7
Volume 171, number 1 PHYSICS LETTERS B 17 April 1986 QUASI-YANG-MILLS STRUCTURE FOR THE OPEN BOSONIC STRING L. BAULIEU and S. OUVRY 1,2 CERN, CH-1211 Geneva23, Switzerland Received 17 December 1985 Using the description of bosonic open strings by a double p-form, the ghost-auxiliary field spectrum is derived algebraically, as well as the BRS symmetry, of the free bosonic string theory. A quasi-Yang-Mills structure emerges, where lagrangians and possible anomalies are related to invariant exterior forms of higher rank. The passage from the free case to the interacting one necessitates the understanding of vertex operators defined so as to ensure some geometrical Jacobi-Bianchi identities. In a recent paper [1], Neveu, Nicolai and West have found that the fundamental objects of the boson- ic string theory are built with double K-form gauge fields, ~K K and f~K K+ 1, where K stands for any posi- tive or nul integer and indicates the rank of the forms associated with the up or down antisymmetrized Virasoro indices (see ref. [1] and ref. [2] for the string excitation contents of ~pK K and ~bKK+I ). A covariant formulation of the free open string theory has been obtained in this way, altogether with some additional features concerning the ghost sector of the theory [1,2]. In this letter we start from the hypothesis that dpKK and dpKK+l are actually the fundamental classical fields of the bosonic string theory. Since these fields are K-form gauge fields, it is thus possible to use geometrical methods [3] to build directly the BRS symmetry of the theory, and derive at once its ghost and antighost spectra, as well as the associated Stueckelberg auxiliary fields. Once this is done, a lagrangian can be built, including BRS invariant gauge fixing terms, and a classification of anomalies becomes possible by solving Wess-Zumino consis- tency conditions. In this approach, lagrangians and 1 On leave of absence from Division de Physique Th6orique (Laboratoire assoeid au CNRS), IPN, F-91406 Orsay, France. 2 On leave of absence from LPTHE, Universit6 Pierre et Marie Curie, F-75230 Paris, France. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) anomalies are simply related with invariant exterior forms of higher rank. As a first step, we consider the free case. Then we shift to the unified notations of Ogawa and Kato [4], and analyze the interacting case. The passage from the free case to the interacting one necessitates the understanding of vertex operators defined as to en- sure some Bianchi identities. In this way the unique- ness of the string interaction appears as an equivalent to the uniqueness of the Chern-Simons form of rank 3. More precisely, the definition of the way two strings interact together point by point will emerge from the requirement that geometrical identities such as Jacobi identities must hold true. This allows in turn for the notion of a generalized Chern-Simons formula, lead- ing to the determination of the interacting theory from the free one. We will see that the structure which emerges is a quasi-Yang-Mills one, and that the determination of lagrangians and anomalies will be algebraically trivial. Furthermore, it will be possible to introduce p-form gauge string objects, the lowest excitations of which are pointlike p-form gauge fields. The latter are known to play a central role in the mechanism of anomaly cancellation [5]. Let us briefly recall the notations of ref. [ 1]. The variances of ~K K and ~KK+I are determined by the action of the operators d, D, K and r/which are built from the structure of the Virasoro algebra and act on products of fields as differential operators. Given a tensor C~b , with a upper and b lower antisymmetrized 57

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Page 1: Quasi-Yang-Mills structure for the open bosonic string

Volume 171, number 1 PHYSICS LETTERS B 17 April 1986

QUASI-YANG-MILLS STRUCTURE FOR THE OPEN BOSONIC STRING

L. BAULIEU and S. OUVRY 1,2

CERN, CH-1211 Geneva 23, Switzerland

Received 17 December 1985

Using the description of bosonic open strings by a double p-form, the ghost-auxiliary field spectrum is derived algebraically, as well as the BRS symmetry, of the free bosonic string theory. A quasi-Yang-Mil ls structure emerges, where lagrangians and possible anomalies are related to invariant exterior forms of higher rank. The passage from the free case to the interacting one necessitates the understanding of vertex operators defined so as to ensure some geometrical Jacobi-Bianchi identities.

