Volume 216, number 1,2 PHYSICS LETTERS B 5 January 1989

SYMPLECTIC GEOMETRY AND GEOMETRIC QUANTIZATION FOR THE OPEN BOSONIC STRING IN THE BRST FORMALISM

Yue YU Centre of Theoretical Physics, CCAST (World Laboratory), Beijing, P.R. China and Institute of High Energy Physics, Academia Sinica, P.O. Box 918, Beijing, P.R. China

Received 19 July 1988

The symplectic structures for the open bosonic string are given by the method developed by Tuclzyjev and Chen and Yu and a geometric quantization for the string in the BRST formalism is proposed here. The conformal anomaly is cancelled if the curva- ture of the BRST vacuum bundle over G~ = G o / H vanishes.

It has been affirmed in many works [ 1,2 ] that the conformal anomalies may be regarded as curvatures of the holomorphic Fock bundles over Diff S~/S ~ (or Super-DiffS~/S ~ ). For the sake of cancelling the anomalies, the other geometric structures over DiffSt/S~ (or Super-DiffSl/S~ ), for example, a holomorphic ghost vacuum bundle [ 2 ] or a canonical line bundle [ 1 ], must be incorporated. The conformal anomalies are shown to dis- appear when the curvatures of the product bundles representing the BRST excitations for strings equal to zero. In the present paper, we wish to construct symplectic geometry describing the dynamics of the open bosonic string in the BRST formalism and to quantize the string by means of geometric quantization [ 3 ] thereby veri- fying that the cancelling condition of the conformal anomaly may be reflected by the vanishing curvature of the holomorphic BRST vacuum bundle over G0 = G o / H where Go is a conformal transformation group and H a subgroup of Go generated by the generator Lo.

Let ~", a = 1, 2, be coordinates on the two-dimensional world sheet M, g"a(~) the metric on M and qU(~) open bosonic string coordinates. The configuration bundle Q for the string may be described by the local coor- dinates (~"; q~'(~), g"P(~)). As was done in ref. [4], in the phase bundle Y with the local coordinates (~"; q,'(~), g"P(~), H~'~(~) H~y(~) ), a two-form is defined by

og= f dkS,(~H ''~ ^ ~qj, + ~H~ ^ ~gP~) , (1) s

where 27 is an initial curve. The classical dynamics of the string is geometrically brought about by reducing the quotient bundle ]~ Y of the one-jet bundle of Y to its lagrangian submanifold N generated by the generating function ~ [ 4 ], where the lagrangian submanifold is defined by

dtol N = 0, dim N = ½ dim ]~ Y. (2)

Now, we require the string theory to possess reparameterisation invariance in M. Thus, this requirement and the definition (2) provide a unique reduction of 09 [ 5 ]:

H,,'~=v/g g-aOaq~, H~=v/g(F'~-I '~at~ '~)-½x/~ gar(F'~agP~-F~g"P) , (3)

where

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Volume 216, number 1,2 PHYSICS LETTERS B 5 January 1989

F~:, = ½g"( Oag.,, p + O~g/~,, - Opga,,) , (4)

and the q~'(~) satisfy

O,( x ~ g"/~O/~qU) =0, T,~ = O, qO~q- ½g,~g~O~qO~q=R~- ½g,~R =O . (5a,b)

The generating function of N is

5~= ½x/~ g'~/~O~qO/~q+ x ~ g . (6)

In eqs. ( 3 )- ( 6 ), R,a denotes the Ricci tenor and R is the curvature scalar on M. Eqs. (5a), ( 5b ) and ( 6 ) give just the equation of motion, the constraints and the lagrangian in the string dynamics. But it is unsuitable that we wish to perform a geometric quantization program for the classical theory starting with the two-form ( 1 ) since the numbers of the constraints (Sa) are less than the degenerate degrees in 09. For this reason, we must fix either all or part of reparameterisation symmetry so that there is a two-form, either nondegenerate, or whose degenerations manifest just the constraints.

