Picture changing operators in closed fermionic string field theory

Physics Letters B 286 (1992) 256-264 North-Holland PHYSICS LETTERS B

Picture changing operators in closed fermionic string field theory

R. Saroja ] and Ashoke Sen 1.2 Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

Received 5 April 1992

We discuss the appropriate arrangement of picture changing operators required to construct gauge invariant interaction vertices involving Neveu-Schwarz states in heterotic and closed superstring field theory. The operators required for this purpose are shown to satisfy a set of descent equations.

A closed bosonic string field theory based on non-polynomial interaction has been constructed recently [ 1- 4 ]. In this paper we shall discuss the construction of gauge invariant field theory for heterotic and type II superstring theories based on the same principles. However, our analysis will be confined only to the Neveu-Schwarz sector of the theory; the construction of a similar field theory including Ramond sector states involves extra complications due to problems involving zero modes of various fields. Although the field theory for fermionic strings is not complete without Ramond sector, we can get some insight into the theory just by looking at the field theory involving the Neveu-Schwarz states. In particular, since for the heterotic string theory, all the bosonic fields come from the Neveu-Schwarz sector, the field theory involving Neveu-Schwarz states can be used to study the space of classical solutions, as well as the geometry of the configuration space of the theory.

We shall, for convenience, restrict our discussion to heterotic string theory only, the analysis for superstring theory proceeds in a similar manner. Let Yd denote the Hilbert space of states in the Neveu-Schwarz sector in the - 1 picture. These are the states created by the matter operators, and the ghost oscillators bn, cn, fin, 7n acting on the state e - ~(°)l 0) = l g2), where 0 is the bosonized ghost, related to fl, 7 through the relations fl= e-~0~, 7=eOr/. Here r/and Care fermionic fields of dimensions 1 and 0 respectively. 10) denotes the SL(2, C) invariant vacuum in the combined matter ghost theory. As in the case of bosonic string theory, a general off-shell string state I~ ) is taken to be a GSO projected state in ~ annihilated by c f f - (Co-go) /xf2 and L ~ - - - ( L o - /So)/x/2, and created from 112) by an operator of total ghost number 3. Following the construction of bosonic string field theory, we shall look for an action of the heterotic string field theory of the form

gN--2 S ( ~ ) = ½ ( 7 J l Q B b f f l ~ ) + ~ ___.~_. {g.,N}, (1)

N=3

where {AI...AN} and [AI.. .AN ] are multilinear maps from the N-fold tensor product of ~¢ to C (the space of complex numbers) and )¢ respectively, satisfying relations identical to the corresponding relations in bosonic string field theory:

{AI...AN} =-- ( -- 1 ) n~ +1 (ell ] [A2 ...AN] ) , ( 2 )

{m, . . .Au}= ( - 1 )(m+,)( . . . . + ' ){A, . . .A i_ ,Ai+ lAiAi+2 . . .Au} , (3 )

1 E-mailaddress: [email protected]. 2 E-mailaddress: [email protected]

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Volume 286, number 3,4 PHYSICS LETTERS B 30 July 1992

{A~ ...( L~ Ai)...AN} =O= {At ...( bff Ai)...AN} , (4)

N ( - 1)"~{ (QaA,)A2...AN)= Z ( -- 1 )z~=~nJ+'){A, ...(QaAi)...A~.)

i=2

- ~., ~ (-1)"(~i'~'tJk~){AiAj,...Aj,,,_2cff[A~...A~,,_l]}. (5) m,n>~3 {jk},{il}

m +n=N+2

Here ni denotes the ghost number of the state [A~). The sum over {it}, {J'k) in eq. (5) runs over all possible divisions of the set of integers 2, ..., N into the sets { it) and {jk). ( - 1 )a~,~k~) is a factor of + 1 which is computed as follows. Starting from the ordering QB, A2, ..., AN, bring them into the order Aj ...... Ajm_~, QB, Ai, .... , A,o_,, using the rule that QB is anticommuting, and A~ is commuting (anti-commuting) if n~ is odd (even). The sign picked up during this rearrangement is ( - 1 ),~,~.tJk~. Using eqs. ( 2 ) - ( 5 ) and the nilpotence of the BRST charge QB, one can show that S(7/) given in eq. ( 1 ) is invariant under a gauge transformation of the form

gN--2 b f f a l ~ ) = Q , bff I A ) + ~ (N-2)-------~ [ ~ttU-2A] ' (6)

N=3

where IA) is an arbitrary GSO projected state in o~ created from IE2) by an operator of ghost number 2, and annihilated by cff and Lff.