In a recent paper [1], Neveu, Nicolai and West have found that the fundamental objects of the boson- ic string theory are built with double K-form gauge fields, ~K K a n d f~K K+ 1, where K stands for any posi- tive or nul integer and indicates the rank of the forms associated with the up or down antisymmetrized Virasoro indices (see ref. [1] and ref. [2] for the string excitation contents o f ~pK K a n d ~bKK+I ). A covariant formulation of the free open string theory has been obtained in this way, altogether with some additional features concerning the ghost sector of the theory [1,2].

In this letter we start from the hypothesis that dpKK a n d dpKK+l are actually the fundamental classical fields of the bosonic string theory. Since these fields are K-form gauge fields, it is thus possible to use geometrical methods [3] to build directly the BRS symmetry of the theory, and derive at once its ghost and antighost spectra, as well as the associated Stueckelberg auxiliary fields. Once this is done, a lagrangian can be built, including BRS invariant gauge fixing terms, and a classification of anomalies becomes possible by solving Wess-Zumino consis- tency conditions. In this approach, lagrangians and

1 On leave o f absence f rom Division de Physique Th6orique (Laboratoire assoeid a u CNRS), IPN, F-91406 Orsay, France.

2 On leave of absence from LPTHE, Universit6 Pierre et Marie Curie, F-75230 Paris, France.

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

anomalies are simply related with invariant exterior forms of higher rank.

As a first step, we consider the free case. Then we shift to the unified notations of Ogawa and Kato [4], and analyze the interacting case. The passage from the free case to the interacting one necessitates the understanding of vertex operators defined as to en- sure some Bianchi identities. In this way the unique- ness of the string interaction appears as an equivalent to the uniqueness of the Chern-Simons form of rank 3. More precisely, the definition of the way two strings interact together point by point will emerge from the requirement that geometrical identities such as Jacobi identities must hold true. This allows in turn for the notion of a generalized Chern-Simons formula, lead- ing to the determination of the interacting theory from the free one. We will see that the structure which emerges is a quasi-Yang-Mills one, and that the determination of lagrangians and anomalies will be algebraically trivial. Furthermore, it will be possible to introduce p-form gauge string objects, the lowest excitations of which are pointlike p-form gauge fields. The latter are known to play a central role in the mechanism of anomaly cancellation [5].

Let us briefly recall the notations of ref. [ 1 ]. The variances o f ~K K a n d ~K K+I are determined by the action of the operators d, D, K and r/which are built from the structure of the Virasoro algebra and act on products of fields as differential operators. Given a tensor C~b , with a upper and b lower antisymmetrized

57

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Volume 171, number 1 PHYSICS LETTERS B 17 April 1986

Virasoro indices, one has, by definition [2] :

[m 1 ...ma ] C [m 1 ...ma ] [dC] [nl. . .nb+l ] = L[nx -n2...nb+ 1 ]

+ aWp[nl tml c~__m2""ma] n2...nb+l]

l b V p t-,[ml...ma] [ntn2 '~p...n/~+l] '

[DC] [m I . . .ma- 1 ] = f r,[P m I ...ma - 1 ] [nl...nb] ~ - P ~ [ n l ...n b ]

qt",[P m I . . .ma- 1 ] + b W[nlp ~ qn2...nb ]

