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On the “Covariantized light-cone” string field theory
Volume 190, number 1,2
PHYSICS LETTERSB
21 May 1987
ON T H E "COVARIANTIZED L I G H T - C O N E " S T R I N G FIELD T H E O R Y S. UEHARA t Department of Mathematics, King's College, Strand, London WC2R 2LS, UK
Received 7 February 1987; revised manuscript received 23 February 1987
We construct a new BRS charge in which the string length parameter and another bosonic parameter are naturally involved as dynamical variables in a string theory. Using the BRS charge we propose another gauge invariant action of covariant string field theory.
The first attempt to construct a Lorentz-covariant formulation of interacting bosonic string field theory was [ 1 ] to covariantly extend the light-cone formulation [ 2 ] through the Parisi-Sourlas mechanism [ 3 ]. Later the covariant bosonic string field theory was explicitly constructed by a different method [ 4-6 ]. In the latter formulation, the interaction terms were constructed referring to the joining and splitting type interactions in the light-cone formulation and the string length parameter a, which was not introduced to construct the free part of the action only, was found to be necessary. That is, a string field should be a functional of X u ( a ) (string coordinates), c(a), g(a) (Faddeev-Popov ghosts) and o~ (string length) in their formulation. Since dynamical coordinate (or m o m e n t u m ) variables at the first quantized level become parameters of a field at the second-quantized level, we may expect a different (BRS) firstquantization, where there exists a dynamical variable which corresponds to the string length parameter a, from that of Kato and Ogawa [ 7]. Furthermore according to the original attempt [ 1 ] we may expect the third type of first-quantization in which at least two other bosonic dynamical variables than X u, c and are involved. In this letter we construct a new BRS charge in which two other bosonic (dynamical) variables and two other Faddeev-Popov ghosts variables are incorporated with the usual variables X u, c and ¢. On leave from Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima 725, Japan. 76
And using the BRS charge we propose a new gauge invariant action of covariant string field theory. Here we discuss about the open bosonic string only and the closed and the superstring cases will be reported elsewhere. Closely related results in the point particle case were reported in refs. [8,9]. As for secondquantization, however, their action cannot be reproduced by taking a naive limit of our string action. There is also a work by Siegel and Zwiebach [ 10] in which they directly Lorentz-covariantize the lightcone formalism [ 2 ] referring to the symmetry group IOSp(D, 2[2). On the other hand, there is an alternative formulation of open string field theory [ 11 ]. The relation between them [ 4,5,11 ] is not clear and we do not make mention of the latter formulation here. Let us now construct a BRS charge following the method ~ la Fradkin and Vilkovsky [ 12 ]. We look for first class constraints of the string action [ 13 ] 1 / - - - m~ s= f f drd~r-~x/-gg OmXUO.Xu, m,n=0,1,
~t=0,1,...,D-1,
(1)
where X u are the string coordinates and gm~ is the metric tensor of the two-dimensional world sheet with g = det gm~. Following the usual procedure we find the first class constraints 0o=½X {n2 + x - 2 ( X ' ) 2 } ,
(2)
01 ---Tr.X' ,
(3)
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 190, number 1,2
PHYSICS LETTERS B
0 . . . . . 7~g ,
21 May 1987
(4)
where ..... -- 0/0o. and r~u and ~gm n a r e the momentum of X u and gin., respectively,
~,=_,<_, ,,/_g gO° o°x, .
(5)
QB= i do.[Co(bo +GeP, +Ca(ba + C r O r 0
-iCo(C6C, + C I C o ) - i C , ( C I C I +C~Co)],
(12)
Note that we regard gin. as independent dynamical variables here, however, we may regard ~"--× g~" (thus det gm.= _ 1 ) as independent variables. In that case era. should be replaced by Ore,
where Co, CI, Ca and Cr are Faddeev-Popov ghosts while Co and C1 are anti-ghosts and {Co(z, o.), Co(z, o.')}=8(o.-o.') ,
(13)
0,, -- (n~)m( - m o m e n t a
{ C l ( r , o.), Cl('C, o . ' ) } = ~ ( o . - o . ' )
(14)
o f g °m) .