Thus, in order to construct a gauge invariant action, we need to construct multilinear maps {A1...AN} from ~t gN t o C, satisfying eqs. ( 2 ) - ( 5 ). We look for an expression similar to the one in bosonic string theory [ 3,5 ]:

{A,.. .AN}=- ~ d2U-6z( f~X)o(bf fPA,) (O) . . f~N)o(bf fPAx)(O)KN) . (7) R(N)

Here P is the projection operator dLoco, f }N) is a conformal map that maps the unit circle into the ith external string of the N-string diagram associated with the N-string vertex, f } N) O (b ff PA~) denotes the conformal trans- form of the field (bff PAi) under the map f }N). The r ~ are the modular parameters characterizing the N-string diagram. The region of integration R ~N), and the N-string diagram are chosen in such a way that at the boundary 0R (N) o f R (N), the N-string diagram is identical to an m-string diagram and an n( = N + 2-m)-s t r ing diagram glued by a tube of zero length and twist 0. One particular example of such N-string diagrams is provided by polyhedra with N faces, the perimeter of each face being equal to 2n. In this case the z~ correspond to independent parameters labelling the lengths of each side of the polyhedron. The region of integration R {N) ove r z is such that it includes all such polyhedra with the restriction that the length of any closed curve on the polyhedron constructed out of the edges should be greater than or equal to 2~z. (Such polyhedra were called regular polyhedra in ref. [ 2 ] ).

Finally we turn to the description of the operator KN appearing in eq. (7). For bosonic string field theory KN was given by rl 2N-6 / • ~i=1 ~ ~/il B), where

(q~ IB) = I d2z[q~Jb(z) + rT'~ zg(e) ] ' (8)

q~j, 0~, ~ are the Beltrami differentials, which tell us how the components g~Z, gee of the metric, induced on the sphere by the N-string diagram, changes as we change z ~ [3 ]. In the case of fermionic string theories, such a choice of Ks will give vanishing answer for {AI...AN} due to ghost number non-conservation. This is remedied by introducing appropriate factors of picture changing operators [6-8 ] in the definition of Ks [9 ]. We shall make the following choice of Ks:

2N--6 KN= X C~ r)^~cNzN-6-~), (9)

r~0

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where C(N r) is an r-form (in the ( 2 N - 6 ) - d i m e n s i o n a l moduli space spanned by r ~) operator, given by

(¢'~(r) ~ r ' I • ~/v ~,...i~= ( q , , l B ) , k = l

(10)

qb tur), on the other hand, is an r-form operator satisfying the "descent" equation

d t E ) ~ ~)= [QB, tib~ r+') } , (11)

where d rE) denotes derivative with respect to z~, acting on the explicit z dependent factors in the expressions for tP~ ~), and [ } denotes a commuta tor (ant i -commutator) for odd (even) r in eq. ( 11 ). ~ o ) takes the form

qb~ °) = ~ A t~)( r ' .... , " I ~ 2 N - 6 ) X ( w } °t) ('~'))...X(w/~¢~__)2('~)) , (12)

where X(z) = { QB, ~(z) } is the picture changing operator [ 6 ]. The sum over o~ in eq. (12) runs over a finite set o f values. For each value of ~ we have a set o f points w ~") (z) .... , w ~U~2 (r) on the N-string diagram, and a weight factor A t~) (r) , satisfying the normalization condition X ,A ~) ( z ) = 1. The ~,~), on the other hand, are constructed as linear combinations of products o f X, 0( and ( ( P ) - ~ ( Q ) ~

