Alternative ghost structures for superstring field theory

Nuclear Physics B335 (1990) 589—609 North-Holland

ALTERNATIVE GHOST STRUCTURES FOR SUPERSTRING FIELD THEORY* Gary KLEPPE and R. Raju VISWANATHAN Department of Physics, University of Florida, Gainesville, FL 326]], USA Received 13 March 1989

The superfield bosonization of Martinec and Sotkov, and Takama is analyzed in terms of the representation theory of the super-reparametrization algebra. A large set of super-reparametrizatiofl invariant operators is shown to exist in the superbosonized theory. A four-parameter family of invariants is identified whose form resembles that of the BRST charge. The condition of nilpotency identifies the true BRST charge. The picture-changing operator in the new representation is also constructed.

I. Introduction In a recent paper [1], we have initiated a program for the construction of a string field theory which has as its fundamental principle invariance under the reparametrization group [2]. In the free case the theory has been explicitly constructed. General prescriptions were given for the construction of operators invariant under reparametrizations and super-reparametrizations. The requirement of reparametrization invariant equations of motion in string field theory was found to lead uniquely to the BRST operator. A catalog of other invariant operators which exist in the free theory was constructed in both the bosonic and supersymmetric string theories. In this approach, the nilpotency of the BRST operator is not required a priori. The operator is constructed as the unique generalization of the Klein—Gordon operator to a dynamical string field operator which is reparametrization invariant both before and after normal ordering. In the bosonic theory, the invariant operator used as the equation of motion may be constructed from the left-moving part of the generator of reparametrizations by multiplying by a measure with the appropriate transformation properties. The new fields introduced to construct the measure are identified with the Faddeev—Popov ghosts of the first-quantized theory. The split-up into left and right movers introduces a possible c-number anomaly in the algebra of the left- (and right-) moving *

This work has been supported in part by the Institute for Fundamental Theory and the United States Department of Energy under contract DE-FGO5-86-ER40272.

0550-3213/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

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generators. The equation of motion is invariant only if this anomaly cancels between matter and ghost contributions. The construction in the supersymmetric theory is similar. The representation theory of the super-reparametrization algebra is more complicated. The usual superstring ghosts form linear representations of this algebra. It is often convenient in superstring field theory, however, to work in terms of bosonized ghost fields. The usual bosonized ghosts in superstring theory [3] transform nonlinearly under superreparametrizations. It would be more useful for calculations in superstring field theory to find a bosonization scheme in which the fields form linear representations of the algebra. Such a “superbosonized” representation was found in the work of Martinec and Sotkov, and Takama [4-6]. In this paper we analyze this superbosonization in terms of the representation theory of the super-reparametrization group. This superbosonization prescription is shown to be the unique one which preserves the commutation relations and transformation properties of the ghosts. The spectrum of L0 is found to contain states of arbitrarily negative eigenvalues in the superbosonized representation. However, it is possible to impose a subsidiary condition on the winding modes which restores equivalence between the usual and superbosonized ghost state spaces. We then consider the construction of invariant equations of motion for the string field theory. We find that a large class of invariant dynamical operators exists in the superbosonized theory. In particular, there is a four-parameter family of invariants whose coordinate part is of the correct form to be the BRST operator. The further requirement of nilpotency is necessary in order to uniquely identify the BRST charge. The resulting BRST charge is not quite the same as the one given in ref. [4]. The role of the other invariant dynamical operators in superstring field theory is not yet clear. The existence of the extra states in the spectrum of L0 corresponds to the well-known fact that the state space divides into spaces with different “picture numbers” corresponding to the existence of different “pictures.” The picture-changing operator provides a mapping between states in different pictures. The construction of the picture-changing operator using our techniques is given. We find a four-parameter family of weight-zero operators. The usual picture changer is a BRST-invariant combination of these. It is possible that other combinations of these operators are also BRST invariant; if they do exist, they would be of interest for constructing interacting superstring field theories. 2. Super-reparametrization representations in this section we review the representation theory of the super-reparametrization group which we will utilize in the following sections. The treatment follows closely our previous paper [1], and any details which are left out here may be found in that paper.

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A field a(a) (either commuting or anticommuting) is defined to transform covariantly under reparametrizations a ö with weight Wa if it has the transformation law —~

da

“v”

(1) For an infinitesimal reparametrization (2) the transformation law (~)becomes ~)F= —(fa’+waf’a).

(3)

The algebra of the reparametrizations (2) is [~~((i),

~g(~2)]

=

8fg~gf(C1f2).

(4)

Classically, if a(a) and b(a) are two covariant fields with weight Wa and Wh, then the product a(a)b(a) is covariant with weight Wa + Wh. The only other covariant bilinear combination which may be formed is wha’b w0ab’, which transforms with —

weight w~,+ W~,+ 1. Quantum mechanically the products of interest are normal ordered products; the effect that the normal ordering has on the transformation properties of the product must be examined in each case. Another representation of the reparametrization algebra is provided by a so-called “bosonized” field; that is, a commuting field ~(a) transforming inhomogeneously: da

-

da

(5)

or, for the infinitesimal transformation (2), 8~~)4=—~(f~’+wf’).