(6)
Furthermore, the BRS charge given by Kato and Ogawa [ 7] is constructed by using ~o and ~ but not ~,, (or om,) [ 14]. In other words dynamical degrees of freedom of ~ " are frozen in their gauge, i.e., goo__ _ 1 and~ °1 = 0 (orgoo= --gll = -- I andgol =0). So we make some of those frozen degrees of freedom of gin,, recover here. We melt two of the degrees of freedom ofg,,n which should correspond to the string length parameter and a "time" (or "energy") [ 1 ]. Since the line element of the world sheet is
.
Ca and Cr are o.-independent due to (11 ): Ca=Ca(O,
CT=CT(0 •
(15)
The hamiltonian is given by
0
= ido" [½x a-~(n2+x-2X '2) 0
ds 2 =good'r 2 +2go~ dz do.+glldo. 2 -ia-'(C6C, +CiCo)-ia-ECoCa] . = g o o { d r 2 + 2 ( g o l ]goo) dr do.
- (~do.)2},
(7)
and the string length parameter and a "time" may be o.-independent, we will make the following variables recover:
1i
OL(r) ---- - -
do. x / - g l l ( r ,
o')/goo(Z, o.)
(8)
do. x/-goo(Z, o.) .
(9)
0
Hence, we have the following first class constraints to construct a BRS charge:
¢o,
Here the variable a - ~Co is determined in order that such terms of the hamiltonian consisting of only the original variables X u, gmn coincide with those derived from the action (1). (Note that go~ is fixed to be zero.) As for the third term on the RHS of (16), only the o.-independent component Yo(Z) of ~'o fo(z)- -
do" Co(z, o.) ,
(17)
0
o
T(z) = -
(16)
01, ~a=--n,~, ¢~-~r,
(10)
where ¢~=0~=0.
Then we have a BRS charge
contributes to the hamiltonian because a-2C,~ is o.independent. The field equations are, of course, derived from - i t b ( - - i ( 0 / 0 z ) q0 = [H, ~b], • = X u,CA,CA
(A=0,1, a,T).
(18)
Some of the field equations are & = J'-- C'a -- ~ ' r ,
(19)
(11) I'--OL-2I " = 0 ,
77
Volume 190, number 1,2
PHYSICS LETTERSB
I = X u, Co, C1, 6"o, CI •
(20)
The field equations can be solved with the open string boundary conditions X'=CI=CI=CI=O,
at a = 0 , n .
(21)
The solutions are T(z) = To, a ( z ) = a o , nr(z) =n-'
iPo ,
21 May 1987
Co(V, a) = ~
1
eo
v
1
+Tn~°O~exp(-inafflz)
(22~)
i Cl(z, a) - - x/~ n~o cn exp( - i n a f f I z) sin n a ,
(22a, b)
(22m)
(22c)
with the commutation relations i q U " = d i a g ( - l ,
1 ..... 1)) [pU, X ~] = -- irf" ,
1 ao + (1 i de ~X ao_2 (re2 + X' 2K2 ) n,~ = -n 0
[a~, a ~ ]
~lW(Jn+m, 0,
=n
[~o, ao] = - i , --ioeff 2(C6C1 + C] C'o) - 2 i a - 3C'oC,~) *,
c°sna'
[7"o, To] = - i ,
(22d)
[c., om] =a.+,., o, 1
1
C,~(z)= ----~c~,
Cr(r)= --cr,
1
1
~',~(z) = ~
Cr(z) = ~
(22e, f)
0~ + x/~ oto2CoZ,
1
Or,
(22h)
K
Xu(r, a) = x " +
(22g)
{ca, 0.} = {cr, (T}= 1.