Besides the descent equation ( 11 ), the • ~0 are also required to satisfy the following boundary conditions. As has been indicated before, the boundary OR ~N) or R tu) consists o f several pieces; each piece corresponds to gluing two vertices with less number o f external states (m and n = N + 2 - m , say) along one o f the external strings from each vertex with a certain twist 0. We demand that on such a component o f 0 R tu),

~(N")Ia~,N,= ~ ~mr--s)̂ qS~s). (13) s=O

Here •tur) [OR (N) denotes the component of tp ~U r) tangential to OR tN). Finally, ~ ~) should be invariant under permutation o f the external strings. Besides these restrictions, the choice of q~ ~u ~) (i.e. the quantities A ~"), w} ") , and the corresponding quantities appearing in the expressions for • ~,~) for r>~ 1 ) is completely arbitrary ~2.

We shall now show that the quantities {A1...AN} constructed this way are non-zero in general, and satisfy eqs. ( 3 ) - ( 5 ) with [A I...A N_ l ] defined through eq. (2). To see that {A 1...AN} is non-zero in general, we only need to note that the contribution to the right-hand side of eq. (9) from the r = 2 N - 6 term is of the form

2N-- 6 (rl, IB) Z A('~)(r)X(w}'~))...X(w~v%)2). (14)

i=O ot

Besides providing the appropriate factors o f (qjl B) as in the case of bosonic string theory, we now also have the correct number of picture changing operators. Thus, if each IA,) is created by a ghost number 3 operator acting on It2), at least the contribution from the r = 2 N - 6 term in eq. (9) to {AI...AN} will be non-zero. As we shall now see, the other terms are necessary for {./11...AN} to satisfy eq. (5).

Verification ofeqs. ( 2 ) - ( 4 ) with the definition o f {AI...AN} given in eq. (7) is straightforward, so we turn to the verification o f eq. (5). Using eq. (7), both sides o f eq. (5) may be expressed as integrals (over r i) of appropriate correlation functions in the conformal field theory. In the left-hand side of eq. (5), we may express (Q~/~) as the contour integral of the BRST current around the location o f the operator b~-A~. We may now deform the BRST contour and shrink it to a point, in the process picking up residues from the locations o f various other operators. The residues from the locations of the operators b ff Ag (i/> 2 ) give rise to the first set o f

~1 The appearance of extra factors proportional to 0( when the locations of the picture changing operators are moduli dependent, was discussed in ref. [ 10].

~2 Although we shall try to choose ~N r~ in such a way that they are continuous inside R (N), this is not a necessary constraint, since, as can be seen from eq. ( 11 ), a discontinuity of ~ ~r) inside R (~') may be compensated by a 0-function singularity in ~ ~.,~+ ~ ).

258


terms on the right-hand side of eq. (5). We are now left with terms proportional to [ Qa, KN]. Residues from the terms proportional to b(z), 6(g) in (~hl B) generates a factor ofj 'd2z [ q,- =T(z) + 0r= eT(z) ] - T! 1 ). Thus we may write,

[QB, C(Nr)}= T( I ) A C(N r- l ) , (15)

where C~N ~) has been defined in eq. (10). Since insertion of a T}*) inside a correlation function generates the z ~ derivative of the correlation function, with the derivative acting on the implicit r ~ dependence of the correlation function due to the dependence of the string diagram on r, we see that the terms proportional to [ QB, C~ ~) } in [QB, Ku] give rise to a term of the form

f dZN-6~" d ( l ) ~ , (16) R (N)

where

N 2N--6 (~) (N2N-- 6-- r) I ( 1 7 ) ~ = ( - 1 ) " ' l--I (f!N)obcA,(O)) Y~ C~r-~)^ /=1 r= l

and d (I) denotes the z derivative acting on the implicit z dependence of the correlation function but not on the explicit z dependence appearing in q~(N r) . On the other hand, terms proportional to [QB, q)~v r) } in [QB, KN] may be analyzed using the descent equation ( 11 ), and gives an answer similar to eq. ( 16 ) with d (~) replaced by d (E). Thus if d-=d (~) + d (E) denotes the total z derivative, the contribution from the term proportional to [QB, KN} that appears in the analysis of the left-hand side ofeq. (5) may be written as,

f d2N-6"C d . ~ = I d2N-7T.~ . (18) R (N) OR (N)

