Page 1

Nuclear Physics B307 (1988) 348-364

North-Holland, Amsterdam

OPEN BRST ALGEBRAS, GHOST UNIFICATION AND STRING

FIELD THEORY

Laurent BAULIEU

Laboratoire de Physique Th~orique et Hautes Energies, Paris, France

Eric BERGSHOEFF* and Ergin SEZGIN

International Center for Theoretical Physics, Trieste, Italy

Received 7 March 1988

Geometrical aspects of the BRST quantization of charged antisymmetric tensor fields and

string fields are studied within the framework of the Batalin and Vilkovisky method. In both

cases, candidate anomalies which obey the Wess-Zumino consistency conditions are given.

1. Introduction

The quantization of actions invariant under gauge symmetries with a BRST

algebra which closes only up to classical equations of motion [1] has recently

attracted some interest [2-5]. In order to associate to such symmetries a nilpotent

BRST operator, Batalin and Vilkovisky have developed a lagrangian formalism

which builds on earlier developments in a hamiltonian framework [6]. They associ-

ate for each one of the classical, ghost and antighost fields, collectively denoted by

q~, an anti-field if*. This doubling permits the construction of a nilpotent BRST

operator, s, which acts on all fields, ~ and q~*, together with an s-invariant local

action S(q~, ~*(~,)). The anti-fields are not quantum fields. They are to be eliminated

through the choice of a local gauge function + a(~) by using in the action and the

transformation rules the constraint ~*= ~-1/~¢. After elimination of the anti-

fields one finds an action, S(~, ~*(~)), which is gauge fixed and BRST invariant,

and contains, in general, higher-ghost interactions. The elimination of the antifields

leads to new BRST transformations which are not necessarily nilpotent. However,

Batalin and Vilkovisky have shown formally that only the unphysical quantities

depend on the choice of qj-l(¢), and that the quantum theory generated by

S(~, q~*(~)) leads to a unitary and gauge invariant S-matrix provided the Ward

identities which follow from the new BRST invariance can be enforced order by

* Supported in part by INFN, Sezione di Trieste, Trieste, Italy

0550-3213/88/$03.50©Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

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L. Baulieu et al. / String field theory

349

order in perturbation theory [1]. In general, the action S(q~,~*(~)) cannot be

obtained by a Faddeev-Popov type procedure.

The construction of a nilpotent BRST operator by doubling the degrees of

freedom is reminiscent of a trick which has been widely used in mathematics, in

particular by Alain Connes in the framework of noncommutative geometry. Con-

sider for instance a non-nilpotent square matrix Q. Clearly the 2 x 2 block matrix

is nilpotent. Thus by a doubling of the representation space, a non-nilpotent

operator has been transformed into a nilpotent one.

The aim of this paper is to show that, despite the doubling of the degrees of

freedom in the Batalin-Vilkovisky formalism, there exists a geometric structure

which is of the same type as that encountered in the ordinary BRST formalism for

the gauge theories of forms [7]. By geometric structure we mean the formulation of

the BRST transformations as curvature constraints in an enlarged space such that

the classical and ghost fields are unified, a feature which is allowed by defining a

grading which is the sum of the ghost number and the ordinary form degree [7]. A

well known motivation for such a geometrical formulation is that it simplifies the

classification of anomalies through descent equations [7, 8]. In this paper, we shall

show that the geometric formulation does indeed exist in two examples: the

Freedman-Townsend model [9] which is the simplest gauge theory of a charged form

with degree larger than one, and string field theory [10-12]. In particular, in the

former example we will find candidate anomalies, and in the latter example we will

be led to a reinterpretation of string fields with negative ghost number, which have

been previously interpreted as antighosts, as anti-fields and to the introduction of

new degrees of freedom which are the true antighost string fields.