- -½(a - l) Vkl [mt t~ lm2" ' 'ma-1] ~[n l ...nb]

t-'r,1 [ml ...m a ] ""~J [nl ...n b ] = (L0 + rn + n - 1) r ' [ml ""ma] ~[nl . . .nb] '

m = ~ m i , n = ~ n i ,

[r/C] [ml""ma-1] r [ q m l " " m a - l ] (1) [nl...nb+ 1] =r /[n lq"[nz . . .nb+l] ,

where the L n are the generators of the Virasoro algebra, which reads

[Ln,Lm] = VnmPLp, [ L _ n , L _ m] = - V n m P L _ p ,

[ L m , L _ n ] = WmnPL p + WnmPL_p + r/mnL(m) ,

L (m) = 2L 0 + !~ (m 2 _ 1). (2)

The properties which express that the Virasoro algebra is a Lie algebra are the commutation relations

d 2 = D 2 = d D - Dd = 2r/K = 0 . (3)

We define now ~K K and dpKK+I to be normal-ordered products of creation, a +, and annihilation, a, operators. The VJK K and @K K+I of ref. [1] are simply given by the action of these operators on the vacuum 10).

Ghost spectrum. The ghost spectrum is obtained by adding ghost and antighost degrees of freedom to the classical indices of ~K K and ~K K+I . We Proceed in analogy with the device used to introduce ghosts and antighosts for a p-form gauge field in ref. [3]. ~K K and $K K+I become

~K K = ~OKK + ~K_I(1 ,0) K + ~OK(0,1) K+I

~K_2(2,0) K + ~K_I(1 ,1) K+I + ~K(0,2) K+2

K g

+ ~ ~d "'" K+q g=3q= 0 ~K-g+q(g-q ,q ) ,

~K K+I = dpKK+I + ~bK-l(1,O) K+I + OK(0,1) K+2

+ $K_2(2,0) K+I + ~bK_l(1,1) K+2 + ~bK(0,2) K+3

+ .... (4)

In any object XK"-g+q(g_q ,q)K'+q, q and g - q stand, respectively, for the antighost and ghost num- bers. The ghosts ~K-g(g,O) K and dPK_g.(g,o)K+l are those introduced by Neveu, Nicolai and West [ 1 ]. The other ghosts (q/> 1), are of a new type. The compatibility of this new set of ghosts with unitarity will be shortly verified by writing a BRS invariant and gauge fixed lagrangian with ghost number zero, func- tion of the above fields, and reproducing the known classical, gauge fLxed lagrangian of ref. [ 1 ]. A defini- tion of the statistics of all fields must be given. This is achieved through the prescription that when g is even (respectively odd) XK"-g+q(g-q,q)K'+q has physical, respectively unphysical, statistics. In other words, the grading is defined as the sum of the ghost and antighost numbers.

Free theory, d and D change the ranks of the form on which they act. Following the general construction of gauge theories, we must construct the field strengths of ~K K and ~K K+I . They must be of the form

GK+IK = d~K K + D~0K+IK+I + .. . .

FKK = d~bK_l K + Dq~KK+I + ... ,

where the dots stand for terms needed to ensure that some Bianchi identities hold true, i.e., that (d + D)G and (d + D)F vanish when F = G = 0. The solution is

GK+I K = d~K K + Dt~K+IK+I + 2r/~bKK+l ,

FK+IK+I = dSKK+I _ D~bK+IK+2 _ K~K+IK+I (5)

and the Bianchi identities read

dGK+I K = DGK+2 K+I - 2r/FK+I K+I , (6)

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Volume 171, number 1 PHYSICS LETTERS B 17 April 1986

dFK+IK+I = -DFK+2 K+2 - -KGK+2 K+I • (6 cont 'd)

B R S symmetry . We introduce the BRS operators s and-g, with ghost numbers (1, 0) and (0, 1) and define

d = d + s , D = D + - g , ~--~?, K ' = K . (7)

s and -~ act as differential operators on all the fields introduced above, their grading being equal to one. To determine the action of the operators s and-~ on the fields, we impose the following horizontally con- ditions on the field strengths [3]"

G~K+IK ='d"~K K + DI~K+IK+I + 2 ~ K K+I

= GK+IK ,

FK+I K+I ~-adpK K + I - D~K+I K+2 -- ~"~K+I K+I

= FK+IK+I (8)

