Non-linearly extended virasoro algebras: New prospects for building string theories

Nuclear Physics B326 (1989) 222-236 North-Holland Amsterdam

N O N - L I N E A R L Y E X T E N D E D V I R A S O R O ALGEBRAS: N E W P R O S P E C T S FOR B U I L D I N G STRING T H E O R I E S Adel BILAL and Jean-Loup GERVAIS

Laboratolre de Physique Theortque de I'Ecole Normale Superleure* 24, rue Lhomond Fo75231 Parts Cedex 05, France Received 16 January 1989

Converging facts are presented as evidence showing that there exist string theories of a novel type where the recently discovered non-hnearly extended Vlrasoro algebras appear as world-sheet symmetries These new theories are shown to exhibit rather novel properties, in particular, at the Planck mass, they may involve fundamental interactions mediated by massless particles with spins higher than two The generahzed ghost systems lead to stnng-fleld topological actions characterized by new cohomologles For A N = S U s + I , as an example, the structure Is similar to the cohomology of v-dimensional mamfolds, with v = 2 + N ( N + 1)/2

1 Introduction

One of the most distinguishing features of string theories [1] with respect to point parncle theories is the existence of an infinite-dimensional symmetry algebra for the world-sheet dynamics It characterizes the string theory to a large extent Vlrasoro algebra [2] for the bosomc string, super-Vlrasoro [3] for the superstrmg, left super-Vlrasoro and right Vlrasoro × K a c - M o o d y [4] for the heterotlc string [5], and SO o n

In the current developments of two-&menslonal critical models, new types of mfimte-&mensional algebras have been discovered [6-12] They are non-hnear extensions of the Vlrasoro algebra m the sense that, although the latter algebra is part of them, the commutation relations do not close m the usual way commutators may be reexpressed only m terms of symmetnc non-hnear polynomials of the generators These algebras are called non-linearly extended Vlrasoro algebras, higher spin algebras or simply W-algebras In a recent senes of papers [8, 9], we have shown that their properties can be systematically derived by quantlzmg the Toda field theories It ~s the purpose of the present arncle to lnvesngate the posslbdlty that new string theories (which we call W-strings) exist where these W-algebras appear as world-sheet symmetries The actual building of these theories may be quite difficult The basic * Laboratoire assocae h l'Ecole Normale Superleure et 5 l'Universlte Parls-Sud

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223

problem is the following Due to the non-hnearlty of the W-algebras, It is not obvious that tensor products of representations yield new representations Thus, for instance, despite the evidence discussed below, it is not clear that there exists a W-string theory which is as "simple" as the Venezlano model where the representation of the Virasoro algebra with central charge 26 is made from 26 free-field representations with C = 1 On the other hand, the representation theory of the W-algebras has already been developed to a large extent [6-10], using, in particular, our general approach [8, 9] The highest-weight irreducible representations are again characterized by the central charge C of the Virasoro subalgebra and by the elgenvalue of L 0 After more than two decades of developments, the understanding of stung theories has reached a point where one may already discuss the consistency of string theories directly from the existence of representations of this type without using their explicit construction Such is the spirit of the present paper where we shall also make some remarks on the related W-fractal-gravities that may be treated in parallel with W-strings There exists a W-algebra for any given simple Lie algebra The aim of the present paper is to g~ve evidence showing that the familiar picture of bosonic string theories (and fractal gravity) repeats itself for each of them In standard string theories, the consistency conditions uniquely determine the intercept e 0 from the mass-shell condition for physical states, and the crmcal central charge Ccr,t for the conformal reparametrlzatmns of the world-sheet Moreover, the properties of the related ghost fields may be used to unambiguously determine the ghost-grading of the string field which in turn leads to the C h e r n - S i m o n s - W l t t e n action [13] These well-known features are reviewed in sect 2 in a way that is suitable for the generalization to the W-strings which is c a m e d out in sect 3 We concentrate on W-algebras assocmted with simply laced Lie algebras, although we have also derived results for W-algebras associated with non-simply laced Lie algebras [9] The general consistency cond,tions recalled in sect 2, applied to the W-algebras, give C~m's and %'s that are different from the standard ones For instance, m the case of A N = SU n +i, one finds C o n t = N[1 + (3 + 2N)2], c o = ( C c n t - 2 N ) / 2 4 , obtaining, e g , C o n t = 100 and % = 4 for SU 3 The values of Qr~t were already derived before [7, 8] in the conformal field theory context, and the SU 3 values have already been put forward [12] from the construction of a BRST operator associated with the SU~ W-algebra Smce the W-algebras are non-linear, a simple-minded generalization of the Venezlano model might be too naive for W-strings, and we cannot predict the W-string phenomenology, for the time being However, since the intercepts may take arbitrarily high values as the rank increases, one is almost unavoidably led to massless particles with spins higher than 2) This opens the door to a much richer Planck-scale physics than is considered nowadays, with fundamental forces of new types, besides gravity and Yang-Mflls interactions In sect 3, we also discuss the new ghost-gradlngs and write down the corresponding string field theory actions We shall find consistent schemes corresponding to