2. The charged antisymmetric tensor

We consider the case of a 2-form gauge field B 2 -

Lie algebra f¢ of a given Lie group, and possibly coupled to a 1-form field

A = A~dx ~ also valued in f~. The spacetime dimension d is chosen to be 4. The

following classical action has been introduced by Freedman and Townsend [9]

1

- B~ 2 dx A dx ~ valued in the

= f d"x Tr( eu'°"B,,Foo + ½A~A ~),

(2.1)

where F = dA + A A A is the field strength of A and the trace is over the adjoint

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L. Baulieu et aL/ String field theory

representation of the Lie group. The field equations are

F~ = O, 6~P°D~Boo + A ~ = 0. (2.2)

By solving the field equation F = 0 one obtains A = g ~ dg. By substitution into

(2.1) one finds the classical equivalence between the Freedman-Townsend model

and a sigma model on a group manifold. I d is invariant under the following gauge

transformations

8B..= D[j,e.] = 0[,6~1 + [A,, e~],

8A,=O,

(2.3)

where % is a local revalued infinitesimal 1-form parameter. Notice that the field

strength G = DB 2 is gauge covariant modulo the field equation of B:

6G=[F,e.dx~'].

(2.4)

Although the structure of the gauge transformation (2.3) seems to be an Abelian

one, the gauge algebra is degenerate since % = D~6, for any re-valued scalar parame-

ter 6, is a zero mode of the gauge transformations modulo the field equation F-= 0.

Before the work of Batalin and Vilkovisky the quantization of Icl has been

studied [7,13,14], the conclusion being that the following spectrum of g-valued

fields is necessary

B o

B{ 1

BO a ,

uo

B ° B1 Bg

Ho I '

A. (2.5)

In this notation, the upper label shows the ghost number g and the lower label

indicates the degree l of the form. The sum of both labels is the total degree of the

object. The exterior derivative d = dx~ 9/Ox~ and the BRST operator s have g = 0,

l = 1 and g = 1, l = 0, respectively. Therefore both of them are odd operators.

Furthermore they are assumed to anticommute, (sd+ ds)= 0. The baseline of the

large triangle in (2.5) contains the main fields: the original classical field, its ghost,

and the ghost for ghost. We call this sector the geometric sector. The other fields are

not geometrical in the sense that they belong to the antighost sector, and are in one

to one correspondence with the Stueckelberg type fields H displayed in (2.5). The

latter ones will be used as Lagrange multipliers to impose gauge conditions on the

2-form gauge field B 2 and its 1-form ghost B~ and antighost B ll.

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L. Baulieu et al. / String fieM theory

351

The on-shell "BRST operator", s 0, which is nilpotent modulo the field equation

F = 0, is defined by

soB°2 = - DBI, soA = O,

oBl= -DBg,

soB ~ = 0;

soB11 = 1I O, So-Fl 0 = O,

SoBo 2 = 110 1, SoJ-I 0 = O,

SoB ° = ri 1, ,oWo = o.

(2.6a)

(2.6b)

One can verify that s~ = 0 on all fields except B ° for which 2 0

the geometric sector it is meaningful to define the following unified object

soB 2 = -[F, B2]. For

- B ° + B 1 + B0 z. (2.7)

Note that the total degree, l + g, of each field on the right-hand side is equal to 2. It

is also meaningful to unify the operators s o and d into s o + d [7]. One can thus

define the generalized curvatures

N- (d+ So)~+ [A, B] =~, (2.8)

o~=_ ( d + so)A + ~[ A, A].

(2.9)

Using the assumption that ds o + sod= O, one has (s o + d) 2= s 2, and the Bianchi

identities for o~ and N read

(d+ s0)~¢= So~+ [~, ~1,

(d+ s0)~= s2A + [~, A]. (2.10)

One can easily verify that the geometrical BRST transformations (2.6a) are equiv-

alent to the following constraints on the generalized curvatures

ffl= G--- dB 2 + [A, B2],

~-= F = dA + }[A, A], (2.11)

and the breaking of the nilpotency of s o is immediately seen by inserting the BRST

equations, in the form (2.11), into the Bianchi identities (2.10) and isolating the

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L. Baulieu et al. / Stringfield theory

terms with ghost number 2. One obtains [7,14]

sgB2= -[F, B2].