(23)
[C,~ (z) and C r ( r ) are the anti-ghosts corresponding to Ca(r) and Cr(z), respectively, and they do not appear in QB. ] Plugging the solutions into (12) we find QB =Q~° + ~
1
1
C~o+ ~77crTo,
(24)
where Q~O is just the BRS charge given in ref. [7]. Since Q~O does not contain 0,~ and 0T, the condition of nilpotency of QB is equivalent to that of Q~O:
p"z
QzB=0~=~D=26 and a ( 0 ) = 1.
+i g
,
n - an e x p ( - i n a f f l z ) c o s n a , (22i)~
n 0
Co(z, a) = ~
1
4~
Co - ~
1
x/n a~
c.z
(Here a ( 0 ) is, of course, the zero intercept of the leading trajectory.) The oscillator mode part of the Fock space here is the same as that in ref. [ 7 ] except that it depends on ao and To. The ghost zero mode part forms an octet {lO),colO),c,~lO),cTlO),¢oc,~lO)
L + 7 l n~oCn e x p ( - i n a f f z) cos n a ,
(25)
,
(22j)
C~CTIO), CT¢O10>, CoC~CrlO> }, i
C, (z, a) ~ - x/n--7E.o cn e x p ( - i n a g ' z )
where I0 ) is defined by the condition sin n a , (22k~
78
(26)
Oo10) =~,~ 10) = 0 T I 0 ) = 0 .
(27)
Note that because the hamiltonian contains the term - i(zrot~) - l ~oC~, the octet structure does not cause an octet degeneracy of the Fock space, however, the
Volume 190, number 1,2
PHYSICS LETTERSB
21 May 1987
Fock space is, in fact, infinitely degenerate due to the parameters ao and To. Introducing a suitable metric operator G=coCaCr and defining the norm ( 0 [CoCaCrl0 ) = 1, or restricting the ghost zero mode part of the Fock space to only I0 ) , the physical state condition is given by [ 15 ]
501 = Q~°Ao,
QB [phys) = 0.
5~o =PTAo -- Q~O~_ l ,
(28)
This condition leads to the fact that physical states have no ao and To dependence, and the no-ghost theorem is proved in the same way as in ref. [ 7 ]. Now we proceed to the field theory of the string. Introducing a string field ~[ Z, ao, To, Ca, Cr] with Z being the parameters [ 4] Z ( a ) ={XU(a), C(a), C ( a ) } ,
(29)
PrOo - P,~ ~o + Q ~o0- 2 = 0.
500 =PaAo - Q~°A-t ,
8~t_, = P T A _ I - P a ~ . _ l + Q~°.E_2,
PT=--ig-lnO/OTo,
So= J" D Z dao dTo dca dcr ~QBqb ,
PrO t = Pr0o = 0,
(39)
P,01 -QaK°¢o =0, (40)
80, = Q~°A~°~,
41= ¢~[2, - a o , To, -ca, c~-],
(31)
Z= {XU(a),-C(a), - C(a)}.
(32)
The field equation is given by Qnq~=0.
(33)
This action is invariant under a gauge transformation, 5~=Q~A.
(34)
Since the Faddeev-Popov ghost number of the integration measure is - 3 , the string field • has + 1 ghost number and the parameter A has 0 ghost number. In the following we show that the action (30) reproduces the correct spectrum. We expand the string field • and the gauge parameter A in ca and cT: ~--'-01 "~'CaOo'~-CT~gO"~'CTCot~¢--I ,
(35)
A =-Ao +caA_ l +cr.V_, + c r c a ~ - 2 •
(36)
Then the field equations (33) and the gauge transformations (34) are (see eq. (24)) Q~°0, =0,
P~=--ig-lnO/Oao.
Using the gauge degrees of freedom of Ao and A_ 1 we can set q/o = ~u_ l = 0. Then the residual field equations and the gauge transformations are Q~°0, =0,
where
(38)
where PT and P~ are different operators:
we propose a quadratic (free) part of the gauge invariant action of open string field theory (30)
(37 cont'd)
Pa01 - Q ~ ° 0 o = 0 ,
PrO, - QK°go = 0 ,
(37)
50o =P~A6 °) - Q~°A~-°~,
(41)
where the residual gauge parameters A~°) and A~_°~ satisfy PTA 60) = PTA ~ = 0 .
