Sewing string tree vertices using canonical forms

Sewing string tree vertices with ghosts using canonical forms Leonidas Sandoval Junior∗ Department of Mathematics Centro de Ciˆencias Tecnol´ogicas UDESC - Universidade do Estado de Santa Catarina - Brazil

arXiv:hep-th/0104145v1 17 Apr 2001

January 1, 2014

Abstract We effectively sew two vertices with ghosts in order to obtain a third, composite vertex in the most general case of cycling transformations. In order to do this, we separate the vertices into two parts: a bosonic oscillator part and a ghost oscillator part and write them as canonical forms.

PACS number: 11.25

1

Introduction

Sewing of vertices in order to build larger ones is a technique that has been used since the early days of String Theory. In [1], it has been shown how to use the Group Theoretic approach to String Theory [2] in order to perform the sewing of two string vertices taking into account their ghost structure. Here we shall perform the sewing of two vertices with ghosts using another procedure, writing the two original vertices in terms of canonical forms (see the appendix) and then performing the sewing explicitly. This method has been used in [3] and [4] in order to build multiloop string amplitudes for the bosonic string. This is done in order to check the result obtained in [1] and also to present an example of the use of canonical forms in the sewing of vertices. The sewing procedure consists on taking two distinct vertices U1 and U2† , the first with N1 strings and the second with N2 strings, and choosing one string from each vertex to be sewn together. We shall be choosing leg E from vertex U1 and leg F from vertex U2 . The sewn legs give rise to what we call propagator P. This procedure is well explained in [5] and [1]. In order to perform the sewing, we shall divide the so called physical vertex [6] of an open bosonic string into a bosonic and a fermionic part. This is the vertex that has the correct number of ghosts. It can be written as

U



N Y

=   ×

i=1 i6=E

  



N1  1 X 

i h0| exp −

1 X N X 1 Y

r=−1 j=1 s=−1

2

∞ X

i,j=1 n,m=0 i6=j

Ers (Vj )bjs

×





∞ X ∞ N X X

−1 j   aµi n Dnm ΓVi Vj amµ  exp 

N X ∞ N X Y

j eij n bn ,



i,j=1 n=2 m=−1 i6=j



cin Enm ΓVi−1 Vj bjm   

(1)

i=1 j=1 n=−1 i6=a,b,c

where ainµ are bosonic oscillators, and bin and cin are ghost oscillators. The matrices Dnm (γ) and Enm (γ) depend on the cyclings and are given by [3] [4] Dn0 (γ) = ∗

1 √ [γ(0)]n , n

(2)

E-mail address: [email protected]

1

Dnm (γ) = D00 (γ) = Enm (γ) =

m 1 ∂m [γ(z)]n , m n m! ∂z z=0 d 1 , ln γ(z) 2 dz z=0

r

"

(3) (4)

1 ∂ m+1 ∂ γz (γz)n+1 m+1 (m + 1)! ∂z ∂z

.

(5)

Dnp (γ1 )Dpm (γ2 ) + Dn0 (γ1 )δ0m + δ0n D0m (γ2 ) = Dnm (γ1 γ2 ) ;

(6)

These matrices have the following multiplication properties: ∞ X

−1 #

z=0

p=1 1 X

Ert (γ1 )Ets (γ2 ) = Ers (γ1 γ2 ) , r, s, t = −1, 0, 1 ;

∞ X

Enp (γ1 )Epm (γ2 ) = Enm (γ1 γ2 ) , n, m ≥ −1 ;

t=−1

(7)

Ern (γ) = 0 , r = −1, 0, 1 , n ≥ 2 , p=−1 ∞ X

p=2

(8)

1 X

Enp (γ1 )Epm (γ2 ) = Enm (γ1 γ2 ) −

r,s=−1

(9)

Enr (γ1 )Ers (γ2 )δsm , n, m ≥ 2 .

(10)

The last term of equation (1) is the product of N − 3 ghost oscillators bin (N legs minus arbitrary legs a, b and c) and is necessary for the vertex to have the correct ghost number. The coefficients eij n are given by [7] ∞ X

eij n =

ij j km Emn (γm ),

(11)

m=−1

where j γ−1 = Vj ,

(12)

γ0j = exp −Lj0 ln aj0 exp − 

γpj = exp −

∞ X

n=p+1



∞ X

n=1

a ¯jn Ljn

!