(6)

To extend the theory to the supersymmetric case we introduce super-reparametrizations ~ which are the “square roots” of reparametrizations in the sense that [~f(~1)’øg(~2)l = ~

(7)

[8f(o),~g(~)~

(8)

We find Jg’-f’g/2(~L

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The parameters ~ are anticommuting numbers. In the following the parameters of and will no longer be explicitly indicated. The super-reparametrization algebra can be represented by fields a(a) and b(a) having the transformation laws ~ja= —Jb,

~fb= —(fa’+2waf’a).

(9)

From eq. (7) we can see that under reparametrizations a and b transform covariantly with weights w~and Wa + With two such multiplets (a, b) and (c, d), the following products are found to transform according to (9): ~.

with weight

with weight

(ac,±ad+bc)

(lOa)

(waad~w~bc,±Waac’ ~ w~a’c+ (Wa + w~)bd)

(lob)

Wa + W~,

Wa

+ W~+

(lOc) with weight Wa + w~+ 1, where the upper (lower) signs are to be read when a is a commuting (anticommuting) field. Eq. (9) is the only irreducible representation of the super-reparametrization algebra whose components transform covariantly. However, a field 4( a) transforming according to (6) may be used, along with a weight- ~ anticommuting field which we will call s(a), to provide another representation ~f4~

—fs,

—(f~V+2wf’).

~jS

(11)

Note that this representation coincides with (9) in the case when w

=

0.

2.1. FUNCTIONAL GENERATORS

The effects of the super-reparametrization algebra on a functional 1~can be expressed as 6~F=—M~’k, where the functional generators Mf and

—i~~I1~,

~

~

(12a,b)

have the algebra 2EMjg,

[Mi, Mg]

=

lMfg~_f~g,

[M1, ~‘gj

= ~‘~‘fg’-f’g/2’

~

~/~t’g)

=

~ (13)

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The generators Mf and ~ may be written as integrals over a of expressions involving fields and functional derivatives

Mj=f~—f(a)M(a),

~j=f~-~f(a)~((a).

The algebra (13) is then satisfied if and only if the combination (~‘(a),2M(a)) transforms according to (9) with weights (~,2). For example, with a multiplet (4, s) as in (11), the generators have the form

M7s=

~s

da -if ~{(f~’

~ f’s

+ (fs’ +

+Wf’)~

~

~

(14a)

(14b)

This type of multiplet may be separated into so-called left- and right-moving pieces, which are defined by the relations

2 SR+SL

5=

2

,

~

0

~

2

(15a,b)

11

—=i—(sR—sL), Ss 2

(l5c,d)

where ~ ±1. All left movers commute with right movers. The generators Mf and ~ split into pieces containing only one type of mover: =

M1

~

Mt+M),

(16a,b)

The generators for a multiplet of the form (9) can be similarly split into left and right movers only if its weights are (0, ~). It is a remarkable fact that the string coordinates x~ and their superpartners form a multiplet with precisely these weights. Henceforth we will discuss only left movers, and it should be understood that 4~L is similar remarks will hold for right movers. The Fourier expansion of ~L(a)—~o—ap~+1~

~

(17)

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where the modes satisfy

~

~

[P4’4~m]=

(18a,b)

~

The ground state, as usual, is annihilated by 4~for all m > 0. The negative modes create states of positive norm for i~= + 1, or negative norm (ghostlike) states for = 1. Because the generators split into left and right pieces, we should expect to be able to build covariant quantities out of left movers alone. Indeed, we find that the normal ordered exponential :e~’L: transforms covariantly with weight a(w ~a/2). The algebra of the left-moving generators is —

—

[Mt, M~}=

=

iMjLg~_j~g + f~(Af’g +

Bf”g),

(19a)

da A -2iMj~-if ~(~fg+2Bf”g),

(19b)

where A and B are c-numbers. The integrands (~4~lL(a)2ML(a)) again form a doublet of the type (9), provided that A and B are zero. In this case, this doublet may be combined according to (lOa) with a doublet of weights 1, to form a quantity whose integral is invariant. In superstring one starts with thesuperpartners. weight (~,1) doublet (Ps,fields, x’s), 2 are the field stringtheory, coordinates and F’~their With these where x’ the algebra (19) has a c-number term B -id, where d is the dimension of space-time. (A, the other c-number in eq. (19), may be changed by a shift in zero modes, so it is not significant for our purposes.) In our previous paper we presented one possible scheme for cancelling this c-number and introducing a measure of the appropriate weight to construct an invariant. Superbosonization is another possible approach. In sect. 3 we apply the preceding formalism to the analysis of the superbosonized theory. (—

—

~)

3. Superbosonization It is well known [7—10]that the ghosts of bosonic string theory may be described either in terms of the anticommuting variables b(a) and c(a), or the commuting variable 4,(a). (Here and after, all fields will implicitly refer to left-moving pieces without an explicit “L” subscript.) The relations between these ~uantities are b=

:e~:,

c= :e~’:.

