- Email: [email protected]
On a gauge covariant formulation of string field theories
Volume 180, number 1,2
PHYSICS LETTERS B
6 November 1986
ON A G A U G E COVARIANT F O R M U L A T I O N O F S T R I N G F I E L D T H E O R I E S Ju-Fei T A N G and Chuan-Jie Z H U
The GraduateSchool,Academia Sinica, P.O. Box 3908, Beijing, PR China Received 21 July 1986; revised manuscript received 7 August 1986
It is shown that the Neveu-Nicolai-West formulation of the gauge covariant string field theories and that of Banks and Peskin can be obtained by different consistent truncation of the BRST multiplets. A proof is given to show the equivalence of light-cone formulation and the gauge covariant formulation without using the property of trivial cohomology of string differential forms.
1. Recent remarkable progress in superstring theory has revived the interest in constructing a manifestly covariant string field theory. Began with Siegel's formulation of a covariantly gauge-fixed bosonic string [ 1 ], there are a lot o f works to find the original gauge-invariant action of free bosonic strings (see ref. [ 2 ] for a complete listing of references). In particular, three groups [ 3 - 5 ] succeed in constructing a linearized local string field theory from which the formulation of [ 1 ] can be recovered by gauge fixing. Later, a simplified formulation was discovered [6]. The connections between all of these formulations were partially explored [2]. In ref. [ 7], the equiva lence o f the light-cone formulation and the gauge covariant formulation is proved in two different ways. The first proof via the equation of motion depends on an unproved assertion that the space of free strings have trivial cohomology, and it is incomplete. The second proof is a complete one. In this paper, we use the idea of "consistent truncation" of the BRST multiplets introduced in ref. [4] and show that the action ofrefs. [ 3,6 ] can be obtained by different consistent truncation. Via the equation of motion, a proof is given to show the equivalence of light-cone formulation and the gauge covariant formulation without using the property of trivial cohomology of string differential forms. 2. Following ref. [4], Siegel's action of free bosonic string is #
where QB is the BRST charge (see below),
~)= ~ n C Y C , ,
I~U)=l~)+Col~U).
n--I
Because of the nilpotency of Qs, Ss~c~¢~is manifestly invariant under the following BRST transformation:
8BT=Qs~
.
(2)
As in ref. [ 2 ], we break QB into pieces as follows:
QB = C o K - 2~Co + d + ~ ,
(3)
where
K = Lo + L ~h°St -
1 ,
[mC. CinCh+,. n--I
n--I
m=l
- ( n + 2m)CV+ C,nC.+,.] , 6= ~ C n L _ . n--I
~
Z
n--I
m
[mC.++mC~Cm 1
+(n+m)C++,,,C,,,C.] . Then the BRST transformation is 8BO= - 2~)~, + ( d + ~ ) 0 ,
(4a)
8B~=KO-- (d+fi)~.
(4b)
Expanding ~ and ~' according to the ghost number, we have
Ssic~, = ½ J d C o ( ~ [ [QB, CoC'o] }gj) = - ½(01KI0) + (~ul~ I q/), 50
(1)
n
--
~
n
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 180, number 1,2
PHYSICS LETTERS B
here 4 , and 7", have ghost number n. Then the BRST transformation can be written as 8B4p_ I = - - 2 ~ T p + I - } - d 4 p + g 4 p , 5B 7" p = K 4 p - - d T " p+ l - - d T .+ l •
(6)
The idea o f consistent truncation [4] is as follows: Elimination of the field ~ is compatible with the BRST transformation if 8 B~ can also be set equal to zero. From eq. (6), we can see that by setting all 4p_ and 7"p equal to zero with p~< 0, we must have --21)Ti + d 4 o + ~ 4 o , (7)
K 4 o - d T " l - ~ 7" : = 0 .
6 November 1986
multiplets [4] is different from ours. In fact their truncation recovered the action found by Banks and Peskin [ 3 ]. The crucial point to understand this conclusion is to note that the projection operator K (v) in ref. [4] is actually the projector of the maximally symmetrized string differential form (also noted in ref. [4]). That is, we have [~)(K Cp)4 ) ] m...~................ = 0 .