(13)

,

a ¯jn Ljn  , p ≥ 1 ,

(14)

where Ljn are the generators of the bosonic and ghost parts of the conformal algebra.

2

Bosonic oscillator part

Now we shall perform the sewing for the bosonic part of the vertex. The bosonic part of vertex U1 is given by 

V1osc =   

N1 Y

i=1 i6=E





 1 

 

i h0| exp −

2

N1 X

∞ X

i,j=1 n,m=0 i6=j



−1 j  aµi n Dnm ΓVi Vj amµ  .



(15)

Isolating all terms related to leg E, we have the following formula for the modified vertex V1osc :

V1osc =









  N1 N1 ∞  1 X  X  Y  j  −1 µi  exp −  h0| a ΓV V a D  i j nm mµ n i    2  i=1 i,j=1 n,m=0   i6=E

i6=j i,j6=E

2

(16)

 

×E h0| exp  −

N1 X ∞ X



−1 E  aµi n Dnm ΓVi VE amµ  .

i=1 n,m=0 i6=E



(17)

This may be written in terms of a canonical form (see the Appendix) [3]:

V1osc

where Bn1µ

=

N1 X ∞ X



 = 

i=1 i6=E

∞ X   Bn1µ aE i h0| E h0| exp − nµ n=1

−1 aµi m Dmn ΓVi VE

i=1 n=0 i6=E

φ1 =



N1 Y

!

exp(−φ1 )

(18)

(19)

,

N1 X N1 ∞ ∞ X X 1 X E −1 j −1 aµi aµi n Dn0 ΓVi VE a0µ . n Dnm ΓVi Vj amµ + 2 i,j=1 n,m=0 i=1 n=0

(20)

i6=E

i6=j i,j6=E

Similarly, considering vertex V2osc† , we must change †

µF µF aµF m → am = −a−m

(21)

so that1

V2osc †

=









  N2 N2 ∞  1 X  X  Y  −1 j  µi   − exp h0| a ΓV V a D  i j nm mµ  n i    2  i=1 i,j=1 n,m=0   i6=F

i6=j i,j6=F

 

× exp  

which can be written

V2osc†

N2 X

∞ X

i=1 n,m=0 i6=F

N2 ∞ X  Y 2  exp − aµF h0| = i −n Anµ   n=1

i=1 i6=F

where A2nµ = − φ2 =

N2 X ∞ X

i=1 n=0 i6=F

(23)

!

exp(−φ2 )|0iF

(24)

Dnm ΓVF−1 Vi aimµ ,

(25)

N2 N2 X ∞ ∞ X X 1 X −1 −1 j µi a + aF0µ . ΓV V ΓV V aµi D a D j F nm n0 mµ n i i 2 i,j=1 n,m=0 n i=1 n=0

(26)

i6=F

i6=j i,j6=F

1

aµi n



−1 F  aµi n Dnm ΓVi VE a−mµ  |0iF ,







The minus sign in the expression for the Hermitian conjugate is because of the minus sign in the commutation relation of the oscillators: n νj [aµi δn,−m η νµ δ ij . (22) n , am ] = |n|

3

The oscillator part of the propagator is purely a conformal transformation [5]: Posc = VE−1 VF Γ .

(27)

Using the properties of coherent states (see the Appendix), we may represent this as [4] 

Posc =: exp 

∞ X

n,m=0

−1 E aµE −n Dnm VE VF Γ amµ −

∞ X

n=1

or, in terms of a canonical form,

Posc = exp −

∞ X

3 aµE −n Anµ

n=1

× exp −

∞ X

!

Bn3µ aE nµ

n=1

 

: exp − !



∞ X

n,m=1



E  aµE −n anµ : ,

i

h

E 3 aµE −n Cnm − δnm amµ

exp(−φ3 )

(28)

 

:



(29)

with

A3nµ = −Dn0 VF−1 VE Γ aE 0µ ,

(30)

−1 Bn3µ = aµE 0 D0n VE VF Γ

(31)

3 Cnm = −Dnm VE−1 VF Γ

,

(32)

,

−1 E φ3 = aµE 0 D00 VE VF Γ a0µ .