(20a,b)

G. Kieppe, R.R. Viswanathan

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4, satisfy

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the product relation / =

~

—2i sink

\\uh

-~-—~)): e~1)±(°2):.

(21)

Using eq. (21) we may invert eq. (20) as =

—i:c(a)b(a):.

(22)

It is not obvious that the Fock space created by the modes 4,, of 4,(a) is isomorphic to that created by the modes of the fermionic ghosts b(a) and c(a). There is a well-known proof of this equivalence, using Jacobi’s triple product identity to relate the partition functions. Here we give another argument. In either the fermionic or bosonized ghost representation, the full Fock space may be generated by acting with the Virasoro operators on a certain subspace which is referred to as the highest-weight states. These are defined to be those states which are annihilated by L,, for all n > 0. Acting with the other L’s (those with n <0) reproduces the full Fock space. The space of highest-weight states is labelled by the eigenvalues of normal-ordered operators which commute with all the L’s, i.e. which are reparametrization invariant. Using only the fermionic ghosts, the only such operator is the ghost number, defined as [7,91

NG=-f~:b(o)c(a):.

(23a)

With the bosonized ghosts, the only such operator is the zero mode

p4= _f~-~~’(a).

(23b)

These two quantities have the same eigenvalue spectrum, and in fact eq. (22) shows that they are actually identical except for a factor of i. Thus the space of highest-weight states is the same in both representations. The supersymmetric string theory includes additional ghosts /3(a) and y(a). (/3, b) and (c, y) form multiplets transforming according to (9), with weights (~,2) and (—1, -i,), respectively. Thus according to (lOa) we can form an invariant superghost number as —

da Nso=-:f——(/3y+bc):.

(24)

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The superghosts can be rewritten [3] in terms of the bosonized field X(a) and the fermionic variables ~(a) and ~(a) as follows: $=i~’e~,

y=iiex.

(25a,b)

An interesting fact about this bosonization of superghosts is that while (/3, b) and (c, y) form supersymmetry multiplets, the new variables 4,, x~~, and ~ do not; their transformation laws under supersymmetry are nonlinear. For instance, the field x’ transforms as (26) Thus the super-reparametrization invariance of the theory is no longer as simply implemented, and this may cause difficulty in some applications. An alternate bosonization of the conformal and superconformal ghosts has been introduced [4—6]which does not sacrifice the superfield structure of the ghosts. This bosonization is as follows: b=~iue°”,

c=üe’~

(27a,b)

y= :(~‘—aüu)e’~’:,

1 /3= —~-—ie~~.

(27c,d)

In eq. (27), (p, u) and ü) are supermultiplets transforming according to eq. (11). Under reparametrizations, q~ and ~ transform like bosonized fields, i.e. with an inhomogeneous term in as eq. (6). The two multiplets are defined to be conjugate to each other in the sense that (~,

fu(a1),~(a2)}=—2i~(a1—a2), (28a, b) [p(ai),i~(a2)]

{ u(a1), u(a2)}

=

0,

[~(a~), ~(a2)]

{ i~(a1),ü(a2)}

=0,

0,

(28c,d)

=0.

(28e,f)

=

Because all modes of p commute among themselves, the exponentials in eq. (27) have their classical weight, namely ±W~a,where w,~is the coefficient of the inhomogenous term in eq. (6). Since the weights of u and i~must be ~ because of (11), we must have w,pa ~. Also, we must have w~ a in order to maintain covariance of y. Then all the ghosts transform with the appropriate weights. =

—

=

—

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Ghost structures

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It is easy to partially invert eq. (27) to obtain i~=2ia$c,

(29a,b)

~‘=2ia:(y/3+bc):.

We would now like to derive eq. (27) using the results reviewed in sect. 2. Consider a pair of self-conjugate doublets (~, s) and (4,, t), with the transformation laws ØfW

=

(fw’

2wf’),

—fs,

¶~fS=

—ft,

Ø,~t=—(f4,’+2vf’),

0f41=

—

+

(30a,b) (30c,d)

where w and v are c-numbers. As before, we consider left movers only. The fields ~‘, 4,, s and t have the respective Fourier expansions w(a) =~ ~

(31a)

~ ~

(31b)

0—ap~+i ~ nO

,zo

s(a)

=

~

e’~,

e”~,

(31c, d)

[Pw,~,jm]r~~1~m,Q,

(32a,b)

[P~,~mI

(32c,d)

1(a) =

~

t,~

where the modes satisfy ~mSm,n,

[“m’”n}’

{ 5n, 5m}

=

,

{ t,~,i,,, }

=

=

~~m,O’

.