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v-dimensional cohomologles analogous to the ones of ordinary dlfferentml operator m v variables, v depending upon the rank of the group For standard bosomc strings, v is equal to three [13,14], and the Wmtten acnon ~s the corresponding C h e r n - S i m o n s term, as is well known In the case of A u, as an example, we obtain v = 2 + N ( N + 1)/2 It ~s further shown that the natural ansatz for the stnng-fieldtheory acnon ~s the corresponding topological term, that is, e~ther a Chern-Slmons term tor N = 1 or 2 m o d 4 (when v is odd) or an anomaly term f F ~/2) for N = 0 or 3 rood4 (when v ~s even) The particular choice N = 1 of course gwes back the Wltten action [13] An outlook from the present work is finally given m sect 4 Before starting our d~scusslon let us mennon another prehmlnary observation about W-strings It ~s based on the relationship between Toda theories and W-algebras which we pointed out in ref [8] and fully exploited in ref [9] On the one band, Polyakov [15] has shown that the L~ouwlle theory naturally emerges when one integrates over random surfaces, due to the Weyl anomaly, and Gervms and Neveu [16,19,22] have developed a general approach to conformal models assocmted with the V~rasoro algebra which ~s based on the Llouvdle theory On the other hand, we have proven [8, 9] that the W-algebras are related to the Toda theories m the same way as the Llouwlle theory is related to the Vlrasoro algebra Thus one expects that these Toda theories wall emerge from some generahzed Weyl anomaly when one functionally integrates over the two-dimensional degrees of freedom of the W-string We leave this fascinating problem for the future

2. Basic features of the purely bosonic strings As a preparation for the discussion of W-strings, it is useful to first recapitulate basic features of string theories, related to the Virasoro algebra, on the example of the purely bosomc strings

2 1 PROPERTIES A THE CRITICALCENTRAL CHARGE AND INTERCEPT In the conformal gauge, the physical states are specified by the condmons (L 0-co)lphys )=0,

L.Jphys)=0,

n>0

(21)

These so-called ghost-kllhng conditions remove two degrees of freedom lff the Vlrasoro central charge C and the intercept c 0 are chosen as C

=

C c n t ~-

26,

co = 1

(2 2)

It ~s well known that the choice (2 2) is the only one leading to consistent bosonlc

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225

string theories The special properties of the stnngs at these values can be seen m several ways A 1 The appearance of spurious states (see e g ref [17]) One may &scuss them, in a model-Independent way, by only using the representation theory of the Vlrasoro algebra Denote by A the elgenvalues of L 0 At level R = 1 m the strmg Hilbert space there ~s a state IA = 1, R = 1) = L _ l l A = 0, R = 0)

