Gauge transformations in string field theory and canonical transformations in string theory

Physics Letters B 284 ( 1992 ) 36-42 North-Holland PHYSICS LETTERS B

Gauge transformations in string field theory and canonical transformations in string theory

Jnanadeva Maharana Institute ofPhv~tcs. Sacht~alala Ma~g. Bhubane~at 751 005. Indta

and

Sudipta Mukherji ' "l ata Instttute o/Fundamental Research. Homt Bhabha Road. Bombay 400 005 Indta

Received 10 February 1992, revised manuscript recelx cd 5 Aprd 1992

We stud', how canonical transformaltons m first quanttzed string theo~ can be understood as gauge transformations in string field theory We estabhsh th~s fact by working out some examples -ks a b3, product, wc could ~dentff', some of the fields appearing m string field theory with their counterparts m the a-model

1. Introduction

The string theory is endowed with a rich symmetry structure. The Ward identities, derived through the introduction of canonical transformations, unravcl the hidden symmetries of string theor). It was shown [ 1,2] that

Ward identities associated with local symmetries such as general coordinate m~arlance, abehan gauge mvarx- ancc (associated with the antls~ mmetrtc tensor field ) and nonabel lan gauge lnvarlance (in the case of compac- tlfied bosonlc strings), could be dcrived m the hamdtonxan path integral formalism. Furthermore, it was sug- gested in ref. [2] (see also ref [3] ) that massive modes of the string might possess local symmetries and the

existence of such symmetries might be manifested through the appropriate new Ward identities. Indeed, it has been argued [4,5] that all the string states arc gauge particles and most of these gauge symmetries are broken spontaneously leaving only the familiar local symmetries as exact tnvanances of the string theo~'. Recently. it has been proposed that quan tum coherence is mainta ined b,~ two-dimensional target space black holes in string theory due to the existence of an infinite set of conserved quan tum numbers. It has been argued [6] that there is an int imate connection between these conserved currents and the stringy (higher) symmetries.

It is natural to expect that the string field theory, ~s the appropriate scttmg to provide a deeper understanding of the underlying (h idden) symmetry structurc in string theory. Now that vve have a consistent covartant quan- tum field theory of closed bosonlc strings [ 7,8 ], we can start asking these questions. This theoD' is characterized b} an infinite number of fields and it possesses non-hnear gauge invarlanee. It has been shown that the general coordinate translbrmatxons arise as a particular combinat ion of gauge transformations in this string field theo~' and the metric can be related to the mfimte component string field through suitable functional relations [9 ].

In th~s letter we will concentrate on some speofic gauge transformations in string field theo~ involving higher level states and will identify those as canomcal transformations of the target space coordinate X" appearing in first quantlzed string theory. In the a-model language these transformations generate symmetries m the space of

E-mad address mukherj~ fitlfr~ax b~lneI

36 0370-2693/92/$ 05 00 © 1992 Elsevier Soence Pubhshers B V All rights rescr',ed

Volume 284, number 1,2 PHYSICS LETTERS B 18 June 1992

the couplings which appear in the a-model action. Throughout this letter we consider only the l inearlzed version of gauge t ransformat ions m string field theory' and at the end we discuss how this can be extended to take care of full non-l inear gauge transformatLons following the prescript ion of ref. [ 9 ].

This letter is orgamzed as follows. First we set our notat ions through stating some known results about gauge lnvanance in string field theory. In section 3 we focus on some part icular gauge transformations. In section 4 we identlf~ these gauge t ransformat ions with some canonical t ransformat ions on X u of the first quant lzed string theory. This automat ica l ly allows us to ldent l~ ' the fields appearing in string field theory with the coupling constants appearing in the a-model describing the first quantlzed string theorry. We conclude in section 5 with some speculat ions and possible extensions of the present work.

