Reparametrisation -invariant string field theory

Volume 183, number 1 PHYSICS LETTERS B 1 January 1987

REPARAMETRISATION -INVARIANT STRING FIELD THEORY

C.R.R. SMITH and J,G. TAYLOR Department of Mathematics, King's College, London WC2R 2LS, UK and lnternational Centre for Theoretical Physics, 1-34100 Trieste, Italy

Received 15 August 1986

A differential-geometric approach to the closed Bose-string field theory is developed in a framework where the Lorentz and reparametrisation invariances are manifest. The Virasoro anomaly in the quantum theory is cancelled by the minimal introduction of the conformal mode, extending the usual string funetionals defined on the space of embeddings. An action is written down, and this approach to strings is compared with the second-quantised formalism for the single particle.

In order to have a better understanding of the differential geometry underlying string field theory it is possible to follow either the use of BRST structure developed in the string field context by Siegel [ 1 ] or altematively attempt to construct a differential geometry more closely related to the original fields without BRST structure [2]. The former approach has achieved elegance and simplicity at the non- interacting level with the action being solely f ~ Q ~ [ 3-5 ], where Q is the BRST operator and q~ is the string field. Extension of this to the interacting case has also been performed in an elegant manner for the open string [3 ] (with an alternative product, closer to that of the light cone, presented in ref. [6]). Extension of this approach using ideas from non commutative geometry has, however, n o t been as succesful for closed strings, nor have alternative approaches [7,8] resolved the question as to the nature of casual interactions in a covariant approach. In this paper it is proposed to investigate covariant string field theory from the alternative non-BRST aspect. It will be seen that in fact these two approaches may be developed from the same underlying princi- ples. In the process the nature of the anomaly cancel- lation and the need for the conformal mode as well as a careful treatment of the function integral will be clarified. As a prior point of departure similar prob- lems in the field theoretic description of the single particle will be considered,

In order to consider a correct measure on

0370-2693/87/$ 03.50 © Elsevier Science Publishers (North-Holland Physics Publishing Division)

space-time it is necessary to take a foliation. This will be described by a suitably smooth function ~u(X) on the points X of the space-time manifold M, with the surfaces Z~= {X: ~ ( X ) = z ) being space-like in the metric on M. As z varies it is assumed that U~27~ = M. To simplify M will be assumed fiat; a discussion when M is curved will be considered in more detail elsewhere. The standard first-quantised hamiltonian is ½e(PZ+m2), where e is the einbein allowing parametrisation invariance. The appropriate action will then be an integral of the density b e ( - O2/OX 2

- m 2 ) ~ b = 0 , where ~=~b(X, e) is the (complex) particle field. The appropriate measure will be obtained by integrating X over 273 and then integrating overall 27~ [ 2 ]. Thus the action A is

= fdz dX d e ' e ( - [] - m 2) q~

x,~(~- ~(~). (1)

The above action has various features which require investigation:

(a) What is the dependence of A~, on g? (b) What is the role of the variable e? (c) What is the manner in which reparametrisa-

tion invariance is manifest in A~,? (d) How may the BRST invariant action of Siegel

[ l ] be obtained from (1) ? It will turn out that the answers to these questions

are very closely related, and are also very relevant to the analogous action which will be discussed shortly

B.V. 47.


in the string case. The form of (1) is simple enough to allow direct

integration over z, and, under the assumption that ~, is a foliation leads to the field theoretic result

A~,=~ f dX de2~KxdP=A . (1 ')

The form of (1') displays no q/ dependence, so answering (a) above. (b) is also answered by noting that the time-translation generator is independent of the variable e 2. The action A appears as a direct sum of contributions, one from each e 2, and the corresponding connected vauum functional Z¢o,~ [J] can also be expressed as a direct integral fde 2 × Z~ohn [ J (e 2) ]. Thus the theory represents a continuous infinity of identical theories, each labelled by the value of e 2. Moreover on-shell the equation of motion is the usual Klein-Gordon one, and is e- independent. The role of e is therefore that corresponding to a continuous parameter in a theory with, say, rotation invariance. That analogy can be made more precise on recognising the U(1) symmetry • ~ e x p [ i a ( e ) ] ×~b of A; the e-independent solu- tion of K ~ = 0 automatically breaks this symmetry. Thus questions (b) and (c) have been answered.