(2.12)

Thus, as announced earlier, the BRST operator is nilpotent only modulo the field

equation F = 0. This forbids the construction of a BRST invariant gauge fixed

action by addition of a gauge fixing term of the form s o (something) to the classical

action. In ref. [14] a modification of the gauge symmetry has been proposed to cure

this problem. In this paper, however, we shall follow the method of Batalin and

Vilkovisky.

In accordance with the Batalin-Vilkovisky formalism, we thus introduce anti-fields

for all the fields collected in the large pyramid of (2.4) as follows:

B° ~ B2 a, B~I__> BO,

B~ --) Bf 2,

Bo 2 --) B 1 ,

B 2 ~ B4 -3, B ° ~ B41 . (2.13)

It is suggestive to display all fields and anti-fields in a diagram as follows:

B; 1 B° o

B 1 B ° B{ 1 Bo 2

B£ 3 B; 2 B~' B ° B1 B~

(2.14)

The sum of the ghost numbers of any field and its anti-field is -1, in accordance

with the Batalin-Vilkovisky formalism. By duality we have chosen the form degree

of the anti-field of a field with form degree l to be d- l, where d is the dimension

of spacetime. Now that the total degree of each anti-field in the geometric sector is

1, it is natural to combine them with the gauge field A, which also has this property

(since A can be rewritten as A°), to define the unified object:

,zC= A + B21+ B32 + Bf 3 .

(2.a5)

The generalized curvature now reads

~- (d+ s)~+ [d, 21 =~,

~¢= (d+ s)d+ ~[A, ~1,

(2.16)

where the BRST operator s will now act as a differential operator on all fields and

antifields (with the assumption sd + ds = 0), and will be shortly defined such that it

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L. Baulieu et aL / String field theory

353

is nilpotent without the use of any field equation. To see how this can be done by

imposing constraints on the generalized curvatures, we consider the Bianchi identi-

ties they obey:

(d+ s)¢= s2.~+ [.~, .~],

(d+s)o~=s2d+

[~, d]. (2.17)

We see that, in order to have S 2 = 0 on all fields and anti-fields, it is necessary and

sufficient to impose the constraint that all components in ~ with positive ghost

number are zero, and that ~ vanishes:

~3gg = 0, for g > 0, J~= 0. (2.18)

These equations are the "off-shell" generalization of (2.8). By their expansion in

ghost number one obtains the following nilpotent BRST transformations

sB°= -DB I- [B2-1, B2],

- Bg,

sB2o = 0,

sA=0,

sB 2 a = F,

sB3 2 = - DB~,

sB43=-DB~-~[B;1, Bzl].

(2.19)

The BRST transformation rules for the nongeometrical fields, including the Lagrange

multipliers, are as before (see eq. (2.6b)). The ghost number zero part of ~ defines

the physical field strength G which contains the anti-fields as follows

G=G+[B],B;

1] + [B2, Bf2].

The ghost number zero part of the first Bianchi

interesting result that G is BRST invariant:

identity in

(2.20)

(2.15) gives the

We now turn to the construction of a BRST invariant action S which depends on

the fields and anti-fields. Since we have constructed a nilpotent BRST operator s

acting on the fields 0 and ~* the theorem of Batalin-Vilkovisky [1] guarantees the

existence of an s-invariant action S(~, 0") with

as[,,**l as[,,**l

sq~ , s(~* = , sS[~, ~'1 = 0. (2.22)

s(~ = O. (2.21)

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An s-invariant action is then

S=Igeomet~i~al+ fTr[BO3AH°+B4tAIIto+BIAHo: ] ,

(2.23)

where /geometrical is the part of the action which depends only on the geometrical

sector and is given by

/geometrical = Icl ( A, B °) + f Tr[ B~-: A (DB 1 + ½[ B g, B~-'I) + 832 A DB2o]. (2.24)

In order to obtain eventually Feynman type gauges, one may add to S the following

trivially s-invariant term

IF = f d4x(~aH~H ~' + flHoaHlo),

(2.25)

where a and fl are arbitrarily chosen gauge parameters.