(32e, f)

The choice of signs in the commutations above is necessary in order to reproduce the ghost algebra. With this choice of signs, the total anomaly in eq. (19) is proportional to 2+v2. (33) B= ~(d+2) —w We now investigate the question of what quantities may be formed with these fields which will transform as the ghosts. The basic covariant doublets are (er””, ase’~) and (e”~, bte~), for any constants a and b. As usual, these doublets may be combined with the rules (10) to yield additional covariant doublets. There is

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no ordering problem at this stage. The results are e’~e~” (34a)

(as+bt)e~e~ ab[(w+ ~a)t—

(0—

~b)s}e’~e~”~

a(w + ~

+ ~a)+b(v—~b)]abste~e~

(34b) 2b(v — ~b)(e~)’e~_

2a(w

+

~a)e”~(e~”~)’ + abste’~e~

~

.

+ (2b(v

—

~b)

+

1)bt(e~)’e~—2ab(w

(34c)

+

The lighter components of these doublets transform with weights a(w + ~a) + b( v ~b) plus 0, ~, and 1, respectively. Since c is anticommuting, and the lighter component of its multiplet, its multiplet must be (34b) for some a, b. This has the correct weight if —

a(w+~a)+b(v-~b)

=

(35)

-~.

Then we must have (c(a,),c(a2)} =0, which will be true only if and (w+~a)2=(v—~b)2.

2=b2,

a

(36a,b)

Thus we see that a(w + ~a) b(v ~b) — ~. Then eq. (33) is satisfied if the number of space-time dimensions d is ten. We will choose a b. Then (removing overall multiplicative constants) =

—

=

=

c=(t—s)e~”~e~,

We obtain the conjugate doublet constants in (lOa):

(/3,

y=(4,’—u’+2ast)e’~e~.

(37a,b)

b) from (lOa) by taking the opposite value for the

$=ie’~e~9,

b=ia(s+t)ee~’~.

(37c,d)

To make the connection with eq. (27), we define the combinations q=~a+4,,

~=w—4,,

u=s+t,

Substituting eq. (38) into (37), we recover eq. (27).

i~=s—t.

(38a—d)

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We now turn to the question of whether the spectrum of states is equivalent in the superbosonized representation. Actually the question is easier to answer here than in the bosonic theory, since here both before and after bosonization, the theory possesses two fermionic variables and two bosonic variables. However, the superbosonized fields p and ~ both have invariant zero modes. We need to determine the spectrum of eigenvalues for these operators. The superghosts /3 and y satisfy ±13(a)(— in the Neveu—Schwarz boundary conditions of the form f3(a + 2ir) sector, + in the Ramond sector). From eq. (27d) we see that the eigenvalue spectrum of p,~which satisfy these conditions is p,~ i(2n + 1)/2a (NS sector) or in/a (R sector), where n is any integer. Since b and c must be single valued, the modes of u and ü will be half-integer in the NS sector and integral in the R sector. Finally, eq. (27c) does not put any constraint at this stage on the eigenvalues of the zero mode p,~.However, eq. (29b) shows that —(i/2a)p,~,is equal to the superghost number. The other invariant zero mode p,,~generates a set of eigenstates that are not present in the standard representation. Furthermore, in the superbosonized representation, the zero-mode Virasoro generator L 0 has the form =

=

~

=

-~ ~ ,, >0

(nü,u,, + flU,i~n+

+~

~

+

~

(39a)

By comparison, with the usual ghosts, 3—nYn =

,i

~>0 n( b..,,c~+ c~b,1+

—

y_~i3,,).

(39b)

/

Comparing these two expressions, we see that while eq. (39b) is bounded from below, eq. (39a), because of the term ~ is not bounded from either direction. Clearly, the space of states is different in the two representations. Some sort of truncation of the spectrum is therefore necessary if we want to have equivalent state spaces. By restricting our attention to those states in the theory which satisfy — 2a2p~,we get in L 0 a term proportional to N~G,which makes L0 bounded from below. =

4. Construction of invariants Let us now consider the question of what invariant operators exist in the superbosonized theory, in particular, the BRST charge [11, 12], which is normally constructed as a product of doublets. Neglecting ordering effects for the moment, we may combine (c, y) with the doublet (~‘(a), 2M’~(a))using the rule (IOa) to

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form the covariant doublet (P~1(a),Q~1(a)).Then we find P~1=(—3a(t—s)F.x’—~atsw’+~ast4,’ (40) 2 + FE’ =

—

~

+ 2w~”+ ~

3a(t — s)(x’ + {~a(w’—4,’) + 3a2st][F. x’ — sw’

—

2v4,”

—

S’s +

—

2vt’]e~e”~. (41)

+ 2ws’ + t4,’ —

Here ~v and v are the weight parameters of w and 4, respectively, given (from eq. (36)) by a 2

