Constraint algebra of open string field theory in midpoint coordinates

Volume 198, number 2 PHYSICS LETTERS B 19 November 1987

C O N S T R A I N T ALGEBRA OF O P E N S T R I N G F I E L D T H E O R Y IN M I D P O I N T C O O R D I N A T E S

Robertus POTTING, Cyrus TAYLOR and Boris VELIKSON Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA

Received 1 July 1987

We study the canonical structure of string field theory in midpoint coordinates. We find the constraint algebra, which consists of second class constraints, together with dependent first class constraints. Exploiting properties of the relevant operators, we show that the constraint algebra is the same in the interacting theory as in the free theory, at least on Fock-space states. We discuss the gauge transformations generated by the first class constraints, and show that they can be used to gauge away unphysical fields.

Witten has proposed a field theory o f interacting strings [ 1 ]. The Feynman rules for this theory have been derived in a first-quantized formalism [ 2 ]; the gauge fixing of the second-quantized formalism has been studied, but the measure in the path integral has not been clearly defined [3]. This issue can be addressed (by the techniques o f Batalin, Fradkin, and Vilkovisky, for example) ~' once the canonical structure o f the theory is understood. In this letter, we report on the canonical structure and constraint algebra of Witten's theory. We find that the algebra consists of second class constraints, together with dependent first class constraints. Remarkably, the algebra is the same in the interacting theory as in the free theory, at least on Fock-space states.

We begin by recalling the conventional expansion of the string coordinate x " ( a ) :

xU(~)= 1 a x e + i xu~ cos(na) . (1)

In terms of these coordinates, we have

C~o =iS/~Xo,

a.=(i/v/2)(a/axl.I +nxlm) (n¢O) ,

t Address after September 1987: Laboratoire de Physique Th6o- rique et Hautes l~nergies, Universit6 Paris VII, Tour 24, 5e 6tage, 2 place Jussieu, F-75251 Paris Cedex 05, France.

~ See the recent reviews in ref. [4].

L , = ~ m=-o~ : a , _ m a m : , (2)

where the au~ are creation/annihilation operators, and the L , are the generators o f the Virasoro algebra. The BRST operator, Q, expressed in terms of the Ln and the ghost/antighost operators fin, ft. is

1 Q = ~ L . f l _ . - ~ 2 (n-m):fl_.P_mfl.+m:. (3)

n n , m

We express the ghost operators in terms of the hermitian ghost/antighost coordinates

c(a) = Co + c, cos(na) , n 1

g(a) = gn s in(na) (4) 1

by

ft. = (1/x/2)(c . + i ~ / ~ ( . ) ,

fl,,=(1/w/2)(i(,,+~/~Cn) ( n > 0 ) , (5)

with the f l_ . , /~_ , defined by hermitian conjugation of these expressions.

The action for the free string field q~ is

I v = ~ q~.Qq~, (6) d

where

184 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


f A * B = i A ( z J B(Z 2) / ~ T ( z l , Z 2 ) . (7) d z I d z z

The forms A and B in this expression have opposite Grassmann parities to those of the functions A (z ~ ) and B(z2), respectively, because

dz = dc0 dxo f i dx~ (idG dG) (8) 11:1

is Grassmann-odd. The twisted 0-function OT(z l, Z z) is given by

OT(z ' , z 2) =6(X~ --X~)O(C~ +ca)

"~ C m ) ( ~ ( C m -x,, ,)O(c,. +&m) x l - [ O(x;. ~ , 2 - , in even

X ]-[ 5(x~ +x.)6(c.2 t _c2)O(g~_g2) . (9) n even

Our expressions are slightly simplified if we take q~ to be a field of arbitrary ghost number. The re- striction to the physical sector h g q ) = - ½ q~ is trivi- ally taken care of by introducing appropriate projection operators.