Let us now consider a specific component of the boundary OR (N) of R (N) that corresponds to gluing of two lower order string diagrams of rn and n = N + 2 - m external states, with a twist 0. Let r(~) and z(2) be the modular parameters describing these lower order string diagrams. Then,

dZN--7T[0R(N) = d 2 m - 6 " ( ( l ) d2n-62"(2) dO, (19)

(r ) (C~4))o ......... IOR '"=(r to[ B ) E C}S) A C ( . . . . . . i) . ( 2 0 )

s=O al...ar 1

Using the boundary conditions given in eqs. ( 13 ) and (20) it is easy to see that,

r= 1 /o,I, ).,h%-6,5)...,~; 610R (N,

f2m--6 x f12n--6 x x =(qoIB)~ ~=o c}r) ACl)(2rn-6-r)] | V C (s, (~(2n--6--s, m /~ A . (21) )

r= T( I )...T ( 1 ) -- z (2)"'Z (2)

Standard manipulations identical to the one for the bosonic string theory can now be used to show that the contribution to the right-hand side of eq. (18) is identical to the second set of terms of the right-hand side of eq. (5). This completes the proof of eq. (5).

We shall now indicate the basic steps involved in the calculation of Feynman amplitudes in this field theory. The external states are taken to be physical states of the form b 6- ] At ) = cO'V, ( 0 ) [ £2 ) = celY~ (0) [ 0 ) where V, is a dimension (1, 1 ) superconformal primary field in the matter sector. Since the contribution from various Feyn- man diagrams can be brought into the form of the contribution from elementary N-point vertex given in eq. (7)

259


using standard techniques [ 11 ], we shall analyze the contr ibution to A ( 1 .... , N) f rom the elementary N-point vertex only. In analyzing this contr ibution we shall use the familiar expression for r/~/, 0~.e in terms of quasi conformal deformat ions [ 12 ] v ~, g:, and write,

(r/, rB) = dz-~-~7~,b(z)+de~fi6(~) , (22) C

where the contour C encloses all the points z~, as well as the locations of all the operators appearing in ~ N ~) . fV-" (fife) is analytic (anti-analyt ic) inside the contour C but not outside C. We can now deform C and shrink it to a point, picking up residues from the locations of various operators in this process. In the case o fboson ic string theory, the only possible residues are picked up f rom the locations of the vertex operators bffA~ =cJ~. This removes the c8 factors from N - 3 of the vertex operators and generates appropriate measure factors that convert the integration over r ~ to integration over the locations of the ( N - 3 )-vertices. In the present case, however, the integration contour can pick up residues from the locations of the picture changing operators inside q~ ~N ~) also. For any r-form operator C, let us define an ( r + 1 ) - form operator 6C as

(6(9)i,...,,+,=A d z ~ b ( z ) + d g ~ r , 6(g) (~2...~+, , (23)

where A denotes ant isymmetr izat ion in the indices i~, ..., ir+ ~, and C denotes a contour enclosing the locations of all the operators in (9. Using eqs. ( 7 ) - ( 1 0 ) the contr ibution to A(1, ..., N) from the elementary N-point vertex may be written as,

J <( 01 ) > 2N- -6r 1 O s Cff~k(7-k, gk) A(~r--s~v2N--6--r) (24) gN-2 d2N-6r ~:oZ ~os'(r-s)!= . R (N)

where 6 ~ denotes s successive operat ions of 6 and z~ =f!N)(o ). So far our description has been independent o f the choice of coordinates of the moduli space. We can now

simplify our analysis by choosing the moduli parameters r ' to be identical to the coordinates x, ( 1 ~< i ~< 2 N - 6 ) defined as follows:

x,=z(i+l)/2 f o r / o d d ,

In that case, for odd i,

5v z ~ 1 a t z ~ z ( i + l ) / 2 ~ fT i

and, for even i,

fv ~ fr-~=O a t z = z j f o r a l l j ,

Let us now define

r = O r]

and

xi=Zi/2 f o r / e v e n . (25)