(2 3)

which satisfies condition (2 1) with ~0 = 1 It is thus physical and of vanishing norm At the next level, R = 2, the crmcal values (2 2) are such that there is another simdar physical state

]A=I,R=2)=(L_ 2 + 3

(2 4)

2 1)la=-l,R=0)

This goes on at higher levels Thus condition (2 2) is such that states appear which are physical, that IS satisfy condition (2 1), and are generated by powers of the L ,,'s Because of this last property, they have a vanishing norm and decouple from the physical S-matrix, even though they are physical They were called spurious states in the early days of string theory If (2 2) holds, the decouphng from the S-matrix becomes precisely such that the number of degrees of freedom is reduced by two at every level In particular, for the Veneziano model, that is, for the trivial Minkowska background metric, this is the correct counting since the value c o = 1 leads to massless spin-one and spin-two particles Since this argument was first gwen long ago, it has become possible to discuss the existence of spurious states abstractly by means of Kac's formula [18] Consider, in general, the Hdbert space generated by the V~rasoro generators L n, n > 0 apphed to a highest-weight vector with weight A such that ( L 0 - A)IA , 0) = 0, LnIA, 0) = 0, n > 0 Kac's theorem concerns the vamshlng of the determinant of the corresponding metric (inner product matnx) Define

A(r,s,C)= ~(C-Co)-~[(r+s) with

C0 = 1,

C 1 = 25

C~-ZCo + ( r - s ) ~ ]

2 (2 5)

If A = A(r, s, C) for some positive r and s, Kac's determinant vanishes since null states occur at levels larger than or equal to r x s The above particular form of K a c ' s formula was gwen m ref [19] One may &scuss the spurious states of the string directly from Kac's formula Indeed a Kac's zero corresponds to a state, which is generated by the L( )'s and is anmhllated by the L(+)'s Consider now the corresponding states obtained from the Vlrasoro generators of the string theory Since the representation is reducible, these states do not vanish Those which are eigenstates of L 0 with elgenvalue one, are the spurious states we just discussed For

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instance, the values of A on the right hand side of eqs (2 3), (2 4) for the first two spurious states are given by A(1,1, 26) and A(1, 2, 26), respectively This procedure will straightforwardly extend to W-strings A 2 The Brink-Nielsen argument [20] For the Venezlano model, the intercept % can be interpreted as the regularized sum of the zero-point (Caslmir) energies of the string harmonic oscillators At D = D~ = C~r,t spacetlme dimensions, there are D~ - 2 degrees of freedom and the Casimir energy is ( D e - 2) l y ~ n = - - d 4 ( D c - - 2 ) = - - e o

(26)

1

in agreement with eq (2 2)

A 3 Modular tnvartance For the Venezlano model, the partition function of the closed string sector, with D - 2 physical free bosonlc degrees of freedom, is modular mvarlant lff eq (2 2) holds (see e g ref [1]) This modular lnvarlance of course works as well in the more complicated cases, for Instance, after compactlfication of certain directions, or when the Llouvllle mode is Included [29] A 4 Mtrror propertws of Kac's formula andfractal gravtty It is useful to remark that C~,.lt = Co + C1

(2 7)

Then one verifies the following remarkable "mirror" property

A(r,s,C)+A(r,-s, Qm-C)= I(cI-CO)=I=%

(28)

As is well known, the Kac formula (2 5) plays a central role in certain classes of conformally invarlant field theories in two dimensions since for suitably restricted values of r, s, and C, it gives the weights of the primary fields For C < 1 [21], and C > 25 [19, 22], the rational conformal theories have central charges Cp p , = 1 - 6 ( p _p,)Z/pp,,

(29)

for integer p and p ' This formula also enjoys a mirror property

G ,,,+ G ,,, = Gr, t

(210)