(33)

Using now the multiplication rule for two canonical forms (as in the Appendix), we obtain

V1osc Posc =

   

N1 Y



i=1 i6=E

∞ X  4  h0| exp − aEµ h0| i −n Anµ E n=1

∞ X

× exp −

Bn4µ aE nµ

n=1

!

!

 

× : exp − 

exp(−φ4 )

∞ X

n,m=1

i

h

E 4 aµE −n Cnm − δnm amµ

  

:

(34)

with

A4nµ = −Dn0 VF−1 VE Γ aE 0µ , Bn4µ

=

aµE 0 D0n

VE−1 VF Γ

4 Cnm = −Dnm VE−1 VF Γ

φ4 =

+

(35) N1 X ∞ X ∞ X

−1 −1 aµi m Dmp ΓVi VE Dpn VE VF Γ

i=1 m=0 p=1 i6=E

,

(36) (37)

,

N1 ∞ X 1 X −1 j ΓV V aµi D j amµ nm i 2 i,j=1 n,m=0 n i6=j i,j6=E

+

N1 X ∞ X

i=1 n=0 i6=E

+

µE −1 −1 E E aµi n Dn0 ΓVi VE a0µ + a0 D00 VE VF Γ a0µ

N1 X ∞ X ∞ X

i=1 n=0 n=1 i6=E

−1 −1 E aµi m Dnm ΓVi VE Dn0 VE VF Γ a0µ .

4

(38)

Because of the vacuum E h0|, the first two exponentials do not give any contribution, and so we end up with V1osc Posc



 = 



N1 Y

i=1 i6=E

∞ X   h0| exp − Bn4µ aE h0| i nµ E n=1

!

exp(−φ4 ) .

(39)

Now we make use of property (6) of the Dnm (γ) matrices in order to simplify the expressions for the coefficients of V1osc Posc . We then have ∞ X

Bn4µ aE nµ

=

N1 X ∞ X ∞ X

i=1 m=0 n=1 i6=E

n=1

+

∞ X

n=1

φ4 =



 µE a +  0

N1 1 X 2 i=1

i6=j i,j6=E

∞ X

µE + a0 +



Since



N1 X

−1 E  aµi 0  D0n VE VF Γ anµ ,



i=1 i6=E

−1 j aµi n Dnm ΓVi Vj amµ +

n,m=0



−1 E aµi m Dmn ΓVi VF Γ anµ

N1 X

i=1 i6=E



(40)

N1 X ∞ X

−1 E aµi n Dn0 ΓVi VF Γ a0µ

i=1 n=0 i6=E

−1 E  aµi 0  D00 VE VF Γ a0µ .



aµE 0 +

N1 X

aµi 0 =

N X

(41)

aµi 0 =0 ,

(42)

i=1

i=1 i6=E

which is implied by momentum conservation on vertex V1osc , we have the following coefficients for V1osc Posc : Bn4µ =

N1 X ∞ X

−1 aµi m Dmn ΓVi VF Γ

i=1 m=0 i6=E

φ4 =

(43)

,

N1 N1 X ∞ ∞ X X 1 X −1 −1 j µi a + ΓV V Γ aE ΓV V a D aµi D F j n0 nm mµ 0µ . n i i 2 i,j=1 n,m=0 n i=1 n=0

(44)

i6=E

i6=j i,j6=E

So, the effect of multiplying V1osc by Posc is just to change the cycling transformations in vertex V1osc in the following way: VE −→ VF Γ . (45) Now we multiply V1osc Posc by V2osc† , obtaining the composite vertex Vcgh

=

V1osc Posc V2osc†

where Anµ = −

N2 X ∞ X



 = 

i=1 n=0 i6=F

N1Y +N2 i=1 i6=E,F



∞ X   h0| − aµE h0| i −n Anµ E n=1

Dnm ΓVF−1 Vi aimµ ,

!