(11 )
In summary, both the Neveu-Nicolai-West action and the Banks-Peskin action can be obtained by different consistent truncation. It is also possible to obtain the superstring action [ 9,10 ] from the gaugefixed action discovered by a number of authors [ 11,12 ] by this same method.
SO our first consistent trunction is obtained by setting all 7"p and 4p_ ~equal to zero with p ~<0 and with 4 o and 7" i satisfying eq. (7). It leads then to a action of the form
3.Our notation for the light-cone formulation is the same as given in ref. [7]. We expand 4 o and T I as follows:
S=-½(4o
(~0 = ~ ~k k , k--0
I K I 4 o ) + ( 7 ' , 11)[7',)
+(2: 1(-2~7", +d4o+64o)),
(8)
where 2j is a lagrangian multiplier. By using the equation of motion, we have )t~ = T~. Accordingly we obtain the gauge invariant action (also derived by the truncation in ref. [ 8 ] :~) S=-½(4o
(9)
This action is identical with the action of ref. [ 6 ] and the BRST transformation is just the gauge transformation if we consider 4~ and 7"2 as gauge parameters, 8B40=--2~7"2+d41 8B TI = K 4 1 - d T 2 - g
+~4l 7"2 ,
,
In the notation of ref. [3], Ok k is a (~)-form and ~0k k+ J is a (k~ ~)-form. The action in eq. (9) becomes S:
~[~ [ - - ½ ( O l , - k l K l O k k ) - - ( ~ O k k + l l ~ , l ~ k k + l k--O
(10)
We have not seen this preprint until now. It is a pleasure to thank the referee for pointing out this reference to us.
)
(13)
The equations o f motion are KOkk--(d(Ok
(14)
I k+t~tflkk+l) = 0 ,
--21)t,Ok k+ I + (dOk ~ +00k+ 1 k+l) = 0 .
(15)
The gauge transformations are 8Aokk=--2~)Ak l~+J + d A k _ l k + O A k
and the hidden symmetry is the BRST transformation of 4~ and ~-/2 with parameters {~2 and 7"3, and the hidden symmetry is the BRST transformation of 4 2 and 7"3 with parameters 4 3 and 7"4, etc. Thus we recovered the new symmetries discovered in ref. [ 6 ]. The original consistent truncation of the BRST :r
(12)
~ k k+l
k--0
..l_((flkk+l l ( d Û k k s F ( ~ k + l k+l))] •
I K 1 4 o ) - ( 7 " , I~1 7",)
+ ( 7", I ( d + O ) 4 o ) .
~'11~--- ~
8A(flk k+l = K A k k+l - d A k
k+l ,
(16)
i k+l __6Ak~,-+2 .
(17)
Begin with the equation for 0o o and ~0o ~: KOo °-6q~o i = 0 ,
(18)
-24)~o ' +d0o o +~0~ J = 0 .
(19)
After using the gauge freedom to set all components o f ¢ o o with Y k k 2 k > O to zero, (18) becomes a set of two equations, because the left-hand side of (18) has 51
Volume 180, number 1,2
PHYSICS LETTERS B
~kk).k=O while the r i g h t - h a n d side has Ykk2k>0. Thus K~o o = 0 ,
(20)
dq~o ~ = 0 .
(21)
F r o m ( 2 1 ) we h a v e ~0o ~ = d E o 2 w h i c is p r o v e d by Peskin a n d T h o r n as n o t e d in ref. [ 7 ] . So eq. (19) becomes
dOo°+d(¢~ ~ - 2 ~ E o 2 ) = 0 .
(22)
F r o m this e q u a t i o n , we can s h o w that L n O o ° = 0 ( n > 0 ) just as is d o n e in ref. [7]. N e x t c o m e the e q u a t i o n s for ~t ~ a n d ~0~ 2 after setting d0o ° = 0 and ~0o t = 0 ,
- 2O~Okk+~ + d C k + l k+dOk ~ = 0 .