Expansion in ghost number of (8) determines directly the action of s and -g on the classical ghost and anti- ghost fields in ~K K and CK K+I . The parts with ghost number (1 ,0) and (0, 1) read

s ~K K = - d ~0K_I(1,0) K -- D ~K(1,0) K+I

- - 2 7 / ¢ K _ I ( 1 , 0 ) K + I ,

s CK K+I = - d CK_ 1(1,0) K+I + D CK(1,0) K+2

K+I (9a) + K ~K(1,0) '

and

St~KK = - D ~OK(0,1) K+I -- d ~K_I(0 ,1) K

K+I - - 2 7 C g - 1 ( 0 , 1 ) ,

-g CKK+ i = - D ~K(0,1)K+2 + d CK- 1 (0,1)g+ 1

-- K ~bK(0,1) K+I (9b)

For the parts (2,0) and (1,1), one has

s ~K_ 1(1,0) K = - d ~K_2(2,0) K -- D ~K_ 1(2,0) K+I

-- 2r/~bK_2(2,0) K+I ,

s CK- 1(1,0) K+I -- D ¢K_1(2,0) K+2 - d OK_ 2(2,0) K+I

K+I (10a) + K ~ K - 1 (2,0) ,

and

a, K+I +g ~K(1,0)K+I s ~'K(0,1)

= -- d ~ g _ l ( 1 , 1 ) K+I -- D ~0K(I,1) K+2

-- 2r/q~K_l(1,1) K+2 ,

s ~bK(0,1) K+2 ---g ~bK(1,0) K+2

K+2 K+3 = -- d $K-1(1,1) + D ¢K(1,1)

+ K ~0K_ 1 (1,1) K+2 , (10b)

which give the variations of the primary ghosts, and so on. The s-variations in (9) and (10) are identical to the ones derived by Neveu, Nicolai and West, when one replaces ghosts by infinitesimal parameters [ 1 ]. However, the BRS symmetry is not yet entirely de- termined by (8): the action of s on the ghosts (g - q, q) with q /> 1 or o fg on the ghost (g - q, q) with q ~<g - 1, is degenerate [see, for example, (10b)]. One can raise this degeneracy by introducing auxiliary fields for ~K K and $K K+I , which will also play the role of Lagrange multipliers of the relevant gauge fixing conditions for the classical and ghost fields in BRS invariant lagrangians. These fields are defined as

K+I + bK - K+I + K+2 bK K+I = bK(1,1) 1(2,1) bK(1,2)

K g + ~ D ~ K+q

g=3 q=l VK-g+q( l+g-q 'q)

"~KK+2 = CK(1,1) K+2 + ¢K_1(2,1) K+2 + CK(1,2) K+3

+ ..., (11)

and the above-mentioned degeneracy is lifted by defining, for g ~> q I> 1,

s ~K-g+q(g -q , q)K+q = bK_g+q(l+g_q,q)K+q ,

s.h K+l+q - K + l - q WK-g+q(g-q ,q ) - CK-g+q( 1 +g-q ,q ) ,

(12) with

s bK_g+q(l+g_q,q)K+q = 0 ,

K+l+q = 0 (13) S CK_g+q(l+g_q,q )

At this point, the action of s and ~ on all fields has been fully determined. The existence of Bianchi identities implies that the horizontality condition (8)

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Volume 171, number 1 PHYSICS LETTERS B 17 April 1986

defines s and-g such that

~2 = ~2 = d D - Dd - 2 = 0 , (14)

and since ~ = r/and K = K, one has finally

s 2 = s s - - g s =g2 = 0 , (15)

which ensures that s and g are nilpotent operators, as they should. Note that the introduction of the anti-BRS operator-g is tightly linked to the existence of the operator D, since D = D +-g, and not from an a priori mere doubling of the BRS operator s associat- ed to d.

