Volume 213, number 1 PHYSICS LETTERS B 13 October 1988

DILATON VERTICES OF THE CLOSED SUPERSTRING AND THE HETEROTIC STRING ~

Bang-Gui LIU Center of Theoretical Physics, CCAST (World Laboratory), Bering, P.R. China and lnstitute of Modern Physics, Northwest University, Xian, P.R. China

Received 1 June 1988

BRST- and conformally invariant dilaton vertices of the closed superstring and the heterotic string are constructed, whose counter terms are consistent with the dilaton terms introduced in the non-linear sigma model approach of string theory.

Ref. [ 1 ] developed a two-dimensional conformal field theory by investigating two-dimensional quan- tum field theory systems. This powerful technique was applied to the string theories in refs. [2,3 ]. A physi- cal state is described by a physical vertex operator which is conformally and BRST invariant [4,5]. Being conformally invariant implies being confor- mal-anomaly-free. There are two kinds of physical vertices: the integrated vertex and the incoming state vertex.

VB =q~,,O-X~'(z)O:X~(g) exp[ik.X(z, g)] ( k 2 - -

0) describes the zero-mass states of the closed bo- sonic string [2]. But the dilaton vertex V ~ = O=X(z).O=R(~) exp[ik.X(z, z)] has a conformal anomaly and is not a conformal field. Ref. [6] was aware of this, and constructed an anomaly-free and BRST-invariant integrated dilaton vertex which in- cludes ghost fields in it. The counter term is con- nected with the ghost current anomaly. Further on, ref. [7] has constructed an incoming dilaton vertex which is conformally and BRST invariant.

The zero-mass states of the closed superstring are described by Vo=tl~,~DXU(z, O)OX~(z, O) exp[ik. X(z, O, f, 6)] (k2=0) [2]. The dilaton part V ~ = DX(z, O).f)X(f, O) exp[ik.X(z, O, g, 0)] has a con- formal anomaly, similar to the closed bosonic string case.

The same problem can come across for the heter- otic string case.

This project is supported in part by the National Natural Sci- ence Foundation of China.

In this note we plan to investigate the conformal anomaly of V~ and construct afterwards confor- mally and BRST-invariant integrated dilaton ver- tices of the closed superstring and the heterotic string.

The energy-momentum tensor of the superconfor- mal field theory system is given by [ 2 ]

T(z, O) = Tm(z, O) + Tg(z, 0), ( 1 )

where Tm and Tg are defined by

Tm(z, O)= -½DX(z, O)'D2X(z, 0), (2)

Tg = - CD2B+ ½DC DB-~D2CB. (3)

In the above equations, D=0o+00 is the supersym- metrical differential operator, and C=c+Oy and B=fl+Ob are supersymmetrical reparametrization ghost fields. In ten-dimensional spacetime we have

T(z,, O,)T(z2, 02) ~ 3 0~___ Z T(z2, 02) 2 z~2

1 1 012 + g ~ DT(z2, 02)+ - - D 2 T ( z 2 , 02), (4)

z ;Zl2 Z12

where z~2 and 0~2 are defined by

ZI2=ZI--Z2--OI02, 012-~01 - -02 . ( 5 )

The superconformal field ~ of dimension (h,/~) is defined by the following equations:

T(zj, 01 )T(z2, 02, Z2, 02)

1 1 Dq~(2) + 0~2 D20(2) ' (6/ ~ h 0 ( 2 ) + 2z,2 z,'--~

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Volume 213, number 1 PHYSICS LETTERS B 13 October 1988

~( e, , O, )O( z> 0~, z-~, 02)

~k70@2~(2)+ 1 1 - 0j2 :'712 ~ D 0 ( 2 ) + zl--~

I) 20(2) . (6')

The BRST charge QB is defined by

t" dz dO QB = (~ ~-~i Ju( z, 0),

jB(z, O) =C[ Tm(z, O) + ½ Tg(z, 0)] - ]D( CDC B). (7)

Since we always set spacetime dimensions to be ten, we have the following identity:

Q~ =0. (8)

All physical vertices Vphys must satisfy eqs. (6), (6') and (8), and in the same time the BRST invariant,

[QB + QB, Vphys ] =0, (9)

VD = DX(z, 0)'DX(Z, 0) exp[ik.Y(z, 0, g, 0) ] (k 2 = 0) is of dimension ( ½, ½ ), but does not satisfy eqs. (6), (6') . Its operator product with T(z, O) and T(& 0) are given by

T(ZI, 01 ) VOD(Z2, 02, Z2, 02)

i 1 2 z)~ k.DX(g2, 02) exp[ik.X(2) ]

1 0,2 vD(2)+ 1 1 DVD(2)+ 0,2 D2VD(2), + z T z ,-7 z 1 2

(lO)

T( e, , O, ) VD ( z2, 02, &, 02)

i 1 ~ ~-~-2£~2 k.DX(z2, 02) exp[ik-X(2) ]

1 012 VD(2)+ 1 1 I7)X(2)+ 0,2 I)2vD(2)"

(10')

I'D(z, O, £, O) is not conformally invariant because it has a conformal anomaly - li ( 1/z12 ) 2k" f)X exp (ik. X) or ½i(1/~212)k.DXexp(ik.X). To construct a conformally invariant physical dilaton vertex we have to seek a counter term to cancel the anomaly.

C(z, O)B(z, O) has conformal dimension I. It is a promising candidate for our counter term. In fact, we have

T(zl, O, ) (C(z2, 02)B(z2, 0z) )

1 1 1 0,2 C(2 )B(2 )+ 7--11D[C(Z)B(2)]

+ 0~2 D2(C(2)B(2 )). (11) Z12

We see that iCBk,f)Xexp(ik.X) has dimension (½, ½), and contribute an anomaly term ½i(k. I))?) exp(ik-X) when multiplied by T. In the same way ik .DXCB exp(ik.X) contributes an anomaly term - l i ( k . D X ) exp(ik.X) when multiplied by T. Consequently, we obtain our dilaton vertex Vd(z, O, ~, 6).