The next step in the Batalin-Vilkovisky formalism is the elimination of the

antifields by introducing a gauge function +-1(~) of degree 0 and ghost number

-1, and imposing the constraint ~* =6~-:/6~.

gauges for the 2-form gauge field and its 1-form ghosts we choose

In order to obtain covariant

+-'= 8; + 0 8o) + 80-2(0 8 )

(2.26)

Notice that the construction of ~ : necessitates the existence of the antighosts, i.e.

the introduction of the nongeometrical sector. With the above choice of the gauge

function the anti-fields are thus constrained as follows

B21 = dB11 ,

-- e~poO B o ,

Blup ° T

I

O•tJ•O

~ r -- 1

e~ooO B r ,

B2o = ~.~oo( a,B TM + aw°),

B; 3 = 0. (2.27)

Using these equations we can eliminate the anti-fields in the action (2.23) and the

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L. Baulieu et al. / String field theory

BRST transformation rules (2.19). This yields

355

S ( eo , 4, * = 6 + q, = Trfd'x[½A.A" + eu"°"F~.Boo + Dt.B~IO[.B~-]I

+eu"O"B~[ O.B~ -t, OoB~ -1] + O°Bo2DoB 2

-~ H~,( Ot~BP~' .Jc c)~eg ) @ HolOP.B 1

s_B 2 = - DB 1 - [riB{ 1, Bg], s_A = O,

(2.28)

"-BI = -DBg,

s B~ = O,

= HOl , _n°l = o,

sBo 2 = Ho 1, s/]-o 1 = 0,

_sB ° --- H i, _sH 1 = 0, (2.29)

with _sS(4~, ¢* = 6# 1/8~) = 0. The action (2.28) is gauge fixed (i.e. all propagators

are defined) and it is invariant under the BRST transformations (2.29). This action

is characterized by the presence of cubic ghost couplings. It is the same as that

derived by De Alwis, Grisaru and Mezincescu by using a Noether inspired method

[2] and has also been given in [15]. The authors of [2] have checked the formal

unitarity theorem of Batalin and Vilkovisky by investigating the unitarity of the

physical sector of the perturbative theory generated by the action (2.28), through a

generalization of 't Hooft and Veltman's diagrammatical method [16].

As an application of the geometric structure of the BRST transformations which

we have introduced, we can determine consistent anomalies, i.e. we can find

solutions of the Wess-Zumino consistency equation [7, 8]:

S~14( Bg, O -1 g) -,~- d~?2= O.

(2.30)

One solution to this equation follows from the existence of the following object

~6 = Tr(~P',~) , (2.31)

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L. Baulieu et al. / String field theory

which is algebraically equal to

~26 = (d + s ) Tr(d.~cd.~), (2.32)

provided that the Yang-Mills gauge group has no third rank invariant tensor with

two indices that are symmetric. It is indeed obvious from (2.15) and (2.18) that ~26

has a vanishing component with ghost number 2 so that the component with ghost

number 2 in (2.32) determines ~2] and ~2~ as follows

~14 = Tr( AAAB~ + 3B2XAAB2 ) ,

~2 = Tr( AAAB 2).

(2.33)

Since sB,, = O[~B,I + .... the potentially anomalous diagram corresponding to ~l 4 is

simply (T(AL,A~ApO'BoI,)Ip v For anomaly freedom it is necessary that the %~po

structure in this Green function vanishes. Note that even in the absence of chiral

fermions such a structure can be generated through the Feynman rules, since the

action contains the parity odd term B A F. After the elimination of the anti-fields,

£2~4 = Tr(AAABI) remains the solution of the consistency equation for the operator _s

modulo the equation of motion F = 0.

Finally we remark that the BRST quantization of the Freedman-Townsend

model with a charged 2-form in 4-dimensional spacetime which we have per-

formed above can be easily extended to the case of similar models with a

charged p-form in (p + 2)-dimensional spacetime. It is also possible to consider the

case when the 1-form A is a gauge field. The starting point is then the action

j Tr(F~F~'~dP+Zx + Be A F) with the gauge invariance 8Au = Due, 8Bp = Dep_ 1.