Interactions can heuristically be introduced by defining

f d * M * cgocf ~¢(z ' )dz ' ~ ( z 2) d z 2 cg(z3) dz 3

xO(x~o - x ~ +B,.(x3,. - x ' . . ) )

2 l 2 XO(Xo-Xo +B,.(x, .-X~m))

X O ( C o 3 l 2 1 2 3 - D. ( c. - c . ) )5( Co - D.( c. - c . ) )

XO(c~ -D11(c~ -c~))

xl - I [0(x113 -A. , . (x ,: - x , . ) ) ii

1 2 3 x,~(x11 - A . , . ( x , . - x , . ) ) 2 3 1 x ~ ( x . -A11,.(x,,, - x , . ) ) ]

x I 1 [O(c,.3 _ C,..(c11' -c.))~(c,.~ ' - Cm.(C~ - c3 ) ) m

XO(&,. 3 - C , . . ( c . - c . ) ) ]

- E , . . ( c. - c . ) )O( c,,, - E , . . ( c. _g3) ) X R [ O( (73n -1 -2 -1 -2 m

-2 -3 -1 ×6(c,, -E,,11(c, - c , )) ], (10)

where r labels the three strings, n runs over odd positive integers, and m runs over positive even inte-

gets. There is an implicit summation over repeated indices, and

4 m 2 A n m = ( _ _ ) ( m + n - l ) / 2

n (n 2 - m 2 ) n '

~m = , / 5 ( - ) m,~ ,

4 ( --)(m+"-l)/en Crnn - - :g n2 _ m2 ,

D11 2x//~ ( _ )(n-l)/2 7g n

4 ( - ) ( . . . . z)/2 m Emil- rn2 n2 (11)

The 0-functions implement the overlap constraints

x ( r ) ( o ") : x ( r + 1 ) ( ~ _ _ 0 ") ,

c(') ( a) = - c ( ' + ~ ~ ( lr - r r ) ,

¢ ( r ) (O") = e ( r + 1 ) ( n - - 0 ")

( r= 1, 2, 3 cyclically, 0~a~< ½n). (12)

While we have not proven that (10) is equivalent to othe representations of Witten's interaction [5 ], we strongly suspect that this is the case when the expression is well-defined, i.e., on Fock-space states. The only explicit property of (10) which we need, expressed below in eq. (37), follows from the work of Gross and Jevicki [6] and Hlou~ek and Jevicki [7].

Previous studies of the constraint algebra of string field theory have proceeded by constructing a hamiltonian generating translations with respect to x °, the center-of-mass time coordinate [ 8 ]. At the level of the free theory, this seems an obvious choice. It is obvious from the form of (10), however, that Wit- ten's interaction is non-local with respect to x °. As has been noted previously [ 9 ], the interaction is local and possesses a canonical structure with respect to x°(½n). Since the meaning of a "hamiltonian" which is non-local in time is somewhat obscure, we adopt x°(½n) as our choice for the time coordinate of the hamiltonian system.

The transition to center-of-string coordinates is effected by the shift [ 10 ]

E m even, ~ 0

Xm=Xm • (13)

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The jacobian of the transformation is unity. This shift induces the transformation

8/8Xo =8/8x~ ,

8/SXn = 8/8X',, ( n odd) ,

8/Sx , .=8/Sx , ' .+x /~( - )" /28x~ (m even), (14)

and hence

O/0 = O/~,

o~. = ol;, (n odd),

a , , = o G + ( - ) ' " / 2a6 (m even), (15)

and

1 Lo=~ Z °~o2+CZo ~ (--)m/~o~,

m even m even, S0

~ ' , + O~_kO~k-- 1 , k=l

1 LN=2 E Ol' a'±Ol ' --)(N-n)/2an k~0, N n odd

( N o d d ) ,

1 1 m even , M--k k

m even, # 0

( M e v e n # 0 ) . (16)

The coefficient of the (ao) 2 term in the expression for the even-moded L,, is naively divergent. In what follows, we regularize this by noting that

Y= X 1 = 1 + 2 X 1 in even m even > 0

= l + [¢(0) - n(0)] = 0 . (17)

The general form of the results we report below (second class nature of the constraint algebra) does not depend on the specific value of y, but appears to be simplest for y -- O.