For C < 1, (respectively C > 1) the conformal weights are given by (2 5) with r > 0, s > 0 (respectively r > 0, s < 0) C > 1 is the natural region for the Liouville theory [6] (where itS Planck constant is p o s i t i v e ) , which describes the quantum behavior of the Weyl component of the world-sheet metric The mirror properties (2 8), (2 10) thus show that (2 1), (2 2) are natural conditions for the coupling of rational theories with C < 1 to the world-sheet metric, the Vlrasoro generators L n

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being given by the direct sum In tins way, critical systems on a random lattice (fractal gravity [23]) appear as generahzatlons of string theories in the conformal gauge

22 PROPERTIES B STRING FIELD COHOMOLOGY B 1 The mlpotency of the B R S T operator Kato and Ogawa [24] have shown that there exists a nllpotent BRST operator, Q2= 0, Iff eq (22) holds This is the modern powerful way to recover property A 1 and to show that eq (22) is necessary for the consistency of the string quantlzatlon B 2 The ghost system The Vlrasoro generators being fields of conformal spin 2 (up to the central term) they are associated with a ghost pair of spin 2 and - 1 In general, such a ghost pair of spin j and 1 - j has a central charge [25]

Cgh(J) = --211 -- 6j(1 - j ) ] ,

(2 11)

c~(2) = - 2 6

(2 12)

and, in particular,

Thus the total Virasoro generators, L~ + Ingh, have vanishing central charge if the L,, have the critical central charge, C = Ccm = Cgh Moreover, when the ghosts are included, there must be no violation of conformal lnvarlance This relates the intercept % to the ghost number of the physical states Bosonlzlng the ghost system, the ghost Virasoro generators read l,gh = ~

a m a . _ m + V/2 e%na,, - c%28. o

with

~ a 0 -~

(213)

where the a,, are harmonic oscillators ([a., am] = nS._m) The Vlrasoro generators (213) are those of the Coulomb gas formalism [26], or equivalently of the quantum Llouvllle theory [16] corresponding to a Planck constant of the Llouville system equal to h = - 1 / 8 a 0 2 [22] Using eq (213), the mass-shell condition (21) can be rewritten as ( L . +/~gh)lphys ® ghost) = 0

Vn >~ 0,

(2 14)

since the ghost component of the physical states is such that a01phys ® ghost) = - ½1phys ® ghost) Hence the intercept e 0 is automatically contained in the

(215)

/gh,s

/0ghlphys ® ghost) = - e0] phys ® ghost)

(2 16)

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228

B 3 The Chern-Stmons-Wltten actwn The ghost-number of the ghost vacuum, eq (2 15) ~s related to the grading - ½ associated with the stnng field A in W~tten's string field theory [13] with action

I= f(A.O

+ 2A•ASrA),

(2 17)

and the value of - v / 2 a 0 , eq (2 13), is connected with the grading - } of the integration procedure Tins may be seen [28] as follows The overlap 8-functional describing the integration procedure for the ordinary stnng is given by

I[X()]=

1-I

8(X(o)-X(rr-o))

(218)

0~
Wrmng thin functional in the form of a ket in the string Fock space, the authors of ref [28] show that, in D dimensions,

K2.II ) = ½ n ( - 1 ) " ( - D ) l I ) ,

K2.-L2.-L_2.

(2 19)

The ghost part of the integration is defined as I ~h= e x p [ - 3lfp(1,/r)] U{~(

),

(2 20)

a

where the last term is a &functional overlap similar to eq (2 18), and q0 is the ghost field Thanks to the mid-point Insertion. the ghost integration functional satisfies Kgh,rghx = i n ( _ 1)"2611gg), 2n I a /

K ~ =- l ~ - l_gh2.