(42)
Next, since 01 has + 1 (or non-zero) ghost number, 01 with Q~° 01 = PZ0l = 0 can be gauged away by using the degree of freedom of the parameter A~°) [7]. Then the residual equations and transformations are Q~° Oo =PrOo = 0 ,
(43)
600 =Pa2~ °) --QK°A~-°)~,
(44)
with ev2~o) = Q~O. ( co( O/Oco)26°)) = 0 .
(45)
We expand the field 0o, the gauge parameters 2~°), A~_°? and the BRS charge Q~O with respect to the other ghost zero mode Co: Oo =-Xo+CoZ-i ,
(46)
,~6o~= ¢~o~+Co¢~O?,
(47)
a~_o? =~o? +Con~_o~,
(48) 79
Volume 190, number 1,2
Q~O _ coL + ( O/Oco)M + OB •
PHYSICS LETTERS B
(49)
Then 3(- J with P r X - J = 0 can be gauged by using the gauge parameter r/~_°~.Hence we find L)Co = (~BXo = PrXo = 0 ,
(50)
8Zo = p , ~ o ) ,
( 51 )
with
PT~°~ =L~O~ =0B~o~ =0.
(52)
(53)
Thus Zo contains the physical degrees of freedom exactly. Note that at the final step Xo cannot be gauged away by ~o) because we assume that P~ is not invertible (see ref. [ 9 ]). As for interaction terms, not only cubic but quartic terms should be necessary in this open string case. Since ~ has + 1 ghost number and the integration measure of each string has - 3, the three-string vertex V3 and the four-string vertex V4 should have + 6 and + 8, respectively. Since we introduced new parameters, especially a, Tand c,,, the OSp(26,212) symmetry becomes clear and then this formulation would be equivalent to the light-cone formulation owing to the Parisi-Sourlas mechanism [ 1 ]. Details will be reported elsewhere.
80
Note added. After completing this work, I became aware of the paper by Neveu and West [ 16 ], in which they extended their previous work [ 9 ] to the string. They adopted, however, a different gauge fixing at the first-quantized level, and hence they took a different approach to the field theory.
References
Finally we can remove the a-dependence of;~o (with (50)) by using the parameter ~o) (with (52)). Then the remaining field equations are L)~o = ~BXo = PrZo =P~Xo = 0 .
21May 1987
[ 1 ] w. Siegel, Phys. Lett. B 142 (1984) 276. [2] M. Kaku and K. Kikkawa, Phys. Rev. D 10 (1974) 1110, 1823. [3] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744. [4] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B 172 (1986) 186, 195; Kyoto preprint KUNS 829-HE(TH) 86/3, RIFP - 660, 673, 674 (1986). [5] A. Neveu and P. West, Phys. Lett. B 168 (1986) 192; Nucl. Phys. B 278 (1986) 601. [6] I.Ya. Aref'eva and I.V. Volovich, Teor. Mat. Fiz. 67 (1986) 486. [7] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [8] S. Monaghan, Phys. Lett. B 178 (1986) 231. [ 9] A. Neveu and P. West, CERN preprint TH-4547/86. [ 10] W. Siegel and B. Zwiebach, Maryland preprint # 86-195. [ 11 ] E. Witten, Nucl. Phys. B 268 (1986) 253. [12] E.S. Fradkin and G.A. Vilkovsky, Phys. Lett. B 55 (1975) 224. [ 13 ] A.M. Polyakov, Phys. Lett. B 103 (1981 ) 207. [ 14] S. Hwang, Phys. Rev. D 28 (1983) 2614. [15] T. Kugo and I. Ojima, Prog. Theor. Phys. suppl. 66 (1979) 1. [ 16] A. Neveu and P. West, CERN preprint TH-4564/86.