W—

—+—

3 4a

a 2

V

,

3 4a

Unfortunately, ordering eff poil the covariance of these quantities. We can remedy this problem by add.~ arms to Q in order to make it both nilpotent and truly super-reparametrization invariant. To find all possible invariant operators we will simply write down all possible covariant quantities and try to assemble an operator doublet (P(a), Q(a)) for which the integral of the heavier weight component Q = fdaQ(a) is invariant. It is simplest to first determine the form of the lighter weight component P(a) and transform it to get Q. We write down all possible terms .P~ which are weight ~ up to anomalies. The correct P will be some linear combination of these. We then require that P transform as the lighter component of a doublet of the form (9), and that the anomalies cancel. This is accomplished by demanding that the f” and f terms in and the f’ and f” terms in Ø 1P add to zero. These restrictions select out a four-parameter set of solutions. This calculation is described in appendix A. The result is “

Q=AQA+BQB+CQC+DQD,

(42)

where A, B, C and D are any constants and

QA=

—

~(2a2+9)Q1+3aQ2+3aQ3_2aQ4 2)Q

—4a(9+a

QB =

~(

2 —

2a

+

3Q 5+4a

3Q 7+4a

8—54aQ9,

(43a)

8—54aQ10,

(43b)

9)Q 1

3aQ2

—

3Q +4a

2)Q 6—2a(27+2a

—

3aQ3

2)Q 5+4a(9—a

+

2aQ4

3Q 6+4a

2)Q 7+2a(27—2a

/ Ghost structures

G. Kleppe, R. R. Viswanathan

601

Q~=_~(9+2a2)Qi_3(2a2_9)Q2_3(27+2a2)Q3 2)Q 2(2a2 — 9)Q 2(27 + 2a2)Q 2(27 + 2a 4 + 4a 5 4a 6 2(—27 + 2a2)Q 2(27 + 2a2)Q + 4a 7 — 4a 8 + 54aQ11, +

—

2)Q QD

=

—

~(2a2

—

2(2a

—

4a

+

— 9)Q1

3(—27

2— 27)Q

+

2(2a2— 27)Q 4— 4a

2(2a2

—

2)Q 2 + 3(9 + 2a

2a

2(9

+

2(27

+

3

2a2)Q

5 + 4a

27)Q

(43c)

6

2a2)Q

7 + 4a

8 + 54aQ12,

(43d)

where the Q.’s are defined as follows: 2+F’F)+(4,’—~’+2ast)x’F)e’~e°’~,(44a) Q1= —((t—s)(x’ Q 2

=

—a [s”

+

2as’u,’ + 2s(a~”+

Q3

=

—a[t”

+

2at’4,’

Q4= Q 5=

+

2t(a4,”

a2wf2)]e~e~~~P,

(44b) (44c) (44d)

+ ~

2[s’4,’+a(s+t)w’4,’+t’w’]e’~e°’~’, —a

II— atw’~+ ass’z’ + ~

Q6 = [astp< — astt’ t&i”

—

ass’t +

(44e)

at~~.,’4,’]e’”°e~,

(44f)

~

(44g)

ast’t]e”~e’~,

(44h) (44i)

=

[—

=

[—t’~’

=

[—sw” + s’w’ + asS’t}e°~’e”

=

[

—

+ s4,”

—

4, —

t4”

+ t’4,”

—

=

—

[s” + ast~.,”+ btw”]e°”’e’~,

=

—

[t” +

bt4,”

(44j)

astt~]ea~)e~,

(44k) (441)

+ as4,”]e’~e°”~.

We now need to investigate whether any of the invariant Q operators obtained from the above Q(a)’s are nilpotent. Consider a general Q of the form Q =

J~(2cM~(a)

-

y~(x~F(a))+

Q~.

(45)

602 2

Using eq. (19),

=

Q

G. Kieppe, R.R. Viswanathan / Ghost structures 0 is equivalent to the conditions

{Qgh,~)} [Qgh’Y(°)]

=i(~y(a)2_2c(a)c~(a)),

(46a)

=i(c’(a)y(a)-2c(a)y’(a)),

(46b)

iD

da (46c)

The expressions on the right-hand side of eq. (46a) can be evaluated in terms of the new ghosts; for instance, the first one is ifty2

—

2cc’)

=

i:(~’2 — a~’i~u — 2i~ü’— a~”+ 2a2I~~u)e2~:.

(47)

We find that eq. (42) satisfies these conditions for

A

=

B= 1,

~-,

C=

(9+14a2) —

,

24a

D=

(2a2+3) —

,

8a

(48)—(51)

so that this combination (up to an overall constant) is indeed a nilpotent operator. We can of course also derive the expression for the BRST charge in terms of the new ghosts by substituting for the ghosts in the old expression for Q and re-doing the normal ordering. In terms of the old ghosts, we have Qgh=

The terms in

~

I ,.da —:y(yb—2/3’c—3f3c’)—c(4c’b—yf3’—3$y’):. 2.’ 2~r

(52a)

Q expressed in terms of the new ghosts are =

(~~2u + 2au’~’— 2a2i~uu’— au~”)e~,

(53)

=

+(~cp’~’i~ + ~-iY’+ a~q”)e~,

(54)

cc’b = (~ai~cp” — ~a2ä~p’2 + ~üI~’u— ai~’p’),

(55)

y/3c’

(56)

y2b

=

~-

ii”

—

—

üq’~’+ üui~’— 2aüqY’ ea’t~,

a

1 /3y’c

=

—

—(i~” + ai~q’~’ — aI~’ui~)e~.