6 November 1986
(28)
T h e gauge t r a n s f o r m a t i o n s are
~)~¢~k k = d A k ~+l ,
(29)
8A~0k k+, = K A k k+ I _ dA~ k+2 .
(30)
A f t e r c h o o s i n g Ak k+~ = - K Jq~kk+t a n d A k k + 2 = 0 , we h a v e 0kk+~SAC~kk=0 a n d ~0k k+~ + 8 ~ 0 k k+~ = 0 , thus gauge fixed Ok k a n d ~0k k+ ~ to zero. T h i s c o m p l e t e s o u r analysis o f the e q u a t i o n s o f m o t i o n o f free b o s o n i c string t h e o r y a n d p r o v e s the e q u i v a l e n c e o f the light-cone f o r m u l a t i o n a n d the gauge c o v a r i a n t f o r m u l a t i o n o f the free b o s o n i c string. It is e v i d e n t that we d o n o t use the p r o p e r t y o f trivial c o h o m o l o g y o f string d i f f e r e n t i a l forms.
KOl I -d~o~ 2 = 0 ,
(23)
-20~0
(24)
We w o u l d like to t h a n k D . D . W u a n d X.J. Z h o u for d i s c u s s i o n a n d Y i - B i n g D i n g for careful r e a d i n g o f the m a n u s c r i p t .
8~Ol t=~)A~ 2 ,
(25)
References
8A~01 2 = K A I 2 - - S A l 2
(26)
[1] w. Siegel, Phys. Lett. B 151 (1985) 391. [2] T. Banks, M.E. Peskin, C.R. Preitschopf, D. Friedan and E. Martinec, preprint SLAC-PUB-3853 (1985 ). [ 3 ] T. Banks and M.E. Peskin, Nucl. Phys. B 264 (1986) 513. [4] K. Itoh, T. Kugo, H. Kunitomo and H. Ooguri, Prog. Theor. Phys. 75 (1986) 162. [5] W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105. [6] A. Neveu, H. Nicolai and P.C. West, Phys. Lett. B 167 (1986) 307. [7] M.E. Peskin and C.B. Thorn, Nucl. Phys. B 296 (1986) 509. [ 8 ] H. Terao and S. Uehara, preprint RRK 86-3 (January 1986). [9] A. Neveu and P. West, Phys. Len. B 165 (1985) 63. [10] H. Aratyn and A.H. Zimerman, Phys. Lett. B 166 (1986) 130. [ 11 ] A. LeClair, Phys. Len. B 168 (1986) 53. [ 12] H. Terao and S. Uehara, Phys. Len. B 168 (1986) 70.
~ 2 --]-~2
2 --~d~l 1 = 0 •
T h e gauge t r a n s f o r m a t i o n s are
H e r e we r e d e f i n e d 0~ ~ a n d ~0~ 2 to i n c l u d e the fixed gauge p a r a m e t e r s Ao 1 a n d Ao 2, that is, ¢ ~ is ¢~ ~ + ( - 2 ~ A o 2 + d A o ~) and ~0~ 2 is~o~ 2 - . F ( - d A o 2 ) . After choosing A t 2 = _ K - ~~0~ 2 and A ~ 3 = 0, we h a v e ~ ~ + 8.¢~ ~ = 0 a n d ~0~2 --F~A~01 2 = 0. T h i s p r o c e d u r e can be e x t e n d e d to e v e r y pair o f 0k k a n d ~0k k+~ G e n e r a l l y , after fixing the gauge f r e e d o m A . "+ ~ a n d An n+2 ( 0 ~ < n < k ) , we h a v e d 0 o ° = 0 , ~Oo~=0, ¢ . " +8A¢~." = 0 a n d ~o. "+~ +SAgo. "+~ = 0 . T h e e q u a tions for q~k~ a n d ~o~k+~ b e c o m e
KO~ k__d~ k k + l = 0 ,
52
(27)
PHYSICS LETTERS B
6 November 1986
ON A G A U G E COVARIANT F O R M U L A T I O N O F S T R I N G F I E L D T H E O R I E S Ju-Fei T A N G and Chuan-Jie Z H U
The GraduateSchool,Academia Sinica, P.O. Box 3908, Beijing, PR China Received 21 July 1986; revised manuscript received 7 August 1986
It is shown that the Neveu-Nicolai-West formulation of the gauge covariant string field theories and that of Banks and Peskin can be obtained by different consistent truncation of the BRST multiplets. A proof is given to show the equivalence of light-cone formulation and the gauge covariant formulation without using the property of trivial cohomology of string differential forms.