(2 21)

If we combine eqs (2 19) and (2 21), we see that the total K operator gives zero at the critical dimension D = 26 The coefficient - ~ of the mid-point insertion gives the grading - 3 back for the integration procedure of eq (2 17) We have generalized tins argument to the Llouwlle string [27], showing that it is not limited to the Venezlano model considered in ref [28] Tins cancellation of anomalies only depends upon the crltlcahty of the central charge

3. The W-algebras and W-strings 3 1

THE

W-ALGEBRAS

The W-algebras are non-hnear extensions of the Vlrasoro algebra [6-10] In general, every W-algebra is associated with a simple Lie algebra g, and for each independent Caslnur lnvarlant of g of order k there ~s a conformal spm-k generator (Wk) . m the W-algebra These higher-spin generators can be very systematically

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229

o b t a i n e d either by the G K O coset construction based on the K a c - M o o d y algebra [7], for the generahzed minimal series, or from the T o d a theories associated with g [8,9] If g = A N ( = SUN+~) e g there are generators of spin 2 (Vlrasoro), 3,4, , ( N + 1) T h e simplest example, corresponding to SU 3, is the spin 3 algebra [10] for the operators W~2) =- L and Wo) = W

[ Ln, tin] = (l'l -- m ) L n + m q'- l~C/'/(n 2 - 1)3.

m,

(3 1) (3 2)

[L,,, W.,] = (2n - m ) W n + m , [Wn, Wm] = /3(11 -- m ) Z n + m -Jr 36~C/'/(/-/2 -- 1 ) ( n 2 -- 4)3. -m + ( n - - m)[~5(n + m + 2)(n + m + 3) -- 16(n + 2)(m + 2 ) ] L . + m ,

(3 3)

E tmtn-m 210(/'/2--4- 5p~)L.,

(3 4)

where Zn=

m

16 /3 =

22 + 5C

,

O. = n m o d 2

(3 5)

M o r e c o m p h c a t e d algebras have been studied corresponding to arbitrary s~mply laced g [6, 7] and simply or non-simply laced g [8, 9] In the following we will restrict ourselves to the case of simply laced g

3 2 THE GHOST

SYSTEM

In a W - s t u n g theory where the W-algebra [e g eqs (3 1)-(3 5) for g = SU3] replaces the Vlrasoro algebra, a ghost pair of conformal spin k and 1 - k should be assocmted with each spln-k generator The contrlbutmn of each pair to the central charge C is Cgh(k), see eq (2 11), and the total ghost central charge is

C~= - 2 £ [ 1 -6k(1 - k ) ] ,

(3 6)

k

where k runs over the orders of the Caslmlr m v a n a n t s of g (k = e + 1, where e runs over the exponents of g) In ref [7] it was remarked that, by eq (3 6), - Cgh --- Ccrlt equals Cent - - Cgh = U + U ( 1 + 2hc) 2 H e r e h e is the Coxeter number of g N + 1 for A N = S U N + I , 2 N - 2 for 12, 18 and 30 for E 6 , E 7 and E 8 ( N always denotes the rank of g )

(3 7) D N = SO2N ,

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A Bdal, J - L Gervats / String theorw~

3 3 T H E G E N E R A L I Z E D KAC F O R M U L A

The value (3 7) of the critical central charge is also confirmed by the generahzation of the Kac formula (2 5) In the present general scheme, eq (2 5) corresponds to g -- SU 2 For arbitrary simply laced g, it is replaced by

(C-Co) 1

-SNhc(hc+ l) Y~'Npq[(rp+Sp)~ -Co +(rp-Sp) C(-C~CT] Pq

c0 +(rq-sq)

× [(rq+,ql

(3 8)

~¢ denotes the inverse of the Cartan matrix of g, ~:= (r 1, , ru) The rp's and the sp's are integers The above formula was derived in refs [6, 7] for the rational cases We arrived at the present general form in refs [8, 9] The latter articles also deal with non-simply laced g The values of Co and C 1 are Co:N

,

C1 = N ( 1 +2h~) 2

By analogy with eq (2 7) we expect that

(3 9)

Ccrlt should be given by

(3 10)

Ccn t = C 0 -]- C 1 ,

which indeed gives the value (3 7) back In sect 2 we remarked that the intercept e 0 of the ordinary bosonlc string may be derived from the lmrror property of Kac's formula, (property A 4) In the present case, the generalized Kac formula (3 8) does satisfy a generalization of eq (2 8)

zx( , e, c ) +

Ccr,,- C) =

-- CO)