2. G a u g e invar iance in s tr ing f ie ld theory"

Here we set our notat ions by stating some known facts about the gauge symmetr ies in non-polynomial closed string field theory'. The configuration space of the string field ~ is generally taken to be a subspace of the full Hllbcrt space .g of ghost number 3 and is annihi la ted by c6- and L(7. The string field theory action involving ~u has the form

~r~ g ~ , - - 2 I ~ `") S(~)=~(~VlQnth7 It /- ' )+ ,Z ~ t ~ , . (2.1)

Here QB is the BRST charge of the first quantnzed theory, and [ } has been defined in refs. [8,10]. The string field t/v can be expanded in some basis as h o l ~ ) = V-Art I ~ r ) and in terms of component fields the action becomes

S ( ~ ) = Z I l(., ) `' = 2 3,-'" . . . . ~'r, ...¢J,, • ( 2 .2 )

The ,4 ~,'~ ~, are constants which are computable using the techniques of conformal field theoD. The classical equat ions of mot ion derived from (2.1) have the form

Qabc7 I ~v) + ~ g~.-2 ,=3 ( N - 1 ) ~ [ ~ v ` ' - , ] = 0 . (2.3)

where (A~ I [A 2 .-'1`' ] ) = ( - 1 ) . . . . ~{,4n ...A ~ } with n~ being the ghost number of A~. The action is lnvar iant under the infini tesimal gauge t ransformat ion

6(bg tP)=Qnb£ A+ g~-2

,,=3 ( N - 2 ) ~ [ ~V`'-2A] " (2.4)

Here A, the infini tesimal gauge t ransformat ion parameter , is a state with ghost number 2 and is anmhl la ted by c6- and L(7. As before A can be expanded in components as be7 I A ) = ~_,,2,, I q,, ). Then (2.4) can be written as

. . . . . . . . . . . . ;t,, ~,, ...q/ . . . . . (2 5) " , = 2

) where B~; ~ .... : are computable constants. Nonce that the infinitesimal gauge t ransformat ions contain l inear as well as non-hnear pans . To hnear order

all the field dependent parts drop out and hence gauge t ransformat ions change the fields by adding BRST exact slates to it.

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Volume 284, n u m b e r 1 2 PHYSICS LE FI ERS B 18 Junc 1992

3. S o m e specif ic off-shel l l inearised gauge t ransformations

In th~s secuon we will concentrate on some specific gauge t ransformat ions which essentmlly amount to choosing some specific A m (2 4). The idea is to choose A m such a way that it can be ~denufied as a canomcal t ransformanon m the a-model constructed from the first quant lzed s tnng t h e o l . Th~s will allow us to ~dentlfy the off-shell field conf iguranons of string field theory with the couphng constants appearing m the a-model.

For that purpose let us choose the gauge t ransformauon parameter to be

j vt~'P , b~7 IA) d 2¢'k Fj, ,.,, (]~) (c~- i ~ct - ~ 5 1 (i)o~ t - i I / ' ) (3.1)

wnh F,, ,.,,: bcmg symmetr ic under v and p mtcrchangc. Nonce that IA ) =c,7 h~3 IA ) sausfies all the reqmred propcrucs of a well defined state m the Hilbert space, n a m e b . ~t has the correct ghost number and xs anmhflated by cff and Lff. Wc now define the string field conf igurauon revolving rank 2.3. and 4 tensor fields as

f " ~" + ~ " c~" o¢'_2) th7 I W ) = d2~k[~,,.,,G~[(o~ "- ~ o~"_~"_ I c~ - ~ L ~ ~ " ~ °z~'- 1 (~)+d, , , , . c~(~(o~ ~o~_1 -2 -1

_ o~"_~( ~( t )+h , , , (o~" ~c~El( '~(_~-a"_~c~E~c~c'_l) (3.2) +fi~,,,(o¢" ~c~"_~c ,c, - t ~ ,

" ~ ° " ~" o: ° )+ ]l&> +I~,,~..c,(I(o~L,o¢ ,o2", _ , + ~ ' ~ c ~ _ l _, - I ..

Here/~,,,,,,. a~.,,.. ~h.,,. ft . . . 17.,,p. are fields of different tensonal ranks (all of them are not necessanl3, ph3slcal) and .. conta ins all o ther fields which are not ~mportant for our present d i scuss ion

,As m e n t i o n e d earher, at the h n e a n z e d level the gauge t r a n s f o r m a n o n s m string field t h e o D reduce to

fi(bff I ~ ) ) = O n h f f 1.4 > . ( 3 . 3 )

The BRST charge has the following decompos luon in terms of mat ter and ghost V~rasoro generator

O.= Z . . . . . ""+~ Le::~')c,,,: +c.c. . (L .... ( 3 4 ) - :e

It IS nov,' s t rmghtforward to calculate Qt~t~,7 IA ) with A gwen in (31 ). Comput ing that, we get