−

∞ X

n=1

Bnµ aE nµ

!

exp(−φ)|0iF

(46)

(47)

5

Bnµ =

N1 X ∞ X

−1 aµi m Dmn ΓVi VF Γ

i=1 m=0 i6=E

φ =

(48)

,

N1 N2 ∞ ∞ X X 1 X 1 X −1 −1 j j µi a + ΓV V ΓV V aµi D a D j j amµ nm nm mµ i i 2 i,j=1 n,m=0 n 2 i,j=1 n,m=0 n i6=j i,j6=E

+

i6=j i,j6=F

N1 X ∞ X

i=1 n=0 i6=E

+

−1 E aµi n Dn0 ΓVi VF Γ a0µ +

N2 X N1 X ∞ X ∞ X

N2 X ∞ X

i=1 n=0 i6=F

−1 F aµi n Dn0 ΓVi VF a0µ

j −1 −1 aµi n Dnp ΓVi VF Γ Dpm ΓVF Vj amµ .

i=1 i=1 n,m=0 p=1 i6=E i6=F

(49)

Using the multiplication properties (6) of the Dnm (γ) matrices, we get φ =

N1 N2 ∞ ∞ X X 1 X 1 X −1 −1 j j µi a + ΓV V ΓV V aµi D a D j j amµ nm nm mµ i i 2 i,j=1 n,m=0 n 2 i,j=1 n,m=0 n i6=j i,j6=E

+

i6=j i,j6=F

∞ X

N2 N1 X X

i=1 i=1 n,m=0 i6=E i6=F

+

N2 X ∞ X

i=1 n=0 i6=F

j −1 aµi n Dnm ΓVi Vj amµ +

i=1 n=0 i6=E





N1 X ∞ X

−1  F aµi n Dn0 ΓVi VF a0µ +

N1 X

j=1 j6=E







−1  E aµi n Dn0 ΓVi VF Γ a0µ +

aj0µ   . 

N2 X

j=1 j6=F



aj0µ   

(50)

Considering now the identification of legs E and F and the momentum conservation for vertices V1osc and V2osc† , this becomes simply N +N ∞ 1 1X 2 X −1 φ= ajmµ . (51) ΓV V aµi D j nm i 2 i,j=1 n,m=0 n i6=j i,j6=E

Because of the two vacua E h0| and |0iF , the terms Anµ and Bnµ will not give any contribution and we will end up just with the contribution of the phase φ. So we have that

Vcosc









  +N2 +N2 X ∞  1 N1X   N1Y  −1 j  µi   = an Dnm ΓVi Vj amµ  i h0| exp −  2  i=1 i,j=1 n,m=0   i6=E,F

(52)

i6=j i,j6=E

which is the right expression for the composite vertex Vcosc .

2.1

Ghost part

We will now consider the ghost part of the vertex. The ghost part of a physical vertex U with N legs is given by 

N Y

U ghost =  

i=1 i6=E

  





∞ X ∞ N X  X  cin Enm ΓVi−1 Vj bjm  

i h0| exp 

i,j=1 n=2 m=−1 i6=j

6

×

1 X N X 1 Y

r=−1 j=1 s=−1

Ers (Vj )bjs ×

N X ∞ N X Y

j eij n bn ,

(53)

i=1 j=1 n=−1 i6=a,b,c

Making use of the integral of some anticommuting variables βr (r = −1, 0, 1) and η i (i = 1, . . . , N ; i 6= a, b, c), the ghost part of this vertex can be written [4] N Y

U gh =

1 Y

!Z

i h0|

i=1



× exp βr

dβr

r=−1

N X 1 X

j=1 s=−1



N Y

Z

i=1 i6=a,b,c



dη i exp   



Ers (Vj )bjs  exp η i

∞ X

N X

∞ X

i,j=1 n=2 m=−1 i6=j

N X ∞ X

j=1 n=−1





cin Enm ΓVi−1 Vj bjm   

j eij . n bn

(54)

The ghost part of vertex U1 can be written in a similar way. Isolating all terms related with leg E, we obtain

U1gh

=





N1  Y   i h0| E h0|  i=1 i6=E



N1 X ∞ X

× exp   

× exp  βr i × exp  η



N1 X

dβr

r=−1

∞ X

1 X

i=1 i6=b,c

i6=j i,j6=E



Ers (Vj )bjs + βr

1 X

s=−1

∞ X



U1gh where

 = 

φ1 =

N1 Y

i=1 i6=E

  i h0| E h0|

−1 j i eij n Enm (V0j Vj )bm + η

N1 X ∞ X ∞ X

1 Y

Z

+η i

N1 X

∞ X

dβr

r=−1

N1 Y



∞ X

i

i=1 i6=b,c

dη exp φ1 exp 



−1 i  cE n Enm ΓVE Vi bm 



−1 E eiE n Enm (V0E VE )bm  .