It follows that Q can be expressed in terms of the shifted coordinates as

Q=iB O/Ot+ f ' , (18)

where

j~_~_N/~ i ( , eve~n,>0 (__)1 ,2 ,/~xO,)

~((CO+~/~ E (--)m/'Cm) m even, > 0

+2(kod~d,>O(--)(k+l)/2k-~O' )

X ( n odd,> 0 Z (--)(n+l)/2~/~ffn) ' (19)

and l? is independent of t = x/~ x ° (½ x). The vanish- ing of Q2 depends only on the dimensionality of the x u, and it not affected by the shift. Consequently, in 26 dimensions,

Q2 =/~2 = 172 = [/~, I?] + = 0 . (20)

The action for the interacting theory becomes

I = f I~*(iB0t + ~ r )~+2 ~ ~ * ~ * ~ . (21)

Passage to the hamiltonian formulation is effected by introducing

I 1 = 8 I / 8 ~ = - i~/~g5, (22)

where ~ is the twist operator. It follows that

~v = / / + ig]/~q~ ~ 0 (23)

is a primary constraint in Dirac's language [ 11 ]. It is straightforward to verify that the Poisson brackets a r e

{gS(z), / /(z ') } =c~(z-z ' ) . (24)

The 8-function in the right-hand side of this expression is a usual bosonic 6-function multifplied by an odd number of Grassmann 8-functions. The result- ing 6 ( z - z ' ) is both odd [ c ~ ( z ) = - 8 ( - z ) ] and Grassmann-odd.

The canonical hamiltonian

=.[ dz ~ / / - -LP Hc

is only determined up to terms proportional to the primary constraint. Consistency of ~ is ensured by demanding that the primary hamiltonian

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Hp=H~+ ff dz2(z)~,(z) (26)

preserve 7 j ~:

{Hp, ~ , ( z ) } ~ 0 . (27)

Explicitly,

/?qb + ~ , qb +i/~2~ 0 . (28)

To proceed further, we study/~. First, we note that the expression for/~ [eq. (19)] can be expressed in the form

B= ( 8/6y)d+ z~/~c7 (29)

where y and z are related to the even- and odd-moded xk, respectively, by linear, invertible transformations, d and 3 are similarly related to the Ck and the E~. A trivial calculation then shows that

/~¢ = 0=~ ~¢=/~A (30)

on Fock-space states. Thus, Ker(/~) = Im(/~). Since /~:=0 implies Im(/~)~Ker( /~) , it follows that Ker(/~) =Im(/~).

Next, we note that on the space of functions of in- terest, it is possible to construct (not uniquely!) an operator/~ such that [/~, B] + = 1. One such choice is

/~_ ~_L -6Co f dy , (31)

where fdy is the operation inverse to differentiation with respect to y on the position-space representa- tion of Fock-space elements. From this, we can construct the projection operators

~B=BB, ( 1 - ~ s ) =BB. (32)

It then follows that we can decompose

q~= ~o + / ~ , (33)

where

B~o = B ~ ] = 0 , (34)

with a similar decomposition fo r / / . We now return to the analysis of (28). Acting on

it with the two projection operators, it is clear that 2 is determined (up to terms proportional to/~) as

2=iB( I?~ + ~ , qb) (35)

and that there is a secondary constraint

=/~/~(I?~o + ~ , qb) ~ 0 . (36)

Remarkably,

/~(q~* q~) = 0 . (37)

This can be seen by noting that ( ~ , ~ ) (z ) is given by setting d = M = q ) and Cg(z3)=~(z-z 3) in (10). Then using

/~(z) ~ ( Z - - Z 3 ) ---~ - - B ( z 3 ) t ~ ( Z - - Z 3) (38)

one can use the properties of the ~-functions in (10) to show (37). In particular, consider the ghost pieces of the two terms in/~ [eq. (19)]:

CO-FN//~ Z (--)m/2Cm~" l-"~_ c(17t ) (39) m even > 0

and

(_ ) ( .+1 ) /2 a/adn . (40) n o d d > 0

The overlap conditions (12) demand that these an- nihilate • . This can be seen directly evaluating their action on (7), or by noticing that it follows from the result of Gross and Jevicki that fl + ( ½ n) I V3) = 0, together with the argument following (28). Consequently,

~2 =/~B(17"~) ~ 0. (41)

This is the same constraint one would have found if only the free action had been considered.