{ ] . 2 /-" ,* k 2

~' o~ "_ v v E , , _ _ " " ~ " - A F . . , , , , o ~ L l , _ +/,"F,. , . , .¢z"~c~ p ~c'_~c,+L l ,: , ,po~Li , c_ ,c , , c , (_ - / , F, , , , o~'~o~_, ,c,

- k " F , , . " , a " , ( ( , - / . " I ' . . , a " , a " _ , c , ( _ , - I ' . . . . : c ~ " , ~ " ,a"_2c , (_ , -F , , , , . . . , a~" , a _ . c r - , G ( - , P / ; { - - - - 1 - - - - - - , , - -

• , - " • ) l k - 1 , 1 , , , : o~L ,o~"_~c~C,~_ ,c , ( , - i , l~,:,.,.c~"_ ,(~'_, ~C~o~ _ , ~ ( , ) . (3.5)

Usmg ( 3 3 ). ( 3.5 ) m ( 3.2 ) we find that thc gauge t ransformed s tnng ficld nov,' has thc following form m tcrms of components :

bd l~U)= f d 2 " k [ ( ~ , , , , + 1'-" ,.,,..)cd (c~"_,c~" _ , c t - c ~ ' _ ' ~" o¢" ( ~ ) , _~ _,

"" - - ~-i G ) + ( J h , , + k ~ . .... : + k " l , : , , , ) ( ~ r " ~ " ~ c _ 1 c l - c ~ ' _ 1 o ~ ( - i

OC t' t~J' t~ v + ( f i , , - L " F , , , p ) ( a ' l _ ~ c l ~ _ l - -1 - i ~ l C - i )

+ ( ~ , , . , ~ + ~ F . , , , , , ) c ~ ( a S . a _ l - l i ~ - -~ i )

+ (Z,,,,,+c, , . , . . . ' : ,at. ,,o,, . , , =)+ Ilk> _ 1 -

J (3.6)

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Volume 284, number 1,2 PHYSICS LETTERS B 18 June 1992

Hence the corresponding fields t ransform as

k 2 ~L, , , , ,= ~ r,,,,,,,, 6~,,~=k"C,,,,,,+k"r,,,,,,v:,

N i l 2 ' , • ,

I )

Let us now redefine the fields as

k a ~ p,,.,, = ~ . . ~ - ~ h,,.po, m,,. = ,h . . -kP~, , , . -k "a ,u , . .

; a , , , , , , = r , , ,,,,.. , ,~/7,,+,° = # ° r , , + , .

3 7 )

- - ~ . p nu. --I1~+ v + k ~Tm. p ,

3.8)

The t ransformat ion laws of all the fields take now the form

(~p,,vv=O. 6m, . . . . 0 . 6 n , , . = O . dd. . , ,=F,,:~p. . ~h, , , . t , .=l. .F.: . , , . . 3.9)

From (2.3) we see that the hnearlzed equation of mot ion in string field theory is Q a b g [ ~ ) =0 . Using this we can easily find the equations of morion for each component field appearing in the expanmon of tp. The equations corresponding to the fields p,,.p, mj.,. and n,,. are purel,, algebraic in nature and can be adennfied as constraint equations. Hence we can set these fields to be zero at least to th~s order. In that case (3.2) reduces to

f ,_6 [ -c~"_ od'_,~ C,) bo ItP) = d k d , , . p [ ( a " , a " _ , ~"_, +cU'_, 6~ " _ ,a" . ) c , e , , _~

+ k " ( a £ ~ d d ' _ ) c _ ~ c ~ - - d ~ " _ t a P _ ~ ( _ ~ d t ) - k " ( a " _ ~ a " _ ~ G ( _ ~ - ~'_ ~ o2"_ ~ (~ c_ ~ ) ]

+ hu./,o( ( a u . . . . t a " 1~ t' t a " I +"~"~-~ a"-I Od'_l O.'"_l )Clgl

+ x , ~ c ~ ( a £ ~ a L , ~ r _ , c ' , - ~ ' L t d ~ ' a " , d , ) . . ] k ) . (3.10)

Notice that here we have replaced aT..t, and ]~..t,. in terms of d,,.p and h,,.,,, using (3.8) . Substi tut ing (3.10) in the quadrat ic part of the action given in (2. l ). ~t is now trlvml to write down the actmn m terms of component fields dj,.p and h. .v . . For later convenience, we write ( 3.10 ) as

b~; l iP) I = d-'k[d.,.,,lA""(h))+h,,.,,.lB""o(k))].'~ (3.11)

where

A, ' "v=[ (a , ' ,a"_~dt '2 + d ~ , ' , ~ " _ , a C . ) c ~ ( , +k~,(a , '_ ,dd '_ ,c_jc t -c~"_, aP_, (_ , ( , )