By: StY z 8 r~=O a t z = z j f o r j ¢ ( i + l ) / 2 , 8 r i - O a t z = z j f o r a l l j , (26)

5z7 ~ 5zTe= 0 5r,----=l atz=z~/2, fir' a tz=z~forj¢i /2 . (27)

(28)

(29)

so that

260


~ N

( L ~ ' - 6 - ' ~ ) " ' 2~-° - ' - 8C,, ' ~C,.,_6_. ,~ eel(z,, ~,), (30)

where

Cj=c(zo+l) /2) f o r j o d d , C;=~(2j/2) f o r j even . (31)

Then eq. (24) takes the form

2N--6 A4~V 2 N - 6 - r ) A(1 ..... N ) = g N-2 d2N-6x, ~ (L~v "~ >. (32) r=0

By techniques identical to those used in the case of bosonic string field theory one can show that the integral over x, runs over the full moduli space when we add the contribution from all the Feynman diagrams.

Using the relations {QB, b(z)}= T(z ) , {QB, b-(~)}---/~(~), the fact that the insertion of f d2z[t l~ZT(z)+ q~ ~/~(~) ] in a correlation function generates the (implicit) derivative of the correlation function with respect to r ~, and the descent equation ( 11 ) we get,

[QB, 6 " ~ ) } = r d { ~ ) ( 6 " - ~ ) ) + d { E ) ( 6 " d P ~ - ' ) ) • (33)

Using eq. (33) one can easily verify that ¢ ~N ') defined in eq. (28) satisfies the descent equation

d ~ v ' ) = [QB, ~ CN'+') } (34)

where d denotes total derivative with respect to the x,'s. We now turn to the question of comparing the amplitude given in eq. (32) with the one calculated from the

first quantized formalism. In order to do so we shall first prove the following

Lemma. If we have two sets of operators ¢Cur> and ~P~N r), both satisfying the descent equations given in eq. (34), and, if

rb(N °) -- ~ v °) = { Q , , ;(N °) } (35)

for some ZtN °) , then

2N--6 d2N-6Xi ~', < L ~ ) A ((~(N 2N-6-r) --~-Y(N2N--6--r))> : 0 . (36)

r=0

The lemma is proved in the following way. From eq. (35) and the descent equation for • J~'), ~v ") , we get

{QB, ~{U ~) -- ~Ul)} ={QB, dZ~u°)} • (37)

This gives

~ N ') ~(N ')--dZ(N °)4- [ . ( ' ) I - - QB,AN j (38)

for some ;(d ). Repeating this, we get the general equation

~ ~u r) -- ~ v r) =dz~v "- ' ) + [ Q, , X~r) } . (39)

Finally, using the definition of L ~U ') and the commutation relations

a (cP,(z,, ~,)) + 0 [QB, g(z , , 2,)] = ~z~ ~ (6P,(z,, 2 , ) ) ,

O (ceP,) [QB, cgP,] =0 (40) o ( e c g ) , {Q., eg}= ~ , { QB, Cl?i( zi, zi)} =

261


we get

[Q~, L~ ~) } = d L ~ -~ ) (41)

Using eqs. (39 ), (41 ), and the fact that the expectation value of a BRST exact operator vanishes, we can bring the left-hand side ofeq. (36) into the form

// 2N--7 ) f d2N-6xid~ r~=O tOvr) A Z ~ g 2 N - 7 - r ) ( - - l ) r , (42)

which vanishes after integration over x,. This completes the proof of the lemma.