PHYSICS LETTERSB
21 May 1987
ON T H E "COVARIANTIZED L I G H T - C O N E " S T R I N G FIELD T H E O R Y S. UEHARA t Department of Mathematics, King's College, Strand, London WC2R 2LS, UK
Received 7 February 1987; revised manuscript received 23 February 1987
We construct a new BRS charge in which the string length parameter and another bosonic parameter are naturally involved as dynamical variables in a string theory. Using the BRS charge we propose another gauge invariant action of covariant string field theory.
The first attempt to construct a Lorentz-covariant formulation of interacting bosonic string field theory was [ 1 ] to covariantly extend the light-cone formulation [ 2 ] through the Parisi-Sourlas mechanism [ 3 ]. Later the covariant bosonic string field theory was explicitly constructed by a different method [ 4-6 ]. In the latter formulation, the interaction terms were constructed referring to the joining and splitting type interactions in the light-cone formulation and the string length parameter a, which was not introduced to construct the free part of the action only, was found to be necessary. That is, a string field should be a functional of X u ( a ) (string coordinates), c(a), g(a) (Faddeev-Popov ghosts) and o~ (string length) in their formulation. Since dynamical coordinate (or m o m e n t u m ) variables at the first quantized level become parameters of a field at the second-quantized level, we may expect a different (BRS) firstquantization, where there exists a dynamical variable which corresponds to the string length parameter a, from that of Kato and Ogawa [ 7]. Furthermore according to the original attempt [ 1 ] we may expect the third type of first-quantization in which at least two other bosonic dynamical variables than X u, c and are involved. In this letter we construct a new BRS charge in which two other bosonic (dynamical) variables and two other Faddeev-Popov ghosts variables are incorporated with the usual variables X u, c and ¢. On leave from Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima 725, Japan. 76
And using the BRS charge we propose a new gauge invariant action of covariant string field theory. Here we discuss about the open bosonic string only and the closed and the superstring cases will be reported elsewhere. Closely related results in the point particle case were reported in refs. [8,9]. As for secondquantization, however, their action cannot be reproduced by taking a naive limit of our string action. There is also a work by Siegel and Zwiebach [ 10] in which they directly Lorentz-covariantize the lightcone formalism [ 2 ] referring to the symmetry group IOSp(D, 2[2). On the other hand, there is an alternative formulation of open string field theory [ 11 ]. The relation between them [ 4,5,11 ] is not clear and we do not make mention of the latter formulation here. Let us now construct a BRS charge following the method ~ la Fradkin and Vilkovsky [ 12 ]. We look for first class constraints of the string action [ 13 ] 1 / - - - m~ s= f f drd~r-~x/-gg OmXUO.Xu, m,n=0,1,
~t=0,1,...,D-1,
(1)
where X u are the string coordinates and gm~ is the metric tensor of the two-dimensional world sheet with g = det gm~. Following the usual procedure we find the first class constraints 0o=½X {n2 + x - 2 ( X ' ) 2 } ,
(2)
01 ---Tr.X' ,
(3)
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 190, number 1,2
PHYSICS LETTERS B
0 . . . . . 7~g ,
21 May 1987
(4)
where ..... -- 0/0o. and r~u and ~gm n a r e the momentum of X u and gin., respectively,
~,=_,<_, ,,/_g gO° o°x, .
(5)
QB= i do.[Co(bo +GeP, +Ca(ba + C r O r 0
-iCo(C6C, + C I C o ) - i C , ( C I C I +C~Co)],
(12)
Note that we regard gin. as independent dynamical variables here, however, we may regard ~"--× g~" (thus det gm.= _ 1 ) as independent variables. In that case era. should be replaced by Ore,
where Co, CI, Ca and Cr are Faddeev-Popov ghosts while Co and C1 are anti-ghosts and {Co(z, o.), Co(z, o.')}=8(o.-o.') ,
(13)
0,, -- (n~)m( - m o m e n t a
{ C l ( r , o.), Cl('C, o . ' ) } = ~ ( o . - o . ' )
(14)
o f g °m) .