The spectrum of fields and anti-fields for a p-form gauge field is displayed in fig.

la.

Bd 1 B (}

Ba 1 B~ } 1 B] 1 B0 2

B~ ' ~ ... B~d_,,+2 B~' +~ B,; ~l Bp2 2 ... B,;P

B;,'-I " a; ? r,i ... B(' B,;

Fig. la. The field and anti-field spectrum for the p-form gauge field. A vertical line between B -1 and

B ° would be a symmetry axis for the fields on the right hand side and the anti-fields on the left hand

side. The base line represents the geometrical sector. The upper half plane contains all the antighost fields

and their anti-fields. The figure is for even p. For odd p the top line would be (Bo o Bj 1) instead of

(Bd I B°), The objects with positive ghost number g are fields on even layers and anti-fields on odd

layers.

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3. String field theory

357

3.1. THE GEOMETRICAL SECTOR AND THE ALGEBRAIC STRUCTURE OF THE NILPOTENT

BRST SYMMETRY

Open string field theory has a quasi-Yang-Mills structure in the free as well as the

interacting case [10,12]. Moreover, it bears some resemblance with the theory of

charged forms because it exhibits the phenomenon of ghost for ghost and also

because in the interacting case the gauge transformations admit zero modes modulo

a field equation which is a vanishing field strength [17]. The problem of building a

nilpotent BRST operator in the interacting case has been addressed in ref. [17]

where some auxiliary fields have been introduced, and also in [3, 4] where Bochichio

and Thorn have applied the Batalin-Vilkovisky method. We will study again this

problem in the light of a geometric analysis similar to that we have used for the

charged 2-form. We will see in particular that the usual interpretation of the

antighost sector of the theory has to be revised.

In order to establish a covariant formalism of the free string field theory, it is

known [10,11] that one must introduce all string fields occurring in the expansion of

a functional x[X~(o), c(o), ~(o)] where X ~ denotes the string coordinate, and c,

are a pair of ghost coordinates. The expansion of X in ghost number determines a

set of string fields Ag[X"], where the ghost number g varies between -oo and

+ oo. A ° is the classical string field, A 1 is the primary ghost, A 2 is the ghost for

ghost, etc. The ghosts A g with negative g have been interpreted as antighosts, while

the Lagrange multiplier string field Hg+I(x") has been associated with each

"antighost" A ~ (g < 0). The following set of fields were therefore considered as the

fundamental fields of the string field theory:

• .. A -g ...

A 3

1"I 2

A-2

H-1

A-1

1-io.

A 0 A 1 A 2 ... Ag ...

... H -g+l ...

(3.1)

The classical action for open string field theory is [11,12]

= f A° * Q A° + Z A ° 3 * A° * A°,

Ic1

(3 .2)

where Q is a nilpotent operator representing the Virasoro algebra in 26 dimensional

spacetime. The associative graded product * and the integration symbol f have

been constructed in [11]. All fields A g have odd grading, say 1. The auxiliary fields

H g have grading 2, and Q has grading 1. The basic properties are: Q(anything) = 0,

and Q(X* Y) = (QX)* Y+_ X*(QY), where the minus sign occurs when Y has odd

grading.

Id[A ] is formally identical to a Chern-Simons form and is invariant under the

following gauge transformation

6A ° = Qe + [A °, e], (3.3)

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where we have used the notation

[A, B] =A* B + (--)abB* A,

(3.4)

where a, b are the gradings of A, B, respectively. From (3.3) we see that e of the

form Dg, for any e', is a zero mode of the gauge transformation, modulo the field

equation F = QA ° + A ° * A ° = 0. But e' itself admits a zero mode of the form De"

for any e", modulo the field equation F = 0, and so on. Hence we see the similarity

between the string field theory and that of the charged forms which we treated in

the previous section.