2a

(57)

All expressions on both sides of this equation are understood to be normal ordered.

/ Ghost structures

G. Kieppe, R.R. Viswanathan

603

For completeness, we give the final form for the integrand of the nilpotent operator Q: Q(a)

=

2üe~rMvl

—

(~‘ aüu)e~~/~’” —

2~uu’— au~”—

—

+

+

2au’~’— 2a

+ üü’u

~

—

~~“ü

—

2ü”

+

2a2ü~’2+ 4aü’~’)e~.(58)

We note that this differs somewhat from the expression given in ref. [4]. As another application of our methods, we consider the construction of the picture-changing operator. This operator has weight zero and is constructed as the anti-commutator of the BRST charge with the field ~(a) (see eq. (25)). The bosonized field x can be written in terms of the new fields as (this can be seen from the operator product y/l)

q~’

i~u

2a

2

(59)

so that the relation e5=üe~_i~~’Sa holds. Also, the fields

=

~‘

(60)

and ~ can be written as

~ue/~a, 2a

~

=

—(2au’ + u~’)e~2°.

(61),(62)

Since the picture changer has a term of the form exF. x’

=

i~e~/SaF. x’,

(63)

we can use our method to write down a general weight-zero operator with this term in it. We note that it is essential for the picture-changing operator to transform without any f’ or f” terms under ~, since all amplitudes calculated with it must of course be invariant under super-reparametrizations. We again find a four-parameter family of operators, this time of weight zero. The independent solutions are (with

G. Kieppe, R.R. Viswanathan / Ghost structures

604

the constant a =

1 for convenience)

=

~

+

P5 — ~

P~=P2+4P5

+

~

—

~P3 + ~

+

(64a)

~P6+ ~P7— ~P8+ ~P9+ ~P~0— ~

P~=P3+ ~-P5 — ~P~+ ~P~— ~P8+ 3D

~P10

(64b)

~P~+ ‘fP10+ ~P~2

(64c)

~

‘

4

+

9P5

+

—

~P7 — ‘~P8+ ~P9 + ‘~P10+~P1~+ ~P~2

(64d)

Here the P’s are given by 2)”e34/2, 2=(e~/ P 2)’(e3’~/2)’, 4=(e”’/

(65a,b)

P1=x’~F(t—s)e~’~ P 2(e3’~/2)”, 3=e’~”

2e3~/2,P

P

P

(65e,f)

2)’e3~/2, 5=st(e~~

P

2(e3~/2)’,

(65c,d)

6=ste”/

2e3~”2,

7 s’te’°” P 2e34~2, 9= :ss’:e’°~

P

2e3~~2,

(65g,h)

8=st’e’°~

=

P

2e3’~~2, (65i,j) 10= :tt’:e~~

P

2e3~2, P 2e3~”2. (65k,l) 11 w” e” 12 4,” e” Any combination of ~A, Pfi, P~and ~D is of course a weight-zero operator. By comparison of coefficients, the usual BRST-invariant picture-changing operator corresponds to the combination =

=

9228 10568 X=lOPA—---~—Pfi+ 603

48 (66)

We do not yet know if other combinations of these four operators exist which are also BRST invariant. It would be of potential interest to find these, if they do exist.

5. Conclusions We have derived superbosonization formulae in superstring theory by making use of the representation theory of the super-reparametrization algebra. The spaces of ghost states in the new and old representations are not identical due to the presence of extra zero modes in the superbosonized theory. This can be resolved by a suitable restriction of the Hubert space. The most interesting difference between the superbosonized theory and ordinary superstring theory is the existence of additional invariant dynamical operators other than the nilpotent BRST charge. The role of

G. Kieppe, R.R. Viswanathan

/

605

Ghost structures

these extra invariants needs further investigation. There also exists the possibility of constructing other picture-changing operators besides the usual one. We wish to thank Pierre Ramond for advisement and many useful discussions. Appendix A EXPLICIT CONSTRUCTION OF INVARIANTS

We now show the details of the construction of the operator P whose superpartner integrates to form an invariant Assuming that P is a commuting operator, it may be built as Q.