There is a deeper understanding of the role of e when question (d) is considered. For besides the U (1) symmetry and that under global translation in e 2 it is possible to modify e so as to be fermionic. Then the time-translation generator eKx is the BRST charge, and the BRST action ABRST of Siegel results [ 1 ]. This action has an explicit gauge invariance d~=eA(X) , since e2=0. This gauge invariance allows explicit verification of the removal of gauge modes in ABRST, and hence that the action has the correct spectrum. The alternative discussion of A given earlier indicates that e is an auxiliary mode, disappearing on-shell. This seems perfectly satisfac- tory, and will now be extended to the case of the dosed bosonic string. Before doing so it is clear that self-interaction terms

IdzdX [ e) q)(X, e)] 5 ( z - ~(X)) de z 4~(x, 2

= f d X d e 2 I~(X, e) l 4

can also be directly written down so as to possess the same invariances as for the kinetic term (1). A simi-

lar extension to strings will be described shortly. The closed string coordinates will be denoted X~'(a) ( # =0 ..... d - l e e S 1 =- [0,2n] with the mode expansion X~'(a)=.S, rXU~ exp(ira) ( reZ) where X~_r= X~UandX~ (a+2~)- - -X a (a). Formally, one may consider the mappings Xu: S 1~ •d as embeddings into Minkowski space-time (~d, t/), i.e. X~e C°°(S1;~d).

A string field is taken to be a real-valued functional ~[XU(a)] on the space of embeddings, i.e. • eC°°(C°°(Si; R) ;•). The "Schrfdinger" represen- tation is defined as usual by the identification of the momentum operator Pa(a) =id/~X(a) on the space of fields. This operator is cononically conjugate to the coordinates in the sense that

[XU(a), P~(#)I = - i d e ( a - a ) .

Expanding Pu(a) into modes,

i 0 Pi,(a) =~--~ ~ - - ~ e x p ( - i r a ) ,

allows construction of the usual creation and anni- hilation operators:

a~ u -- v/~ (PU_r +i rX~) ,

(~u __ x//~(pu +irXU ) ,

with

P~ ~ (i/2zOO/OX~) ~1 u~ .

Writingf(a) =~, the invariance under Diff(S ') (the reparametrisation group) requires

X~'(d) =XU(a), f ( a + 2 r 0 = f ( a ) +2re.

The infinitesimal transformations (corresponding to the Lie algebra diff(S~)) are defined by #=a+ eh (a) + O) ( e 2), whilst the embedding vector transforms as d XU( a) = - e h ( a)X'U( a)Xu( a) -dX~'( a)/ da. A string field qb[XU(a)], taken to be invariant under reparametrisations, has the transformation defined by

de ~ j da-~-x--Z-(~6 X ( a)

= - e r d a h(a)R(a)qb, (2)

with R( a) =-X'~'( a)d/dXF'( a) ~reparametrisation generator. The operator

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R[h(a ) ] - Jda h(a)R(a) =Rh

is well-defined and has mode expansion R h = - F~rhvRr, with Rr - L r - L _ r corresponding to the generator of gauge transformations; {h~)~ Four- ier modes o fh ( a ) . [The notation is that ofref. [9]]. The algebra of the {Rr} is simply [R, Rs] = ( r - s ) Rr+~, which is the classical Virasoro algebra (i.e. no quantum anomaly).

It is convenient to introduce the notion of an inte- ger-valued weight w under the diffeomorphism group Diff(S~). An operator O(a) has weight w under Diff(S ~) (or equivalently, diff(S 1 )) if it transforms as

80(a) = -E[h(a) O'(a) + w h'(a) O(a)] .

The space {XU(a)} in [d] is augmented by a real vielbein field e(a) , e eC°°(S~; R), so that effectively the base manifold on which string fields are defined is [ d + l ] , and topologically Ra×R~R a+~. String fields are written as • = @[XU(a), e(a)] and satisfy the usual smoothness criteria. Since e(a) is to act as vielbein, we require under a reparametrization ~( ~) do 2 = e( a)da 2, which is infinitesimally

de(g) = -E[ h( a)e' ( a) + 2h' ( a)e( a) ] ,

corresponding to a pure w= 2 transformation. The extended string field. ~[XF'(a), e(a)] now

transforms under diff(S ~) as:

8~ = -e fda{h(a)R(a)