+ k " ( a g l ~ C l c _ ) c ~ -d~"~a~'_~d_~g~)-kV(a"_~a"_~c~(_~ - c U ; ~ ~"_ ~(~c_ ~ ) ] (3.12)

and

Bm'/'~r=((o~' ,o~' ,~/ ' , ~ ° , +~';td~"_ od' o~ ~ _ k° "+ " o~"_ _ ~"od ' ) . . . . ~ _~ ~)c~(~ + ~. co ( a _ ~ ~dd' ~c~--d~"l i -~(~) . ( 3 1 3 )

Before ending th~s sectmn we would like to ment ion that instead of choosing (3.1) as gauge t ransformatmn parameter , we can as well take

bff IA)= [d26kF 'u l .p l (k ) (o~U_xc , - -d t_ , ( ' , )a"_ ,d lP_ , [ k ) . (3 14) d

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Volume 284. number 1.2 PHYSICS LETTERS B 18 June 1992

Here F'~,l~ m is antlsymmetrlc under interchange of indices v and p. Following the same prescription as before we can find out how the relevant fields change under this gauge transformation.

4. Canonical transformations in first quantized string theory

In first quantlzed string theor~ one writes down the action in terms of the target space coordinate X" in the presence of background excitations of the string. One can as well write the action in terms of the canonically transformed variable X'" and its conjugate momentum P'". In a-model language, these canonical transformations generate a symmetry in the parameter space of the theory. In this section we will concentrate on some particular canonical transformations and will identify them as the gauge transformations m string field theory discussed in the previous section.

Let us make the following canonical transformation [ 11 ]:

X'~'=X" +OX~OX~,!',,~,, . (4 .1)

We can find out how the parameters in the a-model change under th~s transformation We take the a-model action as follows:

.4 = f d2z[ T( X) + (OX"0XP+ OX"OX")G~q,( X) + D,,,,,,(X) (OX"OX~O2X" + 02X"0¥"~¥ ")

+ tl,,,,j,~,(X) ( 0X"0X"0X"0 .¥"+ OX"OX"OX~'OX ° ) + . . ] . ( 4.2 )

Under (4 .1)

yv~ yorj, 6 T ( X ) = T ( X ) ,,.. . . . . . ,:~o: (4.3)

and

6 [ O,~,(OX,OX,'+ OX,'~X; ) ] = ( OX ~OX"OX'~X,',~!%.~ + O :X"OX"OX,'(",,,: ) G,,,, a " P " "]l?' /) + (OX, ,OX~.¥o~X~! , . . + ~ 2 ¥ 0h 0A ~:,,.: )G,,,,+ (OX"a V',~ V"~X,'~,.',,~. + O~X"~X~.¥,'~,',,~ )(/,,,,

+ (OX"O ¥,'OX"O.¥'(!'~.:, + 02.¥*OXpOX"(!',,,, )G,o, + (0.¥" 0.¥"0X"OX,'~,, . G,,,,. + O ~,'0 ¥"0X"0.¥~,~,,,: G,,,,, ) .

(4 .4)

Comparing (4 .2) and (4 .3) we see that under (4 1) the transformed gravlton becomes

G',,~( ¥)=G,,,,(X)+,~]',,~ T ~ ( X ) . (4 5)

Similarly comparing (4 .4) and (4 .2) we see that the rank 3 and 4 tensors are going to mix with the gravlton as

m;,,,p,,(X)=ll,,,,,,~(X)+2r.~,,,,, ~G~.(X)+g.$~G,,,,.(X). D'j,,,t,( ¥ ) = D . ~ , ( X ) + 2~,,,G,,.( ¥ ) . (4.6)

If we expand now G.,, around the flat background as (;.~=q.,,+h,,,,, we get from (4 .6) (keeping only those terms which are lowest order in the fields)

H'a,,,,.(X) = II.,,~,.(X) + 2¢~ .~,, ~ (4 .7)

and

D;.,,,(X) =D,,.,,(X) + 2;~/,, . . . . (4 .8)