Let G be a reparameterisation group for the open bosonic string, then the configuration bundle Q = { (~; q (~), ~ ( ~ ) , g(~) )), here ~ ( ~ ) is a representative element of an equivalent class in which each metric ~ ( ~ ) may be represented as the product ofanother~ ~a' (~) in the class andg(~) ~ G, i.e. R'~a (~) = g(~)~'~a' (~). The gauge-fixed configuration bundle Q= { (~; q(~), ~"a(~))}. Taking Q as our point of departure, we will get a gauge fixed quantum string theory if we work in the light-cone gauge. But here we prefer another configuration bundle Q in which the reparameterisation symmetry is partly fixed, rather than ~. For example, choosing the familiar con- formal gauge, Q= { (~; q(~), g ~ (~), h (~)) }, here h (~) = exp [ e" (~)L, ] belongs to the conformal transformation group Go which is a subgroup of G and g~a may be turned into exp [p(~) ] r/"a by Go. In this formulation we will see that the degenerations of 690 in the directions/~ are equivalent to the c o n s t r a i n t s Totfl[ conf.gauge = 0.

Instead of the group parameters e'~(~), grassmannian parameters c~(~) called ghosts are introduced by ~"(~) = vc"(~). Here v is a grassmannian constant. This time, in the phase bundle Y associated with ~ = { (~; q(~), g~/~(~), c"(~), g,a(~) )}, a two-form may be defined by

COo = t~lg,~=g~= f d.S,,~ [ 5H u°' ̂ 5qu + 5H~r ^ 5g~'+ iSti~ ^ 5ca+ iStio, zr ̂ ~ ] ~'~ = ~ ' Z"

(7)

where ((~p) is a symmetric traceless matrix with grassmannian value called the antighost. In order to get a lagrangian submanifold lq o f J l Ywith the BRST and conformal symmetries, we introduce a

bundle Y' = Y(g~'P-,g"P) with two-form 69. A lagrangian submanifold lq' of J~ Y' may be defined by eq. (3) and

/-/~ = x/~ g ~ , Ii'~P~'=--½v/gg'~'~v/ggPYc~, (8a)

and

O,~(x/~ g'~aOaq) = 0 , [x/ggaa0.(x/~g"r(ra) + ½w/g g'~Px/~ gYaO,~era] 5c a=O ,

[ v/g g"~'O,,( v/g g~cp) -- ½0,~( x/~ g'~axflg g~ca) ]~(ya = 0 , R,~ a - ½g,~aR= T,~a = 0 ,

- i [ c/j x/g g '~' O~ g,~y - g,~ pgP'~c~ g or x/~ Op gor - x/g g y'~ ~r g,~ ~ ca ] . ( 8b )

Its generating function is

~ ' = £P + ix/~ g";'gy,~ O,~ ( x/~ gPr c ~ ) + ½ ix/~ g"rx/~ g~c~ Oo, ga~ . (9)

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Volume 216, number 1,2 PHYSICS LETTERS B 5 January 1989

NOW, the restriction of J~ Y on lq may be obtained by a substitute g'~P for g~P in (8), (9). Putting g~P into exp [p(~) It/'~p, we have

690 = f dS . (50 ~q ̂ ~Sq + iSg '~p ̂ ~cp), ( 10 ) Z

and

O"O.q=O, OoffOO'Jl- 01 fflO = O, OoffOI "JI- Ol ffl I = O, OoCO-- Ol c l = 0 , O leO- - O o C I = O , ( l l a )

~0 T ,~a = O,~q O aq- ½ ~h~a( Oyq ) 2 - 2iga~, O,~c~- icy7rg,~a = 0 , ( 1 lb)

~o = ½0"qOo, q+ ig, aO'~c a , ( 1 lc)

Eqs. ( 1 1 ) give us a on-shell BRST and the conformal invariant classical string system [ 6 ]. For the open string, the end points of the initial curve must be (0, r, ) and (n, r2). As d~o =0, ~o is independent of the choice 0f27. Thereby, the curve f(^, z) may be chosen:

f ( a , r ) = z = r t O<~a<~Tr, f (~ ,r )=e=Tr z~<~r<<.z2. (12)

The boundary conditions for open strings determine that the Fourier expansions of q, c and g,~a are [6 ]

1 q(a, r) =qo +por+i ~, -- a,. cos m a e x p ( - i n a )

m~O m + o o + o o

c°(a , r )= ~ CmCOSmaexp(-imz), c ' ( a , r ) = - i ~ c m s i n m a e x p ( - m r ) , m = - - o o i n = - - o o

+ ~ + o o

goo(a , t )= ~ g, n c o s m e e x p ( - i m t ) , # O l ( a , t ) = - i ~ # m s i n r n a e x p ( - i m t ) , (13) m = - o o m = - - o o

