardakcl, Berkele'v prepnnt (1986) [13] L Brink and H B Nielsen Phvs Lett 45B (1973) 332 [14] A Ne~eu J H Sch,~arz and PC West, Phvs Lett 164B (1985) 51