+ [h(a)e'(a) +2h'(a)]6/re(a)}~

= - e r d a h(a)R*(a)~=- -eR*h~. (3)

Analogous to the Virasoro operator A(a) -X~(a) 2 + Pu( a) 2, we define

M(a):=q)'(a)2 +P~(a) 2 ,

xP, (o ' ) ---iS/6O(a), (4)

with ~(a) :=In e(a) . The introduction of the ~-field may be justified by an adaptation of an argument due to Polyakow [ 10 ] R* (a) has mode expansion

R~r : Rr AI- Mr -- M - r - 2ix/~ r(flr --f ir) } ,

where { (fir, ~r) } are the usual creation and annihila-

tion operators corresponding to O(a). It is straight- forward to veryify that, given the algebra of {Rr} and {Mr, )~r} the {R*} still satisfy the classical Virasoro algebra ~' * Rr+s. as a [Rr, Rs] = ( r - s ) * This appears consistency condition, since it is the satisfaction of this algebra which guarantees the nature of {R*~} as reparametrisation generators.

One need not appeal to the mode expansion to ensure the closure of the group algebra. Indeed, in terms of the generators R , one has [R~, R~] = -

• wi Rgh'--g'h th an identical result for the usual generators R h [ 2 ]. Now the transformation of M(a) under the reparametrisation group is given by

[ R'~, M( a) ] = h ( a)M'( a) + 2h ' ( a )M(a)

- (1/6z0h"(a) + 4 h " ( a ) ~ ' ( a ) . (5)

The calculation is similar to that forA(a) in the L.C. formalisation; writing

A*(a) =-A(a) - x [ M ( a ) - 4 ~ " ( a ) ] ,

then under diff(S ' ) th is operator transforms as

[R~, A*(o ' ) ] = [Rh, A ( a ) ] --x[R*h, M ~ ( a ) ]

= h(a) [A (a) -xM1 (a ) ] '

+2h ' ( a ) [A(a ) -xMl(a)]

- ( 1 /6 n ) [d -x (4 8 n + 1 ) ] h " ( a ) , (6)

with M~ (a) ---- M (a) - 4~"(a). Therefore there exists a choice of parameter x = d / ( 4 8 n + 1 ) for which the modified Virasoro generator A* (a) has pure weight 2 under the reparametrisation group.

Finally, one can check explicitly the closure of the algebra of A]---fda h(a) A(*a), in analogy with the earlier calculation for R~. The non-zero contributions are:

A* A *~ [Ag, A h ] + K 2 [ M g , Mh] g~ hi ---~

+4xZ{[Mg,f~Z]- [Mh, ¢g]}, (7)

where the first term on the RHS closes onto the usual reparametrisation generator in terms of XU(a), viz. [Ag, Ah] =4Rgh,-g,h [2]. The final result is the closure of the algebra onto a linear combination of the X u and ~ reparametrisation generator.

In order to define a suitable action which has the correct spectrum for the closed string it is necessary to introduce time-like or null foliations x besides the

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foliation ~u. This has to be chosen to be everywhere orthogonal to 0, and such as to eliminate the longi- tudinal as well as the time-component modes of oscillation [ 11 ]. Thus it is necessary to fLx exp [i0(a)] ×X'(a) along some direction, and thus include the further constraint I-I~ 8 (x(X'(a))) . Such an increase of the foliation is to be expected in order to control the dynamics of the string, in comparison to the particle. The associated measure will be denoted by ~m~,, x(X) on the manifold 2:~ with ~,(x) =T.

Following the approach to the single-particle action (1), a comparable action can be written down as

a~,, x= f dT fz ~m~,, x( X) ~O f da

exp[ - ½0(a)]A*(a)¢. (8)

This action can be shown straightforwardly to reduce to the usual light-cone closed string action of Mar- shall and Ramond [2] with ~t(X) = X +, x(X') =X 1-, but with the addition of what looks like a kinetic term for 0 in addition to that for the d - 2 transverse modes X, The presence of the 0" term in A* allows integration by parts of the term in the field equation involv- ing 0, so as to reduce it to

fda e x p [ - ½0(a)] [~2/602(a)+o'z(a)]¢. (9)

The opposite sign of the second term in (9) to that usually present replaces the 0 contribution as a sum of harmonic oscillators ( V ( x ) = k x 2) by a sum of "anti-harmonic" oscillators (V(x)=-kxZ). The latter has non-bound states then, as in the particle case; ~-dependence seems to disappear on-shell. Thus again 0 is an auxiliary field, playing a more impor- tant role than for the particle in that it is essential to cure the conformal anomaly and so preserve reparametrisation invariance.