~5o is expanded as

tbo= d27o~(80e~q^~q+iSg'~PASCp) =Spo^Sqo+i ~ 8Otto ̂ 8Ot~ + i E ~gmA~C--rn" (14) f(a,r) m= 1 m= --oo

L e t f be a C °o function on Y, the hamiltonian vector field Xfassociated withf is defined by

~o(Xr) = - 5f. (15a)

Locally,

Xs = 6f 6 5f 5 + i 2 m ~Po~qo ~)qoSPo-- ,n=l ~ot*n ~Ol*n~,n +im=~--oo~)'~m~C-m +--~Cm ~)('--"-~n " (15b)

Note that as the components of a vector field and a one-form may be grassmannian, the order of the interior product of each one-form a and vector field X on ~ must be determined, say

a ( X ) = X i a i . (16)

For arbi traryfand g belonging to C °o (Y), their Poisson bracket is defined by

~,,g}v.a. = -Coo (Xf, r e ) . (17)

The constraints 7-°~ a = 0 are equivalent to

+oo T°o+~P° l+Z~P° ,= -2 ~ E. e x p [ - i n ( a + r ) ] = 0 , (18)

m= --oo

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Volume 216, number 1,2 PHYSICS LETTERS B 5 January 1989

ioe,~

1 + ~ + ~ L, = - -~ ,,,_~__ ~ ~ . . . . o~,,, ~ m=Z-~ ( n - m )(,+mc_,, =O , (19)

if we extend analytically q(a, v), c(a, z) and e'~P(a, z) from 0 4 a~< n to - n~< a~< n. The/2, are just the Virasoro algebra generators, which obey

{ L , , E , , , } = - i ( n - m ) E , + m , (20)

which correspond to the degenerate directions in do. Note that the first and second terms on the right-hand side of (19) show the generators of the Virasoro algebras in the string and ghost sectors, respectively:

L,, =L,, gh 1 +=~_~ +~ + L , , , L . = - ~ m oo°~ . . . . o~ .... L~h=--m=~_~(n--m)g.+mC . . . . (21a,b,c)

Now we quantize the BRST system (Y, 05o; ~ o = 0) by the method of geometric quantization, choosing a polarization F = { 6/6z*., 8/~g.; m >1 O, n~Z} where

Zo=~(po+iqo), z~=½(Po- iqo) , z*,=ot, , /w/m, zm=ot*Jx /~ , m > 0 . (22)

The Fock space under the polarization F is

o~¢/.-= {f(z,,, c,) , m>~O, n6~-I ( 0~, ,- f l6, .o)f=O} , (23)

where [ 7 ]

1 + ~ - . 0 . . . . . O ~ , , , . - f l ~ . , o , 0L=0L+0f~,., O L = ~ Y~ • ' '~ "~ •

"F=O~o+O~:o=6/Szo+Zo, 0 ..... =O,~r, 0,~,. m> 0 . . . . "r "r = x / ~ Z m ' "F =X/~5 /6Zm, O. (24)

The ghost and antighost operators are

~ F O~,~,=c .... O~,,,=~/~c,., m~Y_, (25)

and the ghost conformal transformation operators are

O~~h= ~ ( n - m ) :c_m~)/Sc . . . . . : . (26) I H = - - ~

In a~F, each s ta tef(z , c) can be written as

f ( z , c ) = f i,..u,u,...j, zi,...zi,,,cj,...cj., (27) m , n = O

where is >/O, s= l, ..., m a n d j , ~ , t= l, ..., n. Thus, the vacuum state of the string may be taken to be the constant state, i.e.,

6/~z, , ,Is)=O, m>_.O. (28)

But for the ghost sector, the vacuum state of the ghost may arbitrarily be appointed in the following states

Ig)N=C-NC-N+~.. . , - - ~ N < ~ + ~ , (29)

by which the creation and annihilation operators of the ghosts and antighosts may be defined:

Oc,,,Ig)N=O m~<--N, "F Oe,,,Ig)N=0 m > ~ N + l . (30)

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Hence, the total vacuum state in ~ - is

10)N= I S ) ® l g ) u , (31)

where N is called the level of the vacuum. Taking the vacuum of level N, one finds that the quantum conformal algebra is not closed:

~/+ , ~/++ I O£ . . . . . . [ c,,, ]--(n--m)O~°+ ={~2D(n3--fln)--+[13n3-(6NZ+6N+l)nl}Jm+.,o. (32)

If we choose the vacuum to be an SL(2, C) invariant state for the string and the ghost sectors respectively, then the conformal anomaly is cancelled at D = 26, f l= 1 and N = 1.