Vd(z, 0, & 0) = [DX(z, O)+ikC(z, O)B(z, O) ]

× [I32(~, 0) +ikC(z, 6)/~(f, 0) ]

×exp[ik.X(z, 0, & O) ] , (12)

which is conformally invariant, i.e. satisfies eqs. (6), (6 ' ) with (h, h ) = (½, ½).

Besides, it can be proved that

jB(ZI, 01 )vd(z2 , 02, Z2, 02)

~ 012 [D2E(2)+DF(2)+OG(2)], (13) ZI2

where E, F and G are defined by

E(2)

= - C D X . ( D X - i k C B ) exp(ik-X) (2), (14a)

F(2)

= I D C D X ( f ) X - i k C B ) exp(ik.X) (2), (14b)

G(2) = [CTm + 1DC CB D - C D2CB

+½DCDBC+¼(DC)2B]exp(ik.X) (2). (14c)

Further we have

~ d z dO , [QB, Vd(2)] = ~--a.~-JBtz, O)Va(z2,02,z2,02)

=D2 E(2) + D f ( 2 ) +[ ) G(2). (15)

Because the right-hand side of eq. (1 5) is a sum of three total derivatives, the integrated vertex of the closed superstring dilaton defined by

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Volume 213, number 1 PHYSICS LETTERS B 13 October 1988

r" V~ST= J dZz d20 Vd(z , O, ,g, 19) (16)

satisfies the following equation:

[QB, V~sv] =0. (17)

In the same way it can be proved that

[O_u, VgST] =0. (18)

Therefore, we have proved that the integrated vertex of dilaton (16) is both conformally and BRST invariant.

The heterotic string consists of left and right sec- tors. The left sector is the ten-dimensional super- string and the right one is the twenty-six-dimensional bosonic string. For the right sector, the energy-mo- mentum tensor T(z) is defined by

?(e) = - ½0e£' 0e£+ g0e/7+ 20- H~, ( 19 )

which satisfies

2 1 7~(e)T(#) = (g_ ~)------------5 T ( # ) + e_ 0~T(~), (20)

and the BRST charge QB is defined by

O~ = - ~ i j~ ( e) , (21)

There is no anomaly present in eq. (20) because spacetime is twenty-six-dimensional.

The heterotic string conformal field A of dimen- sion (h,/7) is defined by eq. (6) and

/~ 1 T(z)A(~) - - d ( f f ) + _ _ 0 J ( ~ ) . (23)

( e - ~ ) ~ z - w

All physical vertices of the heterotic string must satisfy eqs. (6), (23) and (9) with Qn defined by eqs. (21), (22).

Based on the closed bosonic [6] and superstring dilaton vertices, integrated vertex of heterotic string dilaton can be proved to take the following form:

Vdst ---- f d Z z dO (DX+ikCB). (Oe)(- ]ikcb)

Xexp(ik.X) (z, 0, 2). (24)

V~s~ is conformally and BRST invariant since both the Vs%v and bosonic dilaton vertex are.

By means of partial integration, V~sv and V~s, can be changed into the following forms:

V~ST = f d2z[ (0X+ik'~0~0)' (0X+ik.~o)

- ~ ( vfl-cb-ik.~ocfl)

-a(#f l -gb- ik .~gf i ) ] exp( ik .X) (z , f ) , (25)

V~,t = f d2z[ (aX+ik.~o~o)

- 0 ( v f l - cb - i k .~c f l )

- ~a( -c6) ] exp(ik.X) (z, ~). (26)

Because of the ghost current anomaly [8] ~(vfl), ~(cb), a (tyfl) and 0(~6) are not equivalent to zero. On the other hand ~ (ik. ~ocfl)= a (ik.q~efl)= 0 because of the absence of an anomaly for ik. ~ocfl and ik.q~gfl. We note that the bc-current Jt,c = - b c and the fly-current jar=- f ly . Since [2,3]

cgj,,c = O f& = 3x/-g R 2,

Oj/~= Of~= - ~x/g R 2, (27)

where R 2 is two-dimensional world-sheet scalar cur- vature, we have

O( f lv -cb ) = -Oj~, --~Jb~

= - ~x~gR2= 0(fl~-e/~). (28)

Consequently, we obtain

V~ST = f d2z [ (0X+ ik'~0q0" (~£+ i k ' ~ )

+ ¼x/gR 2 ] exp(ik'X), (29)

Vdst = f d2z[ (0X+ik'~0~0)' (OX)

+ ~x/gn 2 ] exp(ik.X). (29')

The counter terms ¼x/g R2 in the above equations are equivalent to those of the closed bosonic string dilaton [6]. It is noted that the counter terms are consistent with the dilaton terms introduced in the non-linear sigma model approach of string theories [91.

In summary, we have constructed the conformally and BRST-invariant dilaton integrated vertices of the

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Volume 213, number 1 PHYSICS LETTERS B 13 October 1988

c losed s u p e r s t r i n g a n d h e t e r o t i c s t r ing, a n d con-

n e c t e d i ts a n o m a l y c o u n t e r t e r m s w i t h t he ghos t cur-

r e n t a n o m a l i e s o f t he c losed u p p e r - a n d h e t e r o t i c

s t r ing, respect ive ly . T h e c o u n t e r t e r m s are e q u i v a l e n t

to each o t h e r for all c losed s tr ings.

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