No further constraints are generated by demanding that ~v2 be preserved by time translations.

The constraint algebra can be summarized as

{~e,(z), ~'l(Z')}=2i~,~(z-z'), {~,(z), ~2(z')}=(1-~B)f~6(z-z'), {~2(z), ~ 2 ( z ' ) } = 0 , (42)

but

{ / ~ , ( z ) , ~ l ( Z ' ) } = { / ~ , ( z ) , ~ 2 ( z ' ) } = 0 . (43)

Thus, 7ti (z) and We (z) are second class constraints, while

F(z) = / ~ u (z) =/~H(z) ~ 0 (44)

is a first class constraint. The structure of this al-

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gebra is rather different f rom that ob ta ined by study- ing the (free) unshif ted theory. There, the algebra is entirely first class. Nevertheless, the physical content appears to be the same. One clue to this is the class of t ransformat ions generated by F ( z ) :

~(z)={Idz'A(z')F(e'),~(z) } = / ~ A ( z ) . (45)

In other words, in terms of the decompos i t ion o f (29) , F(z) generates arbi t rary t ransformat ions of q ~ ( z ) , which can therefore be gauged away. Equiv- alently, it can be set to zero when fixing the gauge. I f we compare the decomposi t ions o f the fields in the shifted and unshif ted theories

=Oo+Co0j

satisfying

kq~o =Bq~l = 0

#o0o =#00, =0

(shif ted)

(unsh i f t ed) , (46)

( sh i f ted) ,

(unsh i f t ed) , (47)

then with the choice o f B o f (31 ), all fields are independent o f Co. It follows that

0~ (z) = 2 i ( , eve, ~ > 0 ( - ) , / 2 8/SX})~,(Z). (48)

Thus, choosing q ~ ( z ) = 0 in the shifted theory is equivalent to choosing 0 1 ( z ) = 0 in the unshif ted theory. This is, o f course, the convent ional choice of gauge for the unshif ted theory [ 12 ].

We expect to report on the quant iza t ion o f this theory in the near future [ 12 ].

This research was suppor ted in part by the Na- t ional Science Founda t ion under Gran t No. NSF- PHY-84-15534. We would like to thank J. Shapiro for useful conversations.

References

[ I ] E. Witten, Nucl. Phys. B 268 (1986) 253. [2] S. Giddings, Nucl. Phys. B 278 (1986) 242;

S. Giddings and E. Martinec, Nucl. Phys. B 278 (1986) 91; S. Giddings, E. Martinec and E. Witten, Phys. Lett. B 176 (1986) 362.

[3] M. Bochicchio, Phys. Lett. B 188 (1987) 330; C. Thorn, IAS preprint IASSNS-HEP-86-1334 (1986); A. Bogojevi6, Brown preprint BROWN-HEP-615 (1987).

[ 4 ] I. Batalin and E. Fradkin, Rev. Nuovo Cimento 9 (1986) 1; M. Henneaux, Phys. Rep. 126 (1985) 1.

[5] E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. B 179 (1986) 57; N. Ohta, Phys. Rev. D 34 (1986) 3785; S. Samuel, Phys. Lett. B 181 (1986) 255; D. Gross and A. Jevicki, Nucl. Phys. B 283 (1987) 1.

[6] D. Gross and A. Jevicki, Nucl. Phys. B 287 (1987) 225. [7] Z. Hlougek and A. Jevicki, Nucl. Phys. B 288 (1987) 131. [8] I. Bengtsson, Phys. Lett. B 172 (1986) 342;

G. Siopsis, Phys. Lett. B. 195 (1987) 541. [9] E. Witten, Nucl. Phys. B 276 (1986) 291;

C. Crnkovi6 and E. Witten, Princeton University preprint (September 1986); C. Crnkovi6, Princeton preprint PUPT- 1033.

[ 10 ] T. Morris, IAS preprint IASSNS-HEP-87/24. [ 11 ] P. Dirac, Proc. R. Soc. London A 249 (1958) 326. [ 12] W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105.

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Constraint algebra of open string field theory in midpoint coordinates