cin Enm ΓVi−1 Vj bjm + βr

i,j=1 n=2 m=−1 i6=j i,j6=E

i=1 n=2 m=−1 i6=E

n,m=−1

Z

∞ X





j=1 n,m=−1 j6=E



N1 X ∞ X

 Ers (VE )bE s 

This may be written in terms of a canonical form2 [4]: 



  cin Enm ΓVi−1 VE bE m  exp 

j=1 s=−1 j6=E N1 X

  N ∞ X ∞ 1  X  X   cin Enm ΓVi−1 Vj bjm  dη i exp      i,j=1 n=2 m=−1

N1 Y

Z

i=1 n=2 m=−1 i6=E





1 Y

Z







∞ X

n=−1

N1 X 1 X



 exp Cn1 bE n

∞ X

n=2

1 cE n Dn

(55)

!

(56)

Ers (Vj )bjs

j=1 s=−1 j6=E

−1 j eij n Enm (V0j Vj )bm ,

(57)

j=1 n,m=−1 j6=E

Cn1 =

N1 X ∞ X

i=1 m=2 i6=E 2

cim Emn ΓVi−1 VE + βr

1 X

Ers (VE )δsn +

s=−1

See the Appendix for the definition of canonical forms for ghosts.

7

∞ X

m=−1

−1 η i eiE m Emn (V0E VE ) ,

(58)

Dn1 =

N1 X ∞ X

Enm ΓVE−1 Vi bim .

i=1 m=−1 i6=E

(59)

Similarly, the physical ghost vertex U2gh† ≡ CF U20gh† can be obtained by making the transformations bin → bi−n , cin → −ci−n

(60)

and by using the Hermitian conjugate |0iF of vacuum F h0|: U2gh†





N2  Y  =  i h0|  i=1 i6=F

Z

1 Y

dβr

r=−1



N2 X ∞ X ∞ X

× exp  

i=1 n=2 m=2 i6=F

 

× exp  −

i × exp  M



i=1 i6=d,g,h

i6=j i,j6=F





i=1 n=2 m=−1 i6=F N2 X

  N ∞ ∞ X 2  X  X  −1 j  i cn Enm ΓVi Vj bm  dM exp    m=−1 n=2   i,j=1 i

bF−m cin Enm ΓVi−1 VF  

N2 X ∞ X ∞ X



N2 Y

Z





∞ X





 Enm ΓVF−1 Vi bim cF−n   exp βr 



−1 j eij n Enm (V0j Vj )bm +

∞ X

∞ X

n=2 m=−1

j=1 n,m=−1 j6=E

N2 X 1 X



j=1 s=−1 j6=F

Ers (Vj )bjs   



−1  bF−m Mi eiF n Enm (V0F VF ) |0iF



(61)

where some terms obtained by following the procedure described above have been annihilated by the Hermitian conjugate vacuum |0iF . This vertex can be written as U2gh†





N2  Y  = i h0|  i=1 i6=F

where

φ2 =

Z

1 Y

dβr

r=−1

N2 Y

Z

dMi exp φ2 exp

n=2

i=1 i6=d,g,h

N2 X ∞ X ∞ X

∞ X

N2 X

!



A2n bF−n exp 


i,j=1 n=2 m=−1 i6=j i,j6=F

+Mi

∞ X

N2 X 1 X

∞ X

n=−1



cF−n Bn2  |0iF

(62)

Ers (Vj )bjs

j=1 s=−1 j6=F


(63)

j=1 n,m=−1 j6=F

A2n = − Bn2

=

N2 X ∞ X

i=1 m=2 i6=F

cim Emn ΓVi−1 VF −

N2 X ∞ X ∞ X

m=−1

∞ X

δnp Epm ΓVF−1 Vi bim .

i=1 p=2 m=−1 i6=F

−1 Mi eiF m Emn (V0F VF ) ,

(64)

(65)

The ghost part of the propagator is given by P = VE−1 VF Γ . 8

(66)

Being a conformal transformation, can be written in terms of the following canonical form [4]: 