Let us now choose

(UCNO) = X( Z3 )X( z4)...X( zlv) . (43)

Using the fact that qb~N°) has the form Z,A~")({x~})X(wl"~)...X(w~v~2) with S,A~"~= 1, we see that ~ o ) _ ~N °) is BRST exact. Thus by the above lemma, we can replace qb~ r) by ~ ) in the expression for A (1 ..... N) given in eq. (32). A set of ~N r) satisfying the descent equation are given by

N (~P~N~))iJ...s~ = f i O{(Zuk+')/2) 1-I X(zj) for ikoddand3<~(ik+l) /2<~N-3foral lk

k=l j=3 j~ (il+ 1 )/2

= 0 otherwise.

Using this we get the following expression forA ( 1 ..... N):

f N--3 f i N A(1, . . . ,N)=g ev-2 [I dez, l~,(z,,2, ) l-[ (cgg(z , ,<)X(z , ) ) i= 1 l= 1 I=N--2

(44)

N--3 1-[ ( P,(z,, e~/[X(z,)-c(z,)0g(z~)]) , l=3

(45/

where d 2 z t --- d~, ̂ dzi. In the above equation, f~(z,, Z,) corresponds to the integrated vertex operator in the - 1 picture [6 ]. Also,

I?,(zi, g i ) [X(z , ) - c (z~)O~(z , ) ]= {Q~, f'~(zi, z , )~ (z , )} -0 ( l?,(z,, g,)c(zi)~(zj) ) -O( l~,(z~, g,)g(z~)~(z~) ) (46)

correspond to integrated vertex operators in the zero picture. Finally, ~(zi, ~,)c(z,)g(g~)X(z,) correspond to unintegrated vertex operators in the zero picture. Thus we see that the right-hand side ofeq. (45) has precisely the form expected from the analysis of the first quantized theory.

We now turn to the problem of determining ~-~) satisfying eqs. ( 11 ) and (13). We shall discuss one particular construction, but one should keep in mind that this construction is in no way unique, and there are (infi- nitely) many other choices possible. In order to prescribe tb} r) in a way that avoids the divergences associated with collision of picture changing operators [ 13 ] ~3, we shall find it more convenient to take the N-string diagram not just a regular polyhedra with N faces, but regular polyhedra with N faces with tubes of a certain fixed length lo attached to each of the faces; together with diagrams corresponding to such regular polyhedra with ml, m2 ..... mr+ ¢ = N + 2 r - (mr +. . .+m~) faces, joined by tubes of length It ..... /~ (0~</~<2/o) and twist 0t .. . . , Or (0 ~< 0, < 2~r). (Such vertices have been used in ref. [ 17 ] for a different purpose. ) We shall denote by R~ N) the component of R {N) for which the corresponding string diagram is a regular N-hedron with tubes of length lo attached to its faces. R c (N) will denote the component of R Cx) for which the corresponding string diagram is given by two or more such regular polyhedra connected by tubes of length ~< 2/o.

We shall first discuss the construction of • ~91 for points inside R e iN) . In this case the picture changing oper-

~3 Removal of such divergences in open string field theory has been discussed by several authors [ 14-16].

262


ators are taken to be at the mid-points of the edges of the polyhedron. Since the number of edges ( 3 N - 6 ) of an N-hedron is larger than the number of picture changing operators ( N - 2 ) , there are several possibilities. We average over all configurations, with the weight factor for a given configuration being proportional to U if(si) , where s, is the length o f the ith edge, the product over i runs over all the N - 2 edges containing the picture changing operators, a n d f ( s i ) is a smooth function of s~ satisfying the constraint

f(si)=O fors~<r / , f ( s ~ ) = l fors~>~2~/, (47)

where r/is a small but fixed number. This construction guarantees that the picture changing operators are always inserted at the midpoints of the edges which have length >1 r/, and hence two picture changing operators never collide inside R~ (N) ~4. This completely specifies ~¢o) inside R~ N) .