A similar relation to the BRST approach as that for the single particle may also be developed. In par- ticular the ghost number o f various objects follows from the global invariance of (8) under 0 ~ 0 + 2 , ¢ ~ exp ( - ¼2) @. For the ghost number of ¢ set equal to -½, then Q = e x p ( - ½0) A* has ghost number 1. Q is the usual BRST operator if 0 is regarded as the bosonised ghost, with exp( -½0) replaced by :exp( -½0): as the fermionic ghost. The usual gauge invariance of (8) then results.

It is interesting to note that interactions can be writtten down straightforwardly in any foliation ~t, and are given by fusion of strings lying in the same hypersurface 27~. Thus the 03 interaction is

A~,,x= f dz fza I-I ~m~,,x (Xi)~Oi i=1

× ¢(0,, x1) ¢(02, x2) ¢(03, x3)

/ ~(x1 +x2 - x3)a(01 + 0 2 - 0 3 )

+ (cyclic perm) +c .c . (10)

The O-function in X and 0 is defined in a reparametrisation invariant manner similar to that of Mar- shall and Ramond [ 2 ] by integrating over the two parameters of the self-intersection point of the string X3, but now with measure exp(½0) at each point. Higher order interactions may be also constructed following ref. [ 2 ], if so desired.

In conclusion this letter claims to show that an explicitly reparametrisation invariant action in terms of string fields on loop space can be written down with the minimal extension of the conformal mode. In this way we seem to be close to building a covariant field theory of the Polyakov string. Extension to the superstring is of interest, as well as analysis of the ~t-,x-dependence of A~,~in (10) (question (a) above). We hope to return to this elsewhere.

We would like to thank C. Hennaux and M. Nouri- Moghadem for useful discussions during this work. One of us (CRRS) would like to thank the SERC for fmancial support whilst this work was being completed.

Finally we should like to remark on some similar studies which we only received after this work was completed [ 12 ].

References [ 1 ] W. Siegel, Phys. Lett. B 151 (1985) 391,396. [2] C. Marshall and P. Ramond, Nucl. Phys. B 85 (175) 375;

IC Bardakci, Covariant gauge theory of strings, Berkeley preprint UCB-PTH-85/33; N. Bralic, String reparametrisation and the geometry of path space, Prog. Theor. Phys. (Suppl), to be published; L. Carson and Y. Hosotani, Line functionals and string field theory, University of Minneapolis preprint UMN-TH- 555/86.

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[3] E. Witten, Non commutative geometry and string field theory, Princeton preprint (October 1985).

[4] A. Neveu and P.C. West, Phys. Lett B 165 (1985) 63. [5 ] A. Restuccia and J.G. Taylor, Construction of a covariaut

theory of strings, submitted to EPS High energy physics Conf. (July 1985, Bari); and Cambridge Supersymmetry Workshop (July 1985).

[6] G.J. Chappell and J.G. Taylor, Phys. Lett. B 175 (1986) 159; K. Itoh, T. Kugo, H. Kunitomo and H. Ooguri, Kyoto Uni- versity preprint KUNS 800 HE(TH) 85/04 (1985).

[7] H. Hata, 1C Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Kyoto University preprint KUNS 829 HE(TH) 86/3, and RIFP-656.

[8] A. Neveu, J. Schwarz and P.C. West, Phys. Lett. B 164 (1985) 51; A. Neveu and P.C. West, Nucl. Phys. B 268 (1985) 125.

[9] C. Rebbi, Phys. Rep. 12 (1974) 1. [ 10] A.M. Polyakov, Phys. Lett. B 103 (1981) 211. [ 11 ] C. Teitelboim, Phys. Lett. B 126 (1983) 49. [ 12 ] G. Munster, Geometric string field theory, DESY preprint

DESY 86-045 (April 1986); S.R. Das and M.A. Rubin, Phys. Lett. B 181 (1986) 81; M. Awada, The gauge-invariant field theories of interacting bosonic strings (1), DAMTP preprint (February 1986).

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Reparametrisation -invariant string field theory