The above results tell us that the conformal invariant BRST Fock space YdDF is a section space on the holo- morphic vacuum of the level- 1 bundle over Y. As was done by Pilch and Warner [ 3 ], under a conformal trans- formation, the operators Zm, +/SZm, C. and 8/8c_., n+Z, m > 0, turn into

' ~ * z,,= a,,,,z,,,+b*~,,, n > 0 , Dl= t

+ ~ - - ( N + I )

c, Z ++ Y+ C~;-Cm ' = Ckm Cm + DI= - -m m = - c ~

5 +c~ 5 N

5 c ' , - y+ b*+''+' + y+ r e = N + I ~ C - - m m = - - o o

8 +oo

5Z'n --,n=~t anm ~ + b.m z,. n > 0

+0o -- ( N + 1 )

k>~-N, c'k= ~+ C~+Cm+ ~ CB,?C,. k ~ < - - ( m + l ) , m = - - N m = --oo

b~.- - - 5 6 + + 5 u 5

8C_m k>~N+l, 8C'_k- ~" b L ' + - - + ~ b ~ - . V - - k ~ N , m = N + l ~ C - m m = - c ~ o ~ C - m

(33)

where the transformation coefficients are obtained by the conformal transformations from q(a) , c~,(a) and ¢"~(a) to q(a' ), c,~(a' ) and ( " a ( a ' ) . The holomorphic connection ( 1, 0) - form F o n G0 = G o / H may be defined by the connection matrix

. ( , ..... ) F}~ ) , k<~-(N+l) , l<~N, (34)

where

y,,,,,= a,?~bl .... FIN) • ++-1 + - ++ + - = c_l,,, Cmk =-- b_kmb,,t k ~ < - ( N + l ) , I<~N. (35) I= I m = - - N r e = N + 1

Since the tangent space _QG~ of G0 is equal to that DiffS~/S t of D i f f S t / S t, at F= / -* = 0, the curvature o f the holomorphic BRST vacuum bundle E over G0i s the same as that over Di f fS t /S t calculated by Pilch and Warner [3]

F(O~,,, 0~,,) = - ½ Tr[y*(O~,,)y(O~,,)--V*(OF,,,)7(O~o)]-Tr[F<N)(O~,)F(-N-')*(O~,,)]

={~D(n3- f ln) - ~ [13n 3 - ( 6 N 2 + 6 N + 1 )n]}~,+m.o • (36)

Comparing (36) with (32), we know that the conformal anomaly in ~¢d~. may be interpreted as the curvature of E. The anomaly is cancelled when the curvature of E vanishes at D = 26, f l= 1.

I benefitted from discussions with S.M. Fei and W.L. Shen.

References

[ 1 ] M. Bowick and S.G. Rajeev, Phys. Rev. Left. 58 (1987) 353; Nucl. Phys. B 293 ( 1987 ) 348; P. Harari, D.K. Hong, P. Ramond and V.G. Rodgers, Nucl. Phys. B 294 ( 1987 ) 556.

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[2] J. Mickelsson, Commun. Math. Phys. 112 (1987) 653; Z.Y. Zhao, K. Wu and T. Saito, Phys. Lett. B 199 (1987) 37; K. Pilch and N.P. Warner, Class. Quant. Grav. 4 ( 1987 ) 1183.

[ 3 ] N.J. Woodhouse, Geometric quantization (Clarendon, Oxford, 1980 ); J. Sniatychi, Geometric quantization and quantum mechanics (Springer, Berlin, 1980).

[4] W.M. Tuclzyjew, Lecture Notes in Mathematics, Vol. 570 (Springer, Berlin, 1975 ) p. 457; W. Chen and Y. Yu, BIHEP preprint BIHEP-TH-88-21, 22.

[5] E. Witten, Nucl. Phys. B 276 (1986) 291; G. Zuckerman, Yale University preprint; C. Crnkovic and E. Witten, in: Newton's Tercentenary Volume, eds. S. Hawking and W.I. Israel; C. Crnkovic, Nucl. Phys. B 288 (1987) 431.

[6 ] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [7] Y. Yu and H.Y. Guo, Phys. Lett. B 216 (1989) 68. [ 8 ] P. Friedan, E. Martinec and D. Schenker, Nucl. Phys. B 271 ( 1986 ) 93.

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