Finally, a classical BRS invariant density lagrangian, function o f the classical gauge fields and of the field strengths can be written as the following sum of order- ed terms:

~, = 1 ~ j (_)K (GK+I K :f dpKX+l _ ~jK x "~ FK K) 2 K

+h.c. (16)

This lagrangian is degenerate and one has to fix the gauge. This gauge fixing problem is trivial since we know the BRS symmetry and the ghost and auxiliary field sectors of the theory. The gauge fixed lagrangian can be written as

gauge. The.gauge used in ref. [1] is a Landau one and is obtained by taking

~GF = ~ S(¢r+q_g(q,g_q+lr+q+l t ~Tg+l O~g~K l<q~g

X CK_g+q~_q,q)K+q+l). (17b)

This gauge implies that all the 4~ vanish. If further- more we want I#KK = 0 for K/> 1, we add simply

J~GF' = ~ S(~K+I(0,0) K+I "[" 77 SK(0,1)K+2~- .(17C) K>0

When the gauge (17b) is used, the classical lagrangian reduces to

1 .C = - ~ Z~ ( - 1 ) K ~JK K "f K ~JK K + h.c. ,

which shows that our system of ghosts, albeit different from the one of ref. [ 1], describes the same physics.

To proceed further, it is convenient to use the unified notations introduced by Kato and Ogawa [4]. From now on, all the indices in ~K K and dpKK+I are contracted by ghost creation and annihilation opera- tors/3+ and/3 and one can define a gauge field X by

£OF = ~ s(Cjx_g,q~_q,q)K+q t OgggK lgqgg

X 0 K-g+q K +q(q ,g-q+ l )

+z 4 lgggK bK+q-g(l+q'l+g-q) +q+l

lC t

l~q~g

X r f +1 I" K+q+l (17a) '-'K-g+q( l +g-q,l +q )

where r/n means that the operator 7/is applied n times. The first term determines a ghost interaction, of the form ~sO (respectively ~ O ' ) and a gauge fixing term. Indeed the action of s on the ghosts ~ and ~ provides terms of the type bO and cO', which thus play the role o f gauge fixing terms, as can be seen after the elimination of b and c through their equation of motion. When ot =/3 = 0, one has a Landau gauge, i.e., a gauge with 6(0) and 6(0') in front of the functional integral measure. For a =/3 = 1, one has a Feynman

X-- ~ (dltml'"ma] /3 "1+ " - - : a+ n,m ~'[nl""na]

X "~+ ...-+ - rh[ml""raa+l /3na+ ml /3ma/3O + l /3nl+ ~" [nl ...na] "'"

-+ ~+ . (18) X/3ml ... ma+l )

One has {~0/30) = 1,8nr n = {~+,/3m) = (~n,/3 +} for n, m t> 1 and all the other anticommutators vanish. Observe that X has an odd grading in the ghost opera- tors/3+ and ~+. We call it a one-form since it starts with ¢/o°~0 . One can also define [1]

-+ m ~ra+VmnP~p ) d =-/3n+(L n + Wren p/3p/3 +

D- /3n (L_n + WmnPlffn+~p + ½~pVmnP/3m),

M=- 2-~° ~1= n/3"~ /3. ,

K--/30 ( L 0 - 1 + ~ n / 3 ~ n n +n~n+/3n) ' (19)

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Volume 171, number 1 PHYSICS LETTERS B 17 April 1986

Q - d + D + K + M . (19 cont 'd)

and then

Q2 = 0 . (20)

The advantage of this notation is that the rules of exteriorcalculus can now be systematically applied.

and Q are defined to be

X - X ( ~ , ~). Q = d + D + K + M ,

d - d + s , D - D + ~ , (21)

where K and M are as previously horizontal. The field strength of X is simply

F = QX, (22)

and the Bianchi identities in eq. (6) become

QF = 0 . (23)