∞ X

Pgh = : exp 

n,m=−1



× exp 

with

∞ X

n,m=2

1 X

s=−1

Multiplying now U1gh by Pgh , we obtain U1gh Pgh =

N1  Y   i h0| E h0| 



× exp 

where

Z

i=1 i6=E

∞ X

n=−1



3 E cE n Gnr br

(67)

(68)

,

Ens VF−1 VE Γ Esr ΓVE−1 VF Γ

3 Fnm = Enm VE−1 VF Γ



n=2 r=−1



∞ X 1 X

E 3 bE −n Fnm − δnm cm :





E 3  cE −n Enm − δnm bm exp

3 = Enm VE−1 VF Γ Enm

G3nr = −



1 Y

dβr

r=−1



 exp Cn4 bE n

(70)

i=1 i6=b,c

∞ X

(69)

,

.

N1 Y

Z



dη i exp φ4 : exp 

4 cE n Dn

n=2

∞ X 1 X

n=2 r=−1



4 E cE : n Gnr br

!

(71)

φ4 = φ1 , G4nr

=

Cn4 =

(72)

G3nr , N1 X ∞ X

(73)

i=1 m=2 i6=E N1 X ∞ X

Dn4 =

1 X

cim Emn ΓVi−1 VF Γ + βr

i=1 m=−1 i6=E

Enm VF−1 Vi bim −

Ers (VF Γ) δsnηi

−1 eiE m Emn V0E VF Γ

m=−1

s=−1 1 X

N1 X

∞ X

Enr VF−1 VE Γ Ers ΓVE−1 Vi bis

i=1 r,s=−1 i6=E

,

(74)

(75)

and some of the terms have been annihilated by the vacuum E h0|. In the expressions above, we have used multiplication properties (7-10) of matrices Enm (γ). The combination of the second and fourth terms in the expression above yields 

: exp 

∞ X 1 X

n=2 r=−1

−



4 E cE n Gnr br

∞ 1 X X

n=2 r,s=−1

∞ X

: exp

4 cE n Dn

n=2





−

1 X

n=2 r,s,t=−1

=

−1 −1 E  cE n Ens VF VE Γ Esr ΓVE VF Γ br +

Using multiplication property (7) we can write this as ∞ X

!

−1 cE n Ens VF VE Γ Est

N1 X

i=1 i6=E

Esr ΓVE−1 Vi bir   .

  N 1  X  E Etr (Vi ) bir  ΓVE−1   Etr (VF Γ) br + i=1 i6=E

9

 

(76)

(77)

what is zero due to the overlap identities of U1gh Pgh . We can now rewrite the combination U1gh Pgh as [4] U1gh Pgh where now φ5 =





N1  Y  = i h0| E h0|  i=1 i6=E

N1 X ∞ ∞ X X

1 Y

Z

+η



i

dη exp φ5 exp 

i=1 i6=b,c

∞ X

N1 X

N1 Y


i,j=1 n=2 m=−1 i6=j i,j6=E i

dβr

r=−1

Z

N1 X 1 X

∞ X

n=−1



∞ X

 exp Cn5 bE n

5 cE n Dn

n=2

!

(78)

Ers (Vj )bjs

j=1 s=−1 j6=E


(79)

j=1 n,m=−1 j6=E

Cn5 =

N1 X ∞ X

i=1 m=2 i6=E

Dn5 =

N1 X ∞ X

1 X

cim Emn ΓVi−1 VF Γ + βr

Ers (VF Γ) δsn + η i

∞ X

−1 eiE m Emn V0E VF Γ

m=−1

s=−1

Enm VF−1 Vi bim .

i=1 m=−1 i6=E

,

(80)

(81)

So the effect of inserting the propagator into U1 G is to make the following change: VE → VE P = VF Γ

(82)

what was expected considering the bosonic oscillator case. Now we multiply U1gh Pgh by U2gh† in order to obtain the composite vertex Ucgh . The only remaining term which is not annihilated by the two vacua associated to legs E and F is the resulting phase φ, so that the result obtained is3   Ucgh