(3 11)

This indicates that the intercept % should be given by

~o = ~4(C1- Co) = ~Nh~(hc + 1)

(3 12)

Note that in all cases, this is an integer As already remarked, the W-algebra generators of a combined system (e g physical + ghosts) are not a priori given by the sum of the Individual generators This IS due to the non-llnearlty of the algebra, see, e g eqs (3 3), (3 4) However, the mirror formula (3 11) indicates that at least L 0 should be simply given by the sum of the L0's of the individual systems Moreover, since the central charges of eq (3 11) add up to Ccm, the addltlvlty of the generators seems to hold for all L.'s as well On the other hand, the fact that Ccnt + Cg h = 0 indicates that this is also true

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231

when one puts a "physical" system and the ghosts together This additlVlty of the L,,'s is to be expected since the Vlrasoro subalgebra is hnear Let us mention that the series of central charges for the rational conformal theories (W-generalized BPZ series or Z N series),

Cpe,=N1-

hc(h c + 1 ) ( p

_p,)2

pp'

,

p,p'~7]

(313)

still satisfy the mirror property (2 10)

Cp p, + Cp _p,= Ccm,

(314)

supporting again the addltlVity of the L,,'s The two mirror properties (3 11) and (3 14) should be interpreted as allowing a consistent coupling of a (W-) minimal model theory (3 13) to the quantum fluctuations of the 2D world-sheet or to a r a n d o m lattice (" fractal W-gravity") This supports the idea, expressed at the end of the introduction, that the Toda theories should play the role of a generalized world-sheet metric for the W-strings 3 4 T H E N I L P O T E N C Y OF T H E BRST O P E R A T O R

The values (3 7) and (3 12) for the critical central charge and the intercept have been confirmed for g = SU 3 Thlerry-Mleg [12] has shown that there exists a nllpotent BRST operator for the algebra (3 1)-(3 5) if and only if C = C c n t = 100 and c o = 4 3 5 T H E A P P E A R A N C E OF SPURIOUS STATES

By analogy with the usual situation, we expect that a physical W-string state obey the ghost-kllhng conditions ( L 0 - %)lphys) = 0, (W(~)),,]phys) = 0,

L,,]phys) = 0, n > 0,

n>0, (3 15)

where IV(k) are the higher-spin generators (W(3) - W for g = SU3) Conditions (3 15) just mean that a physical state is a highest-weight state of the W-algebra with L 0 elgenvalue c o The W-string Fock space will provide a representation of this algebra which remains to be constructed Again, the criticality of the central charge should manifest itself by the appearance of spurious states Now they should obey (3 15) and be created from a highest weight state ]A,0) by application of the L .'s or (IV(k)) ~'s The same reasoning as in A 1 shows that we may foresee what will happen directly from the generalized Kac formula (3 8) For the standard case, the signal of criticality was that A(1, 2, 2 6 ) = - 1 differs from e 0 = 1 by minus two so that a spurious state appears at the second level For W-algebras there is a similar

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232

phenomenon, namely, formula (3 8) leads to

A(F(J),s(J), Ccnt)= -2Egg-~jq

for

rqO'=l,

S(qJ)= l + ~ q j

q

Since 2S,q f~jq and % are both integers, there is again the possibility that spurious states appear at the levels c o + 2~,q%q, confirn-nng that eqs (3 10), (3 12) are indeed the critical values for the W-strings

36 THE BRINK-NIELSEN ARGUMENT AND MODULAR INVARIANCE Trying to generalize the Brink-Nielsen argument [20], one ~mmedlately sees that the critical central charge Ccm (3 7) and the intercept c o (3 12) are related by 214( C c n t -

2N) =

2~(C1 - C o ) = ~o

(316)