A further similarity of the string field theory with the theory of charged forms is

the breakdown of the nilpotency of the BRST transformations when the string fields

A g(g >~ 0) are considered as building the geometric fields. In this case the BRST

symmetry for the geometrical sector is defined as follows [17]

So A°= -QA °- [A °,All,

soAI=-QAI-[A°,A2]-I[A1, A1],

g+l

~ An,A g+l-n ,

n=0

soAg= -QA g+l-

while for the antighost sector one has

soAg = H g+l, So1-lg+l = O, g < O.

In the geometrical sector it is natural to define [17]

g=O

if= (Q + s0)d+~¢ • d.

Then, the geometrical BRST equations (3.5) can be rewritten as

~=F.

Assuming that Q

(3.5)

(3.6)

(3.7)

(3.8)

and s o anticommute, one has (Q + s0) z= s 2 and the Bianchi

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359

identity which is the consequence of the associativity of the star product gives

(Q + So)~= SoZ.~¢ - [~, Y],

(3.9a)

from which, by using (3.8), it follows that

(Q + so)F= s2~ - [zJ, Y].

(3.9b)

By isolating in (3.9b) the terms with ghost number >~ 2 one obtains [17]

s2oAg=[Ag+2, F],

(3.10)

which means that the operator s o is only nilpotent on-shell, i.e. for F = 0.

In order to cure this problem one can introduce anti-fields. But such anti-fields

are already present in the theory if one abandons the interpretation of the fields A g

for g < 0 as antighosts. We shall therefore change the geometrical unification (3.7)

into the following one

+oo

Y'~ A g. .~= (3.11)

g= --oo

In this equation A - g- 1 stands for the anti-fields of A g for any g >/0. The price one

has to pay for this identification is that new objects will be needed for playing the

role of the antighosts. The antighosts are needed in particular for writing a gauge

fixed BRST invariant action in a general gauge. The field strength of the unified

object ~ is

1 ^

(Q + (3.12)

where s is now expected to be an operator which is nilpotent independently of any

field equation, thanks to the presence of the anti-fields in ~. The Bianchi identity

reads

(Q + s)~=sZA + [~¢, ~]. (3.13)

In order to have s 2 = 0 on all components of ~, it is clear from the last equation

that the definition of s must be

~= 0. (3.14)

By substituting (3.14) into (3.12), the transformation rules can be read by a

straightforward expansion in ghost number. For the classical string field one has

sA°=-QAI-[At, A°]-½ Y~. [A-g+l, Ag].

g>0

(3.15)

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L. Baulieu et al. / String field theo O,

If one sets in this equation all the anti-fields equal to zero one recovers the

definition (3.8) of the "on-shell" BRST operator s 0. Moreover, one has the following

generic formula for the action of s on all fields and anti-fields A g, equivalent to

(3.14):

+oc

Y~ A"*A g+l-'.

n= -oQ

sAg=-QA g+l-

(3.16)

One can verify that s 2 = 0 directly from this equation. In fact, this is guaranteed by

construction.

3.2. THE FIELD AND ANTI-FIELD DEPENDENT INVARIANT ACTION

Since we have built the operator s with s z = 0, an s-invariant action S[A g] must

exist, the equations of motions of which yield the definition (3.14) of the BRST

symmetry. The following action

l[,s~]=-sf(~*O~+ 2 ^ •

~'* ~ ~)~=o,

(3.17)

is such that for any value of g

81

8A -g-1

_ _ -ss#g= -(Qja?+~,A) ~+1. (3.18)

Eq. (3.18) is equivalent to the definition of s in (3.14). Now we employ the general

theorem of Batalin and Vilkovisky [1] which states that if an action S(¢,¢*)

satisfies

8s(¢, ¢*)

S¢-- - - - - ,

8¢*

8s(¢,¢*)

8¢

S¢* --

S 2 = 0, (3.19)

then, sS(¢,¢*)=0. Therefore the action (3.17) is invariant under the BRST

transformations (3.14). Another more direct construction of the s-invariant action

(3.17) is the following one. One has the identity

0=fo~,~=f(e+,),[~(e+,)~ +2~'*~^ *~l

=,f[~,(Q +,)~+ ~ ^~ • ~ • ~] . (3.20)

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361

Besides, we already know that s 2 = 0 and thus:

-- fo[ ,o~+ 23 s¢ ^* ~) * ~] =0 . (3.21)

One has therefore identically

sf[~*Q~+23d

^, ~, ~] = 0. (3.22)

This equation proves that not only the part with ghost number zero in

f[~,Q.~+ 2~, ~,.~] is s-invariant, but also that all components of any

given ghost number are s-invariant. The existence of the s-invariant term with

ghost number 1, f[.~ * Q..~+ 2~, .~, ,-~]g=l

s-invariant terms, might be a signal of a potential anomaly in string field theory.