(6)
Furthermore, the BRS charge given by Kato and Ogawa [ 7] is constructed by using ~o and ~ but not ~,, (or om,) [ 14]. In other words dynamical degrees of freedom of ~ " are frozen in their gauge, i.e., goo__ _ 1 and~ °1 = 0 (orgoo= --gll = -- I andgol =0). So we make some of those frozen degrees of freedom of gin,, recover here. We melt two of the degrees of freedom ofg,,n which should correspond to the string length parameter and a "time" (or "energy") [ 1 ]. Since the line element of the world sheet is
.
Ca and Cr are o.-independent due to (11 ): Ca=Ca(O,
CT=CT(0 •
(15)
The hamiltonian is given by
0
= ido" [½x a-~(n2+x-2X '2) 0
ds 2 =good'r 2 +2go~ dz do.+glldo. 2 -ia-'(C6C, +CiCo)-ia-ECoCa] . = g o o { d r 2 + 2 ( g o l ]goo) dr do.
- (~do.)2},
(7)
and the string length parameter and a "time" may be o.-independent, we will make the following variables recover:
1i
OL(r) ---- - -
do. x / - g l l ( r ,
o')/goo(Z, o.)
(8)
do. x/-goo(Z, o.) .
(9)
0
Hence, we have the following first class constraints to construct a BRS charge:
¢o,
Here the variable a - ~Co is determined in order that such terms of the hamiltonian consisting of only the original variables X u, gmn coincide with those derived from the action (1). (Note that go~ is fixed to be zero.) As for the third term on the RHS of (16), only the o.-independent component Yo(Z) of ~'o fo(z)- -
do" Co(z, o.) ,
(17)
0
o
T(z) = -
(16)
01, ~a=--n,~, ¢~-~r,
(10)
where ¢~=0~=0.
Then we have a BRS charge
contributes to the hamiltonian because a-2C,~ is o.independent. The field equations are, of course, derived from - i t b ( - - i ( 0 / 0 z ) q0 = [H, ~b], • = X u,CA,CA
(A=0,1, a,T).
(18)
Some of the field equations are & = J'-- C'a -- ~ ' r ,
(19)
(11) I'--OL-2I " = 0 ,
77
Volume 190, number 1,2
PHYSICS LETTERSB
I = X u, Co, C1, 6"o, CI •
(20)
The field equations can be solved with the open string boundary conditions X'=CI=CI=CI=O,
at a = 0 , n .
(21)
The solutions are T(z) = To, a ( z ) = a o , nr(z) =n-'
iPo ,
21 May 1987
Co(V, a) = ~
1
eo
v
1
+Tn~°O~exp(-inafflz)
(22~)
i Cl(z, a) - - x/~ n~o cn exp( - i n a f f I z) sin n a ,
(22a, b)
(22m)
(22c)
with the commutation relations i q U " = d i a g ( - l ,
1 ..... 1)) [pU, X ~] = -- irf" ,
1 ao + (1 i de ~X ao_2 (re2 + X' 2K2 ) n,~ = -n 0
[a~, a ~ ]
~lW(Jn+m, 0,
=n
[~o, ao] = - i , --ioeff 2(C6C1 + C] C'o) - 2 i a - 3C'oC,~) *,
c°sna'
[7"o, To] = - i ,
(22d)
[c., om] =a.+,., o, 1
1
C,~(z)= ----~c~,
Cr(r)= --cr,
1
1
~',~(z) = ~
Cr(z) = ~
(22e, f)
0~ + x/~ oto2CoZ,
1
Or,
(22h)
K
Xu(r, a) = x " +
(22g)
{ca, 0.} = {cr, (T}= 1.