Furthermore, the BRS symmetry follows directly from

F = F , (24)

once expanded on ghost number. To proceed, one must be able to consider products

of functionals. Given two functional operators X and ×', there are two ways of defining the product of X by X'. The first one corresponds to the usual "trace" operation which will be denoted by X'X': it corre- sponds to the product X+X ' used above. The second operation, denoted by [X, X'], means that some non- trivial overlapping function is used to convert two strings into a third ohe, in order to describe possible interactions. Both products are bilinear and must obey the graded Leibnitz rule under the action o f the operators d and D. Q, x and X' have an odd number of 3 + operators. This means that their grading is one. Now Q is self-adjoint. It follows that the action of Q on ×'x' or [×, x'] must follow the rule

Q(X'X') = (Qx) ' x ' - x ' ( a x ' ) ,

Q[x, x'] = [Qx, x'] - [x, Qx ' ] , (25)

and in particular ½Q[x, X] = - [X , QX]. x now looks like a usual abelian Yang-Mills one-

form gauge field and Q as the usual differential opera- tor. Besides F = QX is a two-form (with grading 2). An invariant lagrangian should be associated to a Q

exact form, function only o f F . Indeed, suppose the existence of I(F) such that

I(F) = Q L , (26)

F being horizontal, (26) implies that

s L = - QL(1,0) , -gL = - Q L ( 0 , 1 ) , (27)

and the action (0ILl0) will then be BRS invariant, since Q annihilates the vacuum. The Bianchi identity QF = 0 and the fact that we restrict ourselves to a bilinear lagrangian in the fields determine the simplest choice

I = F ' F = Q(x .F) = Q ( x ' ( Q x ) ) , (28)

which in turn implies that L = (x 'F) , which is in fact equivalent to (16) when all contractions are made on /3 + operators. We thus recover the result found in ref. [ 1 ], in a purely geometrical manner.

If we count that X and Q have rank one, and F has rank two, L is of rank three, with ghost number zero. To investigate the possibility of anomalies [3], one looks for the solutions A(1,0), with ghost num- ber one, of the Wess-Zumino consistency condition:

s A(1,0) = O A(2,0) • (29)

Such solutions, whenever fermions were to be coupled to the string theory, could correspond to string field theory anomalies. Solutions to (29) do exist in rela- tion with Q exact forms, and a possible candidate is the obvious generalization of the abelian anomaly, associated with the six-form F' (F .F) :

F ' ( F . F ) = Q(x ' (F 'F ) )

A ( 1 , 0 ) = × ( 1 , 0 ) ' ( F ' F ) . (30) More generally, one could consider any Q-exact func- tional of F.

Interacting case. We have seen that an abelian Yang-Mills like structure is hidden in the covariant formulation of the free bosonic string field theory with gauge field X, curvature F = QX, Bianchi identity QF = 0, lagrangian x ' Q x and possible anomaly as- sociated to x ' ( F ' F ) .

To introduce interactions, the problem consists in generalizing Q into a covariant derivative Qe" One possibility is to use the product [ , ] introduced in (25) and covariantize the Q operator by means of

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Volume 171, number 1 PHYSICS LETTERS B 17 April 1986

Qe =Q + [x, ]. (31)

The covariant field strength is defined as

Fc = a x + ½ [x, ×]. (32)

One has

a Fc = - [X, QX]. (33)

One sees that in order to get for the Bianchi identity

Qc Fc = 0 , (34)

the following Jacobi identity must hold true:

Ix, Ix, X]] = 0 . (35)

The geometrical requirement (34) thus severely con- strains the possible interaction described by the over- lapping function corresponding to the bracket [X, X], in the same way as the Lie algebra structure restricts the structure functions of a Lie group.