The gravlton remains unchanged to this order. Now looking at the transformation laws of various fields appearing in string field t h e o ~ under the gauge transformation discussed earlier (see (3 .8) ). it is easy to see that the

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Volume 284. number 1,2 PHYSICS LETTERS B 18 June 1992

canonical t ransformation (4.1) does the same job in the first quantlzed theory if we make the following identification ~ '

d u , , , , = D . , p , h.~.~,=H,, , , , ,~, (4 9)

and

f d26kF,,i.v,,(k) exp(lkx)=(.,,.v)(x) (4.10) 2

In other words, the above Identifications allow us to relate a particular canomcal transformation m the a-model

with a gauge transformation m the string field theory. Finally, notice that instead of (4.1) we can choose a transformation

¥'~'= X " + OX"OA'v~'('vj. (4. I l )

In the language of strmg field theory this would presumably be a gauge transformation with parameters given m (3.14) if we make a relevant field ldentxficatlonal though we have not checked that explicitly.

5. Conclusion

To summarize, we have shown how canonical transformations in first quantlzed string theory can be understood as off-shell gauge transformations of some underlying string field theory. We could show that by taking specific examples and working at the linearlzed level. As a by product, we could make the identification of some of the fields appearing in string field theory wtth their counterparts in the a-model.

Notice that the whole analysis can be performed for one-dimensional strings coupled to gravity theories with ver~ little modification. It is well known that these theories have a two-dimensional target space mterpretatton. Hence the general couplings are functions of two variables, one being the conformal mode of the two-dlmen-

stonal target space metric. It has been noticed earlier that these theories possess an infimte sequence of discrete states, besides the tachyon ,2 [ 13,14]. Hence one can write down a a-model action mvolvmg these states. Con-

sequently the canonical transformations are now suitable redefinitions of the two-dimensional target space co- ordinates X"(/~= l, 2). On the other hand, one can construct a string field theory action [15,16] for these theories. To make the total central charge 26, one needs to introduce a background charge for such non-critical string theories. Hence the Vlrasoro generators and the BRST charge get modified. But the structure of the gauge transformaUons remain the same as (2.4 t. Now if we take the canonical transformations (4. l ) for these two- dtmcnsJonal theories, they wtll certainly have a corresponding gauge transformation parameter in string field theory similar to (3.1). It is known [ 17,6 ] that such canonical transformations are responsible for W~ symmetries for two-dimensional string theories (see also refs. [ 18,19] ). Once one finds the corresponding gauge transformation parameters in string field theory, it will be interestmg to check that explicitly and analyse the consequences following ref. [ 20 ]

Throughout this letter we have considered only the llnearlzed gauge transformations and so we could compare onl) the quadrattc part of the string field theory action with the correspondmg part tn the a-model action. If we want to go beyond the linear level, the changes of the component fields m (3.2) will be completely non-linear. Similarly, for the case of a canonical transformatton in the a-model action, in (4.7) and (4.8) we need to keep higher order field dependent terms. It would be mterestlng to investigate whether similar kmds oftdentlfiCatlons

"' Nouce that I 4 u''p) that comes with the field d~,,, m (3 I I ) ,s nat just the vertex operator (dXU0l ~02Xv+OX"OX~O:X t') (that comes m the a-model with Do,, v) expressed m terms of oscillators but along wtth it there are some momentum dependent terms The similar Is true for [B " ' ° )

.2 In ~ = 1 matrix models it first appeared m ref [ 12 ]

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Volume 284, number 1.2 PHYSlCS LETTERS B 18 June 1992

as ( 4 . 9 ) c a n be m a d e . T h i s c a n be d o n e s y s t e m a t i c a l l y f o l l o w i n g t h e a l g o r i t h m d e v e l o p e d m ref. [ 9 ] .

We h o p e to c o m e b a c k to t h e s e xssues m t h e fu tu r e .

Acknowledgement

W e a r e e x t r e m e l y g r a t e f u l to A s h o k e Sen fo r s e v e r a l d ~ s c u s s l o n s a t all s t ages o f t h t s w o r k a n d fo r h i s c o m m e n t s

o n t h e m a n u s c r t p t . W e w o u l d a l so h k e to t h a n k S u m t t D a s fo r i n t e r e s t i n g d ~ s c u s s t o n s a n d C. S c h u b e r t fo r c o m -

m e n t s o n t h e m a n u s c r t p t .

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Gauge transformations in string field theory and canonical transformations in string theory