+N2  N1Y  = i h0|  i=1 i6=E,F

where φ =

N1 X ∞ X ∞ X

i=1 j=1 n=2 m=−1 i6=E j6=F

r=−1

dMi exp φ

N2 X 1 X

N1 X ∞ X

−1 j i eij n Enm (V0j Vj )bm + η

(83)

i=1 i6=d,g,h

cin Enm ΓVi−1 Vj bjm

N1 X N2 X ∞ X ∞ X

cin Enm ΓVi−1 Vj bjm

j=1 i=1 n=2 m=−1 j6=E i6=F

Ers (Vi )bis

i=1 s=−1 i6=E

N2 X ∞ X

j=1 n=−1 j6=F 3

N2 Y

Z

N2 X ∞ X ∞ X

N2 X

∞ X

−1 j eij n Enm (V0j Vj )bm

i

+M

We write Ucgh ≡ U1gh Pgh U2gh† .

10

N1 X

−1 j eiE m Emn V0E Vj bn

j=1 n,m=−1 j6=F

j=1 n=−1 j6=E

+M

dη

i

i=1 i6=b,c

cin Enm ΓVi−1 Vj bjm +

Ers (Vj )bjs + βr

i

N1 Y

Z

i,j=1 n=2 m=−1 i6=j i,j6=F

N1 X 1 X

j=1 s=−1 j6=E

+η i

dβr

N1 X N2 X ∞ X ∞ X

+βr

1 Y

cin Enm ΓVi−1 Vj bjm +

i,j=1 n=2 m=−1 i6=j i,j6=E

+

Z

∞ X

j=1 n,m=−1 j6=E

−1 j eiF m Emn V0F Vj bn

(84)

where we have set some terms to zero using property (8) and the overlap identities for vertices U1gh and U2gh† . Performing the integrations of the anticommuting variables introduced before, we then obtain

Ucgh









  N +N +N2 X +N2 1 1 N1X ∞ X ∞  1X 2 X  Y  N1Y  −1 j  i   Ers (Vi )bis cn Enm ΓVi Vj bm  × = i h0| exp    i=1  r=−1 i=1 s=−1  i,j=1 n=2 m=−1 i6=E,F

×

×

N1 Y

i=1 i6=b,c

   

N2 Y

i=1 i6=d,g,h

i6=E,F

i6=j i,j6=E,F

N1 X

∞ X

−1 j eij n Enm (V0j Vj )bm +

j=1 n,m=−1 j6=E

   

N1 X

∞ X

N2 X

∞ X

j=1 n,m=−1 j6=F

−1 j eiF m Emn V0F Vj bn +

j=1 n,m=−1 j6=E

N2 X



−1 j eiE m Emn V0E Vj bn 

∞ X

j=1 n,m=−1 j6=F

 

−1 j  eij n Enm (V0j Vj )bm  .



(85)

This expression for the composite vertex matches exactly the one obtained using the overlap identities [1].

3

Conclusions

Using explicit sewing techniques, we have sewn two vertices together in order to become a composite vertex. The calculations have been done with the correct ghost numbers for each vertex and the result has both BRST invariance and the correct ghost counting. It verifies the results obtained using overlap identities and the Group Theoretic method for String Theory.

Acknowledgements This work has been done in the Department of Mathematics, King’s College London, University of London, under the supervision of Professor P. C. West, and finished at the State University of Santa Catarina (UDESC). The author wishes to thank Professor West for suggesting, guiding and correcting this work. My gratitude to CAPES, Brazil, for financial support. G.E.D.!

A

Coherent states and canonical forms

In this appendix we present the concepts of coherent states and canonical forms for bosonic and ghost oscillators [3] [4].

A.1

Coherent states

Given operators an with commutation relations [an , a†m ] = nδnm , a coherent state |f i is defined by4

†

|f i = ef a |0i .

Some of the properties of coherent states are: 1) a|f i = f |f i so that ega |f i = egf |f i ; †

2) ega = |f + gi ; 4

We will not be writing the index n of an until it becomes necessary.

11

(86)

(87)

3) hf |gi = ef

†g

;

†

4) xna a |f i = |xn f i ; 5) 1 =

1 π

R

2

d(Ref )d(Imf )e−|f | |f ihf | ;

6) Defining the tensor product of coherent states as |f1 , f2 , . . . , fn , . . .i =

∞ Y

n=1

†

efn an |0i ,

(88)

we have 

: exp 

∞ X

a†n (Cnm

n,m=1



∞ X

− δnm )am  : |f1 , f2 , . . . , fn , . . .i = |

C1m fm , C2m fm , . . . , Cnm fm , . . .i .