(23)
[C,~ (z) and C r ( r ) are the anti-ghosts corresponding to Ca(r) and Cr(z), respectively, and they do not appear in QB. ] Plugging the solutions into (12) we find QB =Q~° + ~
1
1
C~o+ ~77crTo,
(24)
where Q~O is just the BRS charge given in ref. [7]. Since Q~O does not contain 0,~ and 0T, the condition of nilpotency of QB is equivalent to that of Q~O:
p"z
QzB=0~=~D=26 and a ( 0 ) = 1.
+i g
,
n - an e x p ( - i n a f f l z ) c o s n a , (22i)~
n 0
Co(z, a) = ~
1
4~
Co - ~
1
x/n a~
c.z
(Here a ( 0 ) is, of course, the zero intercept of the leading trajectory.) The oscillator mode part of the Fock space here is the same as that in ref. [ 7 ] except that it depends on ao and To. The ghost zero mode part forms an octet {lO),colO),c,~lO),cTlO),¢oc,~lO)
L + 7 l n~oCn e x p ( - i n a f f z) cos n a ,
(25)
,
(22j)
C~CTIO), CT¢O10>, CoC~CrlO> }, i
C, (z, a) ~ - x/n--7E.o cn e x p ( - i n a g ' z )
where I0 ) is defined by the condition sin n a , (22k~
78
(26)
Oo10) =~,~ 10) = 0 T I 0 ) = 0 .
(27)
Note that because the hamiltonian contains the term - i(zrot~) - l ~oC~, the octet structure does not cause an octet degeneracy of the Fock space, however, the
Volume 190, number 1,2
PHYSICS LETTERSB
21 May 1987
Fock space is, in fact, infinitely degenerate due to the parameters ao and To. Introducing a suitable metric operator G=coCaCr and defining the norm ( 0 [CoCaCrl0 ) = 1, or restricting the ghost zero mode part of the Fock space to only I0 ) , the physical state condition is given by [ 15 ]
501 = Q~°Ao,
QB [phys) = 0.
5~o =PTAo -- Q~O~_ l ,
(28)
This condition leads to the fact that physical states have no ao and To dependence, and the no-ghost theorem is proved in the same way as in ref. [ 7 ]. Now we proceed to the field theory of the string. Introducing a string field ~[ Z, ao, To, Ca, Cr] with Z being the parameters [ 4] Z ( a ) ={XU(a), C(a), C ( a ) } ,
(29)
PrOo - P,~ ~o + Q ~o0- 2 = 0.
500 =PaAo - Q~°A-t ,
8~t_, = P T A _ I - P a ~ . _ l + Q~°.E_2,
PT=--ig-lnO/OTo,
So= J" D Z dao dTo dca dcr ~QBqb ,
PrO t = Pr0o = 0,
(39)
P,01 -QaK°¢o =0, (40)
80, = Q~°A~°~,
41= ¢~[2, - a o , To, -ca, c~-],
(31)
Z= {XU(a),-C(a), - C(a)}.
(32)
The field equation is given by Qnq~=0.
(33)
This action is invariant under a gauge transformation, 5~=Q~A.
(34)
Since the Faddeev-Popov ghost number of the integration measure is - 3 , the string field • has + 1 ghost number and the parameter A has 0 ghost number. In the following we show that the action (30) reproduces the correct spectrum. We expand the string field • and the gauge parameter A in ca and cT: ~--'-01 "~'CaOo'~-CT~gO"~'CTCot~¢--I ,
(35)
A =-Ao +caA_ l +cr.V_, + c r c a ~ - 2 •
(36)
Then the field equations (33) and the gauge transformations (34) are (see eq. (24)) Q~°0, =0,
P~=--ig-lnO/Oao.
Using the gauge degrees of freedom of Ao and A_ 1 we can set q/o = ~u_ l = 0. Then the residual field equations and the gauge transformations are Q~°0, =0,
where
(38)
where PT and P~ are different operators:
we propose a quadratic (free) part of the gauge invariant action of open string field theory (30)
(37 cont'd)
Pa01 - Q ~ ° 0 o = 0 ,
PrO, - QK°go = 0 ,
(37)
50o =P~A6 °) - Q~°A~-°~,
(41)
where the residual gauge parameters A~°) and A~_°~ satisfy PTA 60) = PTA ~ = 0 .