The BRS symLnetry can be now derived from the horizontality o f F c defined by

Fc -~)X + ~-[X,X] =Fc" (36a)

This yields in particular

s x = - Q x(1,0) - [x, x(1,0)] • (36b)

Replacing X by the ~K K and the (gKK+I , and X(1,0) by the corresponding infinitesimal parameters one gets the gauge transformations on the classical fields in the interacting case.

An invariant action should be derived from a Q exact four-form, containing the free action defined in (27) and function only of the field strengths, as in the abelian case. Thinking of the Chern-Simons formula in the Yang-Mills case, one is tempted to introduce

14 = Q(x'(Qx) + ½X" [X, X]). (37)

Indeed, computing the right-hand side of (37) and assuming that the definition of the bracket is com- patible with

X" Ix', X"] = X"" [X, X'], (38)

one gets

14 = F c'F c •

14 is indeed only a function of the field strengths F c and the invariant lagrangian density reads

1 L = x'Q× + ~x" [x, ×] • (39)

In this construction, the key assumption is in the existence of a vertex operator such that the products [ , ] and ( , ) satisfy the Jacobi condition (35) and the cyclicity condition (38). The simplicity of the description of the interacting string theory is then really striking: the algebra determines everything up to the overlapping structure of two interacting strings, and leads to an interaction governed by the Chern- Simons form of rank 3.

Knowing the BRS symmetry from (36), it is then obvious to gauge fix the lagrangian (39) in a BRS in- variant way, as in (17). Besides, as in the free case, anomalies may exist in the interacting theory in as- sociation with Q-exact forms made from Fc, since (35) and (38) ensure the validity of the Chern-Simons formula.

Further string symmetries. The string field X, built with t~KK and #pK K+I has been interpreted as a one- form since its/3 + expansion starts with a linear term. One can also consider an object Xp associated to a system ~K K+p- 1 and (gKK+P. The expansion of ×p starts with terms of the form (/3+)P and we call it a p-form gauge string field. It is not difficult to deter- mine the symmetries acting on Xp [3], by means of building field strengths satisfying some Bianchi iden- tities. The simplest example is for p = 2 when one considers interactions with the basic string field X:

G3 = Q'X2 + ~ ' ( X ' ( ~ ) + ½~" [ ~ , ~ l ) ,

1 Fc = Q'x + ~ IX' X]. (40)

Here X is an arbitrary parameter. Setting G3 and F to be horizontal defines the symmetry. One has, for in- stance

s X2 = - Q x2(1,0) + X X(1,0)'Qx. (41)

The introduction of p-forms Xp in interaction with the fundamental one-form X should in general modify the cohomology of the BRST operator s, a phenom- enon which should play a role when supersymmetry is introduced, in view of the anomaly cancellation mechanism.

It is a pleasure to thank A. Neveu, H. Nicolai, R. Stora and P. West for useful discussions.

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Volume 171, number 1 PHYSICS LETTERS B 17 April 1.986

Note added. While comple t ing this let ter we re-

ceived a paper by Witten [6] where some geometr ica l

considerat ions for the interact ing case leading to

results analogous to ours are made.

References

[1] A. Neveu, H. Nicolai and P.C. West, Phys. Lett. B 167 (1986) 307.

[2] W. Siegel, Phys. Lett. B 151 (1985) 391,396. T. Banks and M.E. Peskin, preprint SLAC-PUB-3470 (1985); M. Kaku, CCNY preprint (1985);

W. Siegel and B. Zwiebach, Berkeley Report UCB-PTH- 85/130 (1985); H. Hata, K. Itoh,, T. Kugo, H. Kunitomo and K. Ogawa, preprint RIFP635-KUNS814 (1985).

[3] L. Baulieu, LPTHE preprint 84/04, Carg~se Lectures (July 1983), in: Perspectives in particles and fields, ods. J.L. Basdevant and M. L6vy (Plenum, New York), Architecture of fundamental interactions, LPTHE pro- print 85/43 (1985), in: Proc. Les Houches Lectures (1985), eds. P. Ramond and R. Stora, to be published.

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