1. Recent remarkable progress in superstring theory has revived the interest in constructing a manifestly covariant string field theory. Began with Siegel's formulation of a covariantly gauge-fixed bosonic string [ 1 ], there are a lot o f works to find the original gauge-invariant action of free bosonic strings (see ref. [ 2 ] for a complete listing of references). In particular, three groups [ 3 - 5 ] succeed in constructing a linearized local string field theory from which the formulation of [ 1 ] can be recovered by gauge fixing. Later, a simplified formulation was discovered [6]. The connections between all of these formulations were partially explored [2]. In ref. [ 7], the equiva lence o f the light-cone formulation and the gauge covariant formulation is proved in two different ways. The first proof via the equation of motion depends on an unproved assertion that the space of free strings have trivial cohomology, and it is incomplete. The second proof is a complete one. In this paper, we use the idea of "consistent truncation" of the BRST multiplets introduced in ref. [4] and show that the action ofrefs. [ 3,6 ] can be obtained by different consistent truncation. Via the equation of motion, a proof is given to show the equivalence of light-cone formulation and the gauge covariant formulation without using the property of trivial cohomology of string differential forms. 2. Following ref. [4], Siegel's action of free bosonic string is #
where QB is the BRST charge (see below),
~)= ~ n C Y C , ,
I~U)=l~)+Col~U).
n--I
Because of the nilpotency of Qs, Ss~c~¢~is manifestly invariant under the following BRST transformation:
8BT=Qs~
.
(2)
As in ref. [ 2 ], we break QB into pieces as follows:
QB = C o K - 2~Co + d + ~ ,
(3)
where
K = Lo + L ~h°St -
1 ,
[mC. CinCh+,. n--I
n--I
m=l
- ( n + 2m)CV+ C,nC.+,.] , 6= ~ C n L _ . n--I
~
Z
n--I
m
[mC.++mC~Cm 1
+(n+m)C++,,,C,,,C.] . Then the BRST transformation is 8BO= - 2~)~, + ( d + ~ ) 0 ,
(4a)
8B~=KO-- (d+fi)~.
(4b)
Expanding ~ and ~' according to the ghost number, we have
Ssic~, = ½ J d C o ( ~ [ [QB, CoC'o] }gj) = - ½(01KI0) + (~ul~ I q/), 50
(1)
n
--
~
n
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 180, number 1,2
PHYSICS LETTERS B
here 4 , and 7", have ghost number n. Then the BRST transformation can be written as 8B4p_ I = - - 2 ~ T p + I - } - d 4 p + g 4 p , 5B 7" p = K 4 p - - d T " p+ l - - d T .+ l •
(6)
The idea o f consistent truncation [4] is as follows: Elimination of the field ~ is compatible with the BRST transformation if 8 B~ can also be set equal to zero. From eq. (6), we can see that by setting all 4p_ and 7"p equal to zero with p~< 0, we must have --21)Ti + d 4 o + ~ 4 o , (7)
K 4 o - d T " l - ~ 7" : = 0 .
6 November 1986
multiplets [4] is different from ours. In fact their truncation recovered the action found by Banks and Peskin [ 3 ]. The crucial point to understand this conclusion is to note that the projection operator K (v) in ref. [4] is actually the projector of the maximally symmetrized string differential form (also noted in ref. [4]). That is, we have [~)(K Cp)4 ) ] m...~................ = 0 .