12

P(a)= >A1P~(a),

(67)

=1

where P1

=

x’

F(t

—

s)e’~e’°~,

P2

~

=

1’)’,

P3

=

e°°~(e~)”,

P

P4

4,

(68a,b) (68c, d)

(e°’~ )‘(e”

=

P

5_st(e)e

6=ste°”(e”~)’,

P7=stee,

P8=st’e”e~,

P9

P10

(68g,h)

: ii’ :e’~”e”~, (68i,j) 4~, (68k,l) P11 w” ~ P12 4,” e~~e0 For to be super-reparametrization invariant, P must transform with weight ~ since we have 2a(w + ~,-a) — ~, the exponentials ~ have total weight so these are the only terms P can have. As already stated, ordering effects will alter the super-reparametrization transformations of these quantities. Useful results are =

: ss’ :e’”~e’°~,

(68e,f)

=

=

=

Q

=

—

=

~,

: w’(a 1)e

(a2):

a

+ a1

—

a2

ea~02),

(69a)

eons),

(69b)

a =

s(a1)s(a2)

=

:4,/(a1)e0~~02): —

:s(a1)s(a2):

a1

—

a2

1 +

a1

—

,

a2

(69c)

1 t(a1)t(a2)

=

:t(a1)t(a2):—

a1

—

a2

.

(69d)

606

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G. Kieppe, R.R. Viswanathan

Ghost structures

These equations hold in the limit where a1 a2 0. The exact expressions of course involve periodic functions. To illustrate how the normal ordering affects the transformation properties, we show one of the calculations; —~

—

~(s

eaw)

[—

+ 2wf~(a))e0~1)—

(f(a)~’(a)

=

—f:~’e°°~:—2wf’e’~—a lim

=

s(a)( _af(aj)s(ai)e0~©o)]

f(a)—f(a1) a—a1

e0~~1)

-~

O1~~’O

_f:w~e0~0:_(2w+a)f~e0~.

=

(70)

Using eq (69) (and various derivatives thereof), we calculate the following results (all quantities are normal ordered): 2+F’F)

~

~fP1= —f((t—s)(x’

(71a) ~

—a(f{s” +2as’~,.,’+ 2s(aw” +a2~~2)]+2f’[s’ + asw’] ~

(71b) Ø~P 3 —a(f[t” + 2at’4,’ + 2t(a4,” =

+ a24,~2)] +

2f’[t’

+ at4,’]

+f~~t)e0~e0~, (71c)

ØjP4

—a~(f[s’4,’ + a(s

=

= (f

Ø1P6

[

—

~

(f [as4,’~

=

Ø~P7=(f[_t~”

+

ass’t

+f’[s4i’

+ t)~i’4,’+ t’w’] +

as&.,’4,’]

+

~f’(t

—

s)w’

—

asit’

—

atw’4,’]

+f’[~(t

—

s)4,’]

—

ass’t

+

s’4,’] +f’{_t~’

—

~s’J

tw’I)e’~e~,

(71d)

—

+af~~t)e0ue0~,

(71e)

—

~af~~s)e0c0e0~,

(71f)

+

+

+

(71g) =

(ii_t’w’

+

s’4,”

—

ast’tJ +f’{~t’ +s~’] +

—

(71h) Ø~P9=(f{_sw”+s’w’+ass’tI _f’[s~c’_ ~s’J

=

(f[ —14,” + t’4,’

=

—(f[s”

Ø~P12

=

—

(f[t”

—

+ asw” + +

astt’I _f’{t~’

—

hto.~”]+ 2f’s’ ~

bt4,” + as4i”]

+

2f’t’ ~

—

pt’] +

~

+

(~ —

~

(71i) (71j) (71k) (711)

G. Kieppe, R.R. Viswanathan

6.

/

Ghost structures

607

We also need to investigate the transformation of P under the reparametrizations We find P

=

—fP’

+

(

—

—

~f’P —f”[(

~A3

~A2

—

—

~A4 +

~A4 + _Ai2)e0~(e0~)t + (—

—

~A3

—

~A9

—

+

~A1()

—

~A6 + ~A7 + ~As)ste0~e0~

A11

+

—

+

—

~)

AilIe0~e0~. (72)

We require, in order to maintain covariance, that the f” and f terms in P and the f’ and f” terms in ~1P be zero. The general solution for the constants A, which satisfy these constraints is “

(73)

P=APA+BPB+CPC+DPD,

where A, B, C and D are any constants and

9 2+9)P —

—(2a

1+3aP2+3aP3—2aP4 3P 2)P 3P 5 + 4a 6 2a(27 + 2a 7 + 4a 8

2)P —

4a(9 + a

—

1—

3aP2— 3aP3+ 2aP4

3P +4a

—

2)P 5+4a(9_a

1 8a —(9

3P 6+4a

2)P +

2(2a2 +

54aP9,

2+9)P

9

P8= ~(—2a

=

—

2a

—

3(2a

2)P 2

—

2(27 + 2a2) P 5 4a 6 2a2)P 8+ 54aP11,

—

4a

2 —9) P 1

2(27

+

3(27 + 2a

9)P

—

—4a

(74b)

7+2a(27~_2a2)P8_54ap1o,

2( —27 +

2) P 3

+

+

2(27

+

2a

4

2a2) P

4a

7 (74c)

18a 2 9)P 2)P —~—(2a 1 + 3(—27 + 2a 2 2(2a2 27) P 2(9 + 2a2) P 4a 5 + 4a 6 2(27+2a2)P +4a 8+54aP12. —