The existence of anomalies in string field theory has been suggested in [18].

Before the elimination of the anti-fields, it is amusing to observe that both terms

in (3.23) are separately s-invariant. This can be easily seen by writing

= - if s~ * ~

* J~]g=l modulo

,'[ =

•

= (3.23)

where we have applied the definition of the BRST symmetry o~= 0. It is indeed

obvious from (3.23) and (3.21) that since I is s-invariant, f(..~ * ~ *-~)g=0 is

s-invariant by itself, and thus also f(s~ * Q-~)g=0-

3.3. THE ANTIGHOSTS AND THE ELIMINATION OF THE ANTI-FIELDS

So far we have built a nilpotent BRST operator which acts on the fields and

anti-fields. Furthermore, we have constructed an invariant action depending on

these fields. In order to eliminate the antifields, and arrive at a gauge fixed BRST

invariant action, a gauge function q;-x must be introduced which has ghost number

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L. Baulieu et al. / String field theory

: : : : : :

g-1

An+l

1 o 1 An+21

Ang+l

A,, g-I

...

An+ 1 An+x An+l

An2

An 1

A2g ,..

A o A1

.. Af Af -~ ' ~I

A~

A 1 .. Agl A-g

... A 2

Al 1 A~ 2

A o A 1

A Jr A2+~{ 1

Agn -1 A g

A~-g A~g 1

Ag -1 A g

Fig. lb. The ghost spectrum for the string field. The base line stands for the geometrical sector. The

upper half plane contains the antighost string fields and their anti-fields. Here, n is an even integer. This

figure is formally an infinite extension of fig. la.

- 1, and which does not depend on the anti-fields. This necessitates the introduction

of anti-ghosts, i.e. fields with negative ghost number, together with their anti-fields.

This means that we must fill the "upper half plane" on top of the base line which

represents the geometrical objects A g stemming from the ghost expansion of the

unified string field ..~, as displayed in fig. lb*. The figure shows that the field

spectrum of string field theory is a formal generalization, for p ~ oo, of the field

spectrum which is displayed in fig. la and which is relevant for p-form gauge fields.

Since ~ itself is obtained from a functional x(X ~, c, g), we are thus led to

introduce an infinite set of functionals xn(X ~, c, ~), where 1 ~< n ~< o0 stands for

the antighost number. By expansion of each one of these functionals on a basis

of ghost creation and annihilation operators one then obtains all the antighosts

A~( - oo ~< g ~< oo, 1 ~< n ~< oo) which fill the upper half plane as displayed in fig. lb.

This new set of string fields and anti-fields must of course be completed with

Stueckelberg type fields/-/,g with the following transformation rules

sA~ = Fig +t, sH~ +1 = O. (3.24)

As in the case of p-form gauge fields, the Stueckelberg fields /7~ are in one to one

correspondence with the fields A~ g of the antighost sector.

In earlier works [3,10,11] the existence of antighost string fields had not been

pointed out. It is in fact possible to obtain the light-cone gauge and the Siegel gauge

by starting from the eovariant formulation of string field theory and considering

only the field spectrum shown in (3.1). The possibility of not considering the full

spectrum which includes what we call the true antighosts and their anti-fields, and

nevertheless obtaining consistent actions in the above mentioned gauges, is com-

parable to the possibility of writing the Yang-Mills action in an axial gauge,

* These antighosts had been introduced in [i2] from other considerations based on the possibility of

defining an anti-BRST operator in string field theory.