(11 )
In summary, both the Neveu-Nicolai-West action and the Banks-Peskin action can be obtained by different consistent truncation. It is also possible to obtain the superstring action [ 9,10 ] from the gaugefixed action discovered by a number of authors [ 11,12 ] by this same method.
SO our first consistent trunction is obtained by setting all 7"p and 4p_ ~equal to zero with p ~<0 and with 4 o and 7" i satisfying eq. (7). It leads then to a action of the form
3.Our notation for the light-cone formulation is the same as given in ref. [7]. We expand 4 o and T I as follows:
S=-½(4o
(~0 = ~ ~k k , k--0
I K I 4 o ) + ( 7 ' , 11)[7',)
+(2: 1(-2~7", +d4o+64o)),
(8)
where 2j is a lagrangian multiplier. By using the equation of motion, we have )t~ = T~. Accordingly we obtain the gauge invariant action (also derived by the truncation in ref. [ 8 ] :~) S=-½(4o
(9)
This action is identical with the action of ref. [ 6 ] and the BRST transformation is just the gauge transformation if we consider 4~ and 7"2 as gauge parameters, 8B40=--2~7"2+d41 8B TI = K 4 1 - d T 2 - g
+~4l 7"2 ,
,
In the notation of ref. [3], Ok k is a (~)-form and ~0k k+ J is a (k~ ~)-form. The action in eq. (9) becomes S:
~[~ [ - - ½ ( O l , - k l K l O k k ) - - ( ~ O k k + l l ~ , l ~ k k + l k--O
(10)
We have not seen this preprint until now. It is a pleasure to thank the referee for pointing out this reference to us.
)
(13)
The equations o f motion are KOkk--(d(Ok
(14)
I k+t~tflkk+l) = 0 ,
--21)t,Ok k+ I + (dOk ~ +00k+ 1 k+l) = 0 .
(15)
The gauge transformations are 8Aokk=--2~)Ak l~+J + d A k _ l k + O A k
and the hidden symmetry is the BRST transformation of 4~ and ~-/2 with parameters {~2 and 7"3, and the hidden symmetry is the BRST transformation of 4 2 and 7"3 with parameters 4 3 and 7"4, etc. Thus we recovered the new symmetries discovered in ref. [ 6 ]. The original consistent truncation of the BRST :r
(12)
~ k k+l
k--0
..l_((flkk+l l ( d Û k k s F ( ~ k + l k+l))] •
I K 1 4 o ) - ( 7 " , I~1 7",)
+ ( 7", I ( d + O ) 4 o ) .
~'11~--- ~
8A(flk k+l = K A k k+l - d A k
k+l ,
(16)
i k+l __6Ak~,-+2 .
(17)
Begin with the equation for 0o o and ~0o ~: KOo °-6q~o i = 0 ,
(18)
-24)~o ' +d0o o +~0~ J = 0 .
(19)
After using the gauge freedom to set all components o f ¢ o o with Y k k 2 k > O to zero, (18) becomes a set of two equations, because the left-hand side of (18) has 51
Volume 180, number 1,2
PHYSICS LETTERS B
~kk).k=O while the r i g h t - h a n d side has Ykk2k>0. Thus K~o o = 0 ,
(20)
dq~o ~ = 0 .
(21)
F r o m ( 2 1 ) we h a v e ~0o ~ = d E o 2 w h i c is p r o v e d by Peskin a n d T h o r n as n o t e d in ref. [ 7 ] . So eq. (19) becomes
dOo°+d(¢~ ~ - 2 ~ E o 2 ) = 0 .
(22)
F r o m this e q u a t i o n , we can s h o w that L n O o ° = 0 ( n > 0 ) just as is d o n e in ref. [7]. N e x t c o m e the e q u a t i o n s for ~t ~ a n d ~0~ 2 after setting d0o ° = 0 and ~0o t = 0 ,
- 2O~Okk+~ + d C k + l k+dOk ~ = 0 .