The LHS can again be interpreted as the zero-point (Casxmxr) energy of Ccn t - 2N sets of harmonic oscillators This indicates the decoupllng of 2N degrees of freedom due to the appearance of spurious states we just noted This decouphng is certainly necessary for the correct counting of the physical degrees of freedom Concerning modular lnvarlance, eq (3 16) ensures that the partition function Z ( r ) - Tr exp[2rr,r ( L 0 - %) - 2rr/'r(L00 - % ) 1

=exp(4~r[%--~4(C--2N)llmr)[(Imr)i/2lrl(r)12]-(D-2U'

(3 17)

lS modular mvarlant at C = Ccn t Of course, modular invarlance can be achieved in many other ways For example, part or all of the free fields could be replaced by the Toda fields [8] generahzang the situation encountered for the Llouvllle stnngs [27, 29] Thus the present argument is certainly more general than the free-field case considered which may be too naive for the W-strings

3 7 T H E W - S T R I N G FIELD THEORIES

Let us finally speculate how the W-string field action might look like In order to do so, we remark that the complete system of the 2N ghost fields can be bosonlzed to ymld N bosonic ghost fields They are equivalent to the N Toda fields [8, 9] at a couphng constant which is such that CToda = C g h = - Ccnt Each ghost pair of conformal spin k and I - k has its bosonlzed Virasoro generator given by eq (2 15) l ~ k = ~Y[ a mk a nk_ m m

(318)

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233

with a o replaced by x/2 a o ( k ) = k - ~

(3 19)

Their contribution to the central charge is 1 - 24[a0(k)] 2 which coincides with eq (2 11) As already discussed in B 2, conformal lnvarlance must be restored when the ghosts are Included, and the total L ' s have to vanish

L.+Y',I.gh k)lphys®ghost ) = 0

Vn>~O

(3 20)

k

For n = 0 this yields

~ {½(aok)2-[,~o(k)12}=-%

=

£ ( a o k ) 2 = 1N

k

(321)

/,

which IS satisfied if a0k = + ~ Vk One thus sees that the intercept is such that the physical part of the string field A has k t h ghost number +_ ½ What is the corresponding grading of the integration procedure'~ This may be seen by generalizing the argument of Gross and Jevlckl [28] recalled in B 3 It is reasonable to assume that the W-string theory wdl be such that eq (2 19) is replaced by

K2,,lI)=ln(-1)"(-Ccnt)lI),

Kv,-L>,-L

2n

(322)

It follows from our general calculation of the K-anomaly in Llouvllle theories [27] that, if we define, for the kth ghost field,

I(k)- exp[tl(1 - 2k) qCk'(l~r)] I-I8(

),

(3 23)

o

the contribution to the K-anomaly [the coefficient of the i n ( - 1 ) n term] is equal to 211 - 6k(1 - k)] It thus follows from eqs (3 6), (3 7) that the integration measure II-[ k I (k) has no K-anomaly This leads to a grading ½ - k of the integration with respect to the k t h ghost field The BRST operator (see e g ref [12]) has a total ghost number 1

Q=c(o2)(Lo-%)+ Y'~ C(0k)[(W~k,)0--%k]

+

,

(324)

/,>3

(where the c(k)'s are the fermlonic ghosts) Its grading is well defined only with respect to the total ghost number, and we expect that only the overall grading (total ghost number) should add up to zero in the W-string-field action What is the

A Bllal,J-L Gervats / Stnng theortes

234

grading of A 9 Most likely, physical states should correspond to A -- b0(2)Aphys,

(3 25)

w h e r e Aphy, satisfies COAphys- 0, and the b's are the antlghost fields The reason is as follows We want the cohomology condition QA = 0 to give ( L 0 - •0)Aphys = 0 as

the only condltton lnvolvmg zero components of the W-generators Indeed the other possible conditions (W(~)--~0k)Aphys=0, k > 2, would Involve the W-string momenta raised to powers higher than two, and cannot appear in the equations of motion With the choice (3 25) the zero-ghost-mode parts of Q shown in eq (3 24) automatically vanish except for k = 2 The overall gradmg of A is thus given by gr(A) = _ ~ + ~ + 1 +