—

—

—

27)P 4

—

—

—

2)P 2 3(9 + 2a 3 2(2a 2(2a2 27)P 4a 7

+ —

(74d)

G. Kieppe, R.R. Viswanathan / Ghost structures

608

The corresponding partners a) can of course be obtained from the transformations of the P ‘s; the integrals of the Q(o)’s are then invariants. We now give a list of other constructions which lead to invariant operators. First, we can use (34a) if we choose the parameters such that the product doublet has weight (-k, 1). This is written schematically as Q(

1’) (e°’~, ase0~~~) ®~(e~, bte”

(e”~e~,(as + bt)e0~)e~),

=

(75)

where a and b are chosen so that a(w + -ia) + b(v -~b) ~. Secondly, we can repeat the process used above to construct (P, but with arbitrary constants a and b such that the total weight is still (~-,l).We must redefine —

=

Q),

P 1=x’.F[(v—

~a)t]e0~~e~,

~b)s—(w+

(76)

and change the exponentials e°’~’to e~ in the other P’s. We can then follow a procedure identical to the one described above to make a doublet (P, The analysis follows closely the previous case, and we will give the result only for the particular case where A11 A12 0. Any operator will be invariant which is a linear combination of ~A and P8, where Q).

=

—

=

2a(w + ~a)(1 + b(v

+ b(a + 2ab(v

—

+ ab(l + 2b(v

—

~b) ~b)

—

~b))P3

2(1

+

+

+ 6(w + ~a))P5 +

—

12(w

b(v

~b))(1

—

2ab(1 + b(v

+

2b(v

—

—

+

(77a)

+2ab(l+b(v—~b))P8—12ab(w+~a)(v—~b)P9,

+2a(w+~a)(1+b(v—~b))P3—2(1+b(v—~b))(1+2b(v—~b))P4 2(v —

—

ab(1 ab(1

+ +

2b(v 2b(v

—

~b))P5

—

2a(b

+ 3(v

—

~b)

—

+b

—

_2ab(1+b(v—~b)—6(v—~b)2)P 8—12ab(w+~a)(v—~b)Pio.

(77b)

G. Kleppe, R.R. Viswanathan

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609

The expression for the ‘s can be obtained by taking the super-reparametrization transformation of eq. (77). For these operators to be invariant, it is necessary that a(W + ~a) + b(v ~b) It can be checked that with the restrictions a b and w + ~-a v +b, the operators ~A and ~B of the last section are obtained. It is also possible to generalize this type of invariant to Lorentz tensors. This is done simply by redefining the P’s once again: Q

—

=

=

—

~.

=

—

P/~=x’~F~[b(v—~b)s—a(w+ P/’= ~g”P

1,

(78a)

~a)t]e~e~,

(78b)

i>l.

Then we may once again follow the same procedure and construct the invariants as before. Finally, there is the possibility of constructing still more invariants by, for example, combining (34a) with (.~‘,2M) according to (lOa) and adjusting the coefficients to cancel the non-invariance caused by ordering effects, if possible. The list of invariants is not exhausted. To conclude this appendix, we briefly describe how the same process can be used to obtain weight-zero objects that change the picture number. As mentioned in the last section, the usual picture changer contains ii ~ can 2a be rewritten as ~ a term of the formorexas e~e’1’with The exponential e~i_P~~’ k (a 1/2a) and l=(a + 1/2a). Then the same procedure as above can be =

=

—

employed to get the solutions (with the choice a

=

1) (64).

References [1] G. Kleppe, P. Ramond and R.R. Viswanathan, Phys. Lett. B206 (1988) 466 Nucl. Phys. B315 (1989) 79

[2] P. Ramond, in Proc. First Johns Hopkins Workshop, Jan. 1974, ed. G. Domokos and S. KOvesiDomokos; C. Marshall and P. Ramond, Nucl. Phys. B85 (1975) 375 [31 D. Friedan, E. Martinec and S. Shenker, Nuci. Phys. B271 (1985) 93 [4] E. Martinec, G. Sotkov, Phys. Lett. B208 (1988) 249 [5] M. Takama, Phys. Lett. B210 (1988) 153 [6] G.T. Horowitz. S. Martin and R.C. Myers. Phys. Lett. B215 (1988) 291; B218 (1989) 309 [7] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge University Press, New York, 1987) and references therein [8] M.E. Peskin, in Proc. Theoretical Advanced Study Institute in Elementary Particle Physics (University of California, Santa Cruz, 1986), ed. Howard E. Haber (World Scientific, Singapore, 1987) [9]C.B. Thorn, Phys. Rep. 175 (1989) 1 [10] C.R. Preitschopf, Univ. of Maryland preprint UMD-EPP-88-206 [11]M. Kato and K. Ogawa, Nucl. Phys. B212 (1983) 443 [12] W. Siegel, Phys. Lett. B151 (1985) 396