6 November 1986
(28)
T h e gauge t r a n s f o r m a t i o n s are
~)~¢~k k = d A k ~+l ,
(29)
8A~0k k+, = K A k k+ I _ dA~ k+2 .
(30)
A f t e r c h o o s i n g Ak k+~ = - K Jq~kk+t a n d A k k + 2 = 0 , we h a v e 0kk+~SAC~kk=0 a n d ~0k k+~ + 8 ~ 0 k k+~ = 0 , thus gauge fixed Ok k a n d ~0k k+ ~ to zero. T h i s c o m p l e t e s o u r analysis o f the e q u a t i o n s o f m o t i o n o f free b o s o n i c string t h e o r y a n d p r o v e s the e q u i v a l e n c e o f the light-cone f o r m u l a t i o n a n d the gauge c o v a r i a n t f o r m u l a t i o n o f the free b o s o n i c string. It is e v i d e n t that we d o n o t use the p r o p e r t y o f trivial c o h o m o l o g y o f string d i f f e r e n t i a l forms.
KOl I -d~o~ 2 = 0 ,
(23)
-20~0
(24)
We w o u l d like to t h a n k D . D . W u a n d X.J. Z h o u for d i s c u s s i o n a n d Y i - B i n g D i n g for careful r e a d i n g o f the m a n u s c r i p t .
8~Ol t=~)A~ 2 ,
(25)
References
8A~01 2 = K A I 2 - - S A l 2
(26)
[1] w. Siegel, Phys. Lett. B 151 (1985) 391. [2] T. Banks, M.E. Peskin, C.R. Preitschopf, D. Friedan and E. Martinec, preprint SLAC-PUB-3853 (1985 ). [ 3 ] T. Banks and M.E. Peskin, Nucl. Phys. B 264 (1986) 513. [4] K. Itoh, T. Kugo, H. Kunitomo and H. Ooguri, Prog. Theor. Phys. 75 (1986) 162. [5] W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105. [6] A. Neveu, H. Nicolai and P.C. West, Phys. Lett. B 167 (1986) 307. [7] M.E. Peskin and C.B. Thorn, Nucl. Phys. B 296 (1986) 509. [ 8 ] H. Terao and S. Uehara, preprint RRK 86-3 (January 1986). [9] A. Neveu and P. West, Phys. Len. B 165 (1985) 63. [10] H. Aratyn and A.H. Zimerman, Phys. Lett. B 166 (1986) 130. [ 11 ] A. LeClair, Phys. Len. B 168 (1986) 53. [ 12] H. Terao and S. Uehara, Phys. Len. B 168 (1986) 70.
~ 2 --]-~2
2 --~d~l 1 = 0 •
T h e gauge t r a n s f o r m a t i o n s are
H e r e we r e d e f i n e d 0~ ~ a n d ~0~ 2 to i n c l u d e the fixed gauge p a r a m e t e r s Ao 1 a n d Ao 2, that is, ¢ ~ is ¢~ ~ + ( - 2 ~ A o 2 + d A o ~) and ~0~ 2 is~o~ 2 - . F ( - d A o 2 ) . After choosing A t 2 = _ K - ~~0~ 2 and A ~ 3 = 0, we h a v e ~ ~ + 8.¢~ ~ = 0 a n d ~0~2 --F~A~01 2 = 0. T h i s p r o c e d u r e can be e x t e n d e d to e v e r y pair o f 0k k a n d ~0k k+~ G e n e r a l l y , after fixing the gauge f r e e d o m A . "+ ~ a n d An n+2 ( 0 ~ < n < k ) , we h a v e d 0 o ° = 0 , ~Oo~=0, ¢ . " +8A¢~." = 0 a n d ~o. "+~ +SAgo. "+~ = 0 . T h e e q u a tions for q~k~ a n d ~o~k+~ b e c o m e
KO~ k__d~ k k + l = 0 ,
52
(27)