+~= ½(N-2)

(3 26)

For the integration procedure we have (restricting ourselves in the following to A N, for convenience of notation) g r ( f ) = Y~k(½ - k ) = - N ( N + 2 ) / 2 Finally, the field strength F = QA + A * A must have a well-defined grading One must have gr(Q) = gr(A) + gr(,k), thus gr(4t) = 2 - N / 2 We see that the Chern-Slmons action (cf ref [301)

, 9 = (n + 1) f f~ dt [ A ( * t Q A + ~rt2A*A)"]

(3 27)

has grading 0, gr(5 °) = 0, provided

n = ¼ N ( N + 1) + ½

(3 28)

Of course this ~s only possible if n comes out as an integer, which is the case for N = 1 or 2 rood 4 Then, the action (3 27) yxelds

F -- QA + A~rA = 0

(3 29)

as a consequence of the W-string field equations of motion If eq (3 26) does not yield an integer value for n, 1 e for N = 3 or 4 mod 4, one can instead construct the topological action

~=

f F(.F) m,

m = J N ( N + 1)

(3 30)

This type of topological action can be gauge fixed [31] by imposing i f - ( ~ - F ) " ' , leading to an action of the type fFff In this game, however, one must introduce a metric, contrary to the odd case Altogether, the cohomologies which appear are similar to the ones of v-dimensional manifolds with v = 2 + N ( N + 1)/2

A Bllal, J I Gervals / String theorws

235

4. Outlook A l t h o u g h we are unable at present to carry out the construction of the W-strings, we feel that the hints presented here for their existence are compelling The striking novel feature is that the intercepts are larger than one and would of course probably d o u b l e if we considered the left-moving and the right-moving modes together The spectrum c a n n o t be predicted at the present time, but unltarlty tells us that Regge trajectories are hnear at the tree level Thus m a n y more tachyons seem to be present By analogy with the usual strings one hopes that all these unwanted particles will disappear f r o m the appropriate sectors of the W-superstrlng theories Moreover, one is led to massless particles with spins larger than one (for the open strings) or two (for the closed ones) Unless they all decouple, this gives a picture of the asymptotic physics where Interactions of a novel nature would take place They would have to turn on as one goes up in energy This may lead to new interesting physics for the c o m i n g very-high-energy accelerators A n o t h e r point IS that the new critical central charges are very large and grow very fast with the rank of the group Thus one goes even further from D = 4 In this connection, let us recall the existence of the "lower critical dimensions" which a p p e a r [27] for the standard strings when one includes the Llouvllle theory They are lower than Ccm and the corresponding theories involve less massless particles than the s t a n d a r d ones Our earlier remarks [8] about the partition functions of the T o d a theories indicate that these lower critical dimensions also exist for the W-strings The corresponding string models may thus be of physical interest Clearly much work remains to be done

Note added in proof Subsequent to the circulation of this article, we have learned that P Howe, A Neveu, and P West have independently also considered W-strings They observed that the hlghered Intercepts could lead to additional difficulties, not considered in the present article, since tachyonlc fields which carry spins lead to states with negative definite n o r m This led these authors to somewhat pessimistic conclusions which we do not share since these tachyonlc states may decouple in super W-string theories

References [1] M B Green, J H Schwarz and E Wltten, Superstrmg theory Vols 1 and 2 (Cambridge Umverslty Press, Cambridge 1987) [2] M A Vlrasoro, Phys Rev D1 (1970)2933 [3] P Ramond, Phys Rev D3 (1971)2415, A Neveu and J H Schwarz, Nucl Phys B31 (1971) 86 J -L Gervms and B Saklta, Nucl Phys B34 (1971) 632 [4] I B Frenkel and V G Kac, Inv Math 62 (1980) 33 G Segal, Commun Math Phys 80 (1981)301

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