(89)

n=1

Symbolically, we can write this property as

† (C−1)a

: ea †

† (x−1)a

Application: since xa a |f i = |xf i and : ea

: |f i = |Cf i .

(90)

: |f i = |xf i, we have †a

xa

† (x−1)a

=: ea

:

(91)

when applied to coherent states.

A.2

Canonical forms

There will be two definitions here for canonical forms. In terms of bosonic oscillators [3] αn , [αn , αm ] = δn,−m , a canonical form O is any operator which can be written as O = exp −

∞ X

α−n An

n=1

!



: exp 

∞ X

n,m=1



α−n (Cnm − δnm ) αm  : exp −

∞ X

!

Bn αn e−φ

n=1

(92)

where φ, An and Bn do not depend on αn or α−n . The product of two canonical forms O1 and O2 is again a canonical form O with ∞ X

An = A1n + Bn = Bn2 +

m=1 ∞ X

1 Cnm A2m ,

(93)

1 2 Bm Cmn ,

(94)

m=1

Cnm =

∞ X

1 2 Cnp Cpm ,

(95)

p=1

φ = φ1 + φ2 +

∞ X

Bn1 A2n .

(96)

n=1

In terms of the ghost oscillators [4], a canonical form C can be defined as any operator that can be written as ∞ X

C = eφ exp

n=2



× : exp 

!

An b−n exp 

∞ X

n,m=−1



× exp 

∞ X



n,m=2

∞ X

n=−1



Bn c−n  



c−n (Enm − δnm ) bm  exp  

n=2 r=−1



b−n (Fnm − δnm ) cm  : exp  12

∞ X 1 X

∞ X

n=−1



cn Gnr br 



Cn bn  exp

∞ X

n=2

Dn cn

!

.

(97)

Again, the product of two canonical forms C1 and C2 will be a canonical form C with coefficients φ = φ1 + φ2 − An = A1n + Bn = Bn1 +

∞ X

n=−1

∞ X

Cn1 Bn2 −

∞ X

Dn1 A2n +

n=2

∞ X 1 X

A2n G1nr Br2 ,

T

1 A2m Fmn ,

n=2 ∞ X

(98)

n=2 r=−1

(99)

T

2 1 Bm Emn ,

(100)

n=−1

Enm = Fnm =

∞ X

1 2 Enp Epm ,

p=−1 ∞ X

(101)

1 2 Fnp Fpm ,

(102)

p=2

Gnr = G2nr + Cn =

Cn2

−

Dn = Dn2 +

∞ X 1 X

2 2 , Fnm G1ms Esr

m=2 s=−1 ∞ 1 X X

2 A2m G1mr Ern

+

m=2 r=−1 ∞ X 1 X

∞ X

(103) 1 2 Cm Emn ,

(104)

m=−1

2 Fnm G1mr Br2 +

m=2 r=−1

∞ X

1 2 Dm Fmn .

(105)

m=2

where AT means the transpose matrix of A.

References [1] L. Sandoval Jr., Sewing string tree vertices with ghosts, hep-th/9909162. [2] A. Neveau and P. West, Group Theoretic Approach to the open bosonic string multi-loop S-matrix, Comm.Math.Phys. 114 (1988) 613. [3] P. Di Vecchia, F. Pezzela, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, N -point g-loop vertex for the free bosonic theory with vacuum charge Q, Nucl.Phys. B322 (1989) 317. [4] P. Di Vecchia, F. Pezzela, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, N -point g-loop vertex for a free fermionic theory with arbitrary spin, Nucl.Phys. B333 (1990) 635 [5] A. Neveau and P. West, Cycling, twisting and sewing in the Group Theoretic Approach to Strings, Comm.Math.Phys. 119 (1988) 585. [6] M. Freeman and P. West, Ghost vertices for the bosonic string using the Group Theoretic approach to String Theory, Phys.Lett. 205B (1988) 30. [7] L. Sandoval Jr., String Scattering, Ph.D. thesis (1995) King’s College London, Univ. of London.

13