Let us now turn to a point inside R¢ (N) which corresponds to two regular polyhedra joined by a tube of length l. Let h be some fixed length ~<lo. For l>~2h, we choose qb(N °) tO be simply the product of (/)~v °) on the two polyhedra. This automatically ensures that ~ ~o) satisfies the boundary condition (13 ) at the boundary l= 2/0 of R (N). For l~< h, we choose the picture changing operators to be on the edges o f the two polyhedra in such a way that • ~v °) is independent of l and is identical to its value at l= 0, where it is given by the previous construction of • ~) for points inside Re (N) . This does not specify • (fl) completely, since a picture changing operator inserted on the right boundary of the tube or the left boundary of the tube at the same angle generates the same configuration when the tube length is collapsed to zero. We remove this ambiguity by assigning equal weight factor to each of these configurations. Finally, in the region h ~< l ~ 2h, (/) (o) is chosen to be a linear combinat ion o f the expressions for (/) (o) for l~< h and l>~ 2h such that the resulting expression smoothly interpolates between the values o f ~ °) for l~h and l>~2h.

This construction can easily be generalized to points inside R~ N) representing string diagrams where more than two polyhedra are joined together by tubes of length ~< 2/o. Ambiguities similar to the one discussed above arise in the region when some of the tubes are of length ~< h, since many different segments of edges belonging to different polyhedra may correspond to the same segment when all tubes of length ~< h are collapsed. Hence if the polyhedron obtained after this collapse has an insertion of picture changing operator on this segment, we need to decide how to distribute it over the various segments before the collapse. One possible consistent way is to distribute it only among the two segments which are at the two extreme ends o f the chain of tubes connecting these different segments, with equal weight factor.

This finishes the discussion on the construction of (/) ~o). Once • ~o) is given, • ~v r) may be obtained by solving the descent equations. We shall not discuss the general case here, but as an example, give a specific solution for (/)y). In this case (/) ~o) may be expressed as

c19~ °) =X(P)X(Q) + ~ f (") ( z ) x ( P (~)) [X(Q (")) -X(R( '~ ) ) ] , (48)

where the sum over o~ runs over a finite set. The corresponding solution for (/)~) is,

cP~)= [d~(P)X(Q)+X(P)d~(Q)]+ ~ df('~'X(P('~))[~(Q('~))-~(R('~))] OL

+ ~f('~){d~(P("))[X(Q(~'))-X(R("))]+X(p(c'))[d~(Q('~))-d~(R('~))]}, (49) o~

~4 Note that we could also have, in principle, chosen the picture changing operators at the vertices, imitating the corresponding construction for open string field theory. But in this case special care (like moving the picture changing operators away from the vertices) is needed to ensure that two picture changing operators do not collide, since for N>_- 5, Re(N) contains configurations where the number of well separated vertices (say, by a distance >~ ~/) is less than N - 2, Such a situation does not occur if we insert the picture changing operators at the midpoints of the edges, since a regular N-hedron contains at least N edges of finite length.

263


@4 ~2) = d ~ ( P ) A d ~ ' ( Q ) - ~ d f (") A d ~ ( P ¢°°) [ ~ ( Q ¢ ° ~ ) ) - ~ ( R ¢ " ) ) l O~

+ ~ f ¢'~)d~(P ('~)) A [ d ~ ( Q ( " ) ) - d ~ ( R ¢")) ] , o~

q~r ) = 0 f o r r > 3 , (49 c o n t ' d )

where d ~ ( P ) = 8~ (P )dP , etc.

To s u m m a r i z e , in this pape r we have d iscussed the cons t ruc t ion o f a gauge inva r i an t f ield theory for the

N e v e u - S c h w a r z ( N S ) sector o f the he te ro t i c str ing theory. This cons t ruc t ion can easily be genera l i zed to inc lude

the N S - N S sector o f c losed supers t r ing f ield theory as well. The R a m o n d sector, however , suffers f rom extra

p r o b l e m s due to the p resence o f zero m o d e s o f va r ious opera tors , and cannot , at present , be t rea ted by the same

me thod . We hope to c o m e back to this ques t i on in the future.

References

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Picture changing operators in closed fermionic string field theory