(42)
Next, since 01 has + 1 (or non-zero) ghost number, 01 with Q~° 01 = PZ0l = 0 can be gauged away by using the degree of freedom of the parameter A~°) [7]. Then the residual equations and transformations are Q~° Oo =PrOo = 0 ,
(43)
600 =Pa2~ °) --QK°A~-°)~,
(44)
with ev2~o) = Q~O. ( co( O/Oco)26°)) = 0 .
(45)
We expand the field 0o, the gauge parameters 2~°), A~_°? and the BRS charge Q~O with respect to the other ghost zero mode Co: Oo =-Xo+CoZ-i ,
(46)
,~6o~= ¢~o~+Co¢~O?,
(47)
a~_o? =~o? +Con~_o~,
(48) 79
Volume 190, number 1,2
Q~O _ coL + ( O/Oco)M + OB •
PHYSICS LETTERS B
(49)
Then 3(- J with P r X - J = 0 can be gauged by using the gauge parameter r/~_°~.Hence we find L)Co = (~BXo = PrXo = 0 ,
(50)
8Zo = p , ~ o ) ,
( 51 )
with
PT~°~ =L~O~ =0B~o~ =0.
(52)
(53)
Thus Zo contains the physical degrees of freedom exactly. Note that at the final step Xo cannot be gauged away by ~o) because we assume that P~ is not invertible (see ref. [ 9 ]). As for interaction terms, not only cubic but quartic terms should be necessary in this open string case. Since ~ has + 1 ghost number and the integration measure of each string has - 3, the three-string vertex V3 and the four-string vertex V4 should have + 6 and + 8, respectively. Since we introduced new parameters, especially a, Tand c,,, the OSp(26,212) symmetry becomes clear and then this formulation would be equivalent to the light-cone formulation owing to the Parisi-Sourlas mechanism [ 1 ]. Details will be reported elsewhere.
80
Note added. After completing this work, I became aware of the paper by Neveu and West [ 16 ], in which they extended their previous work [ 9 ] to the string. They adopted, however, a different gauge fixing at the first-quantized level, and hence they took a different approach to the field theory.
References
Finally we can remove the a-dependence of;~o (with (50)) by using the parameter ~o) (with (52)). Then the remaining field equations are L)~o = ~BXo = PrZo =P~Xo = 0 .
21May 1987
[ 1 ] w. Siegel, Phys. Lett. B 142 (1984) 276. [2] M. Kaku and K. Kikkawa, Phys. Rev. D 10 (1974) 1110, 1823. [3] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744. [4] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B 172 (1986) 186, 195; Kyoto preprint KUNS 829-HE(TH) 86/3, RIFP - 660, 673, 674 (1986). [5] A. Neveu and P. West, Phys. Lett. B 168 (1986) 192; Nucl. Phys. B 278 (1986) 601. [6] I.Ya. Aref'eva and I.V. Volovich, Teor. Mat. Fiz. 67 (1986) 486. [7] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [8] S. Monaghan, Phys. Lett. B 178 (1986) 231. [ 9] A. Neveu and P. West, CERN preprint TH-4547/86. [ 10] W. Siegel and B. Zwiebach, Maryland preprint # 86-195. [ 11 ] E. Witten, Nucl. Phys. B 268 (1986) 253. [12] E.S. Fradkin and G.A. Vilkovsky, Phys. Lett. B 55 (1975) 224. [ 13 ] A.M. Polyakov, Phys. Lett. B 103 (1981 ) 207. [ 14] S. Hwang, Phys. Rev. D 28 (1983) 2614. [15] T. Kugo and I. Ojima, Prog. Theor. Phys. suppl. 66 (1979) 1. [ 16] A. Neveu and P. West, CERN preprint TH-4564/86.