g-Loop vertices for free fermions and bosons

Volume 220, number 4

PHYSICS LETTERS B

13 April 1989

g-LOOP VERTICES FOR FREE FERMIONS AND BOSONS F. P E Z Z E L L A i NORDITA, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark SISSA, Strada Costiera 11, 1-34014 Trieste, Italy and INFN, Sezione di Trieste, Trieste, Italy Received 24 June 1988

We construct an N-point g-loop vertex for the conformal theory of free fermions of spin ½with the property of reproducing the N-point correlation functions of the primary fields of the theory on arbitrary Riemann surfaces. It turns out to be written in terms of the Szeg6 kernel. The same construction is then performed for free real scalar fields, checking the bosonization rules of the free fermionic theory, by the use of the covariant operatorial formalism.

Many different approaches have been developed by now for computing multiloop scattering amplitudes in string theories. They include the functional integral technique both in the covariant [ 1 ] and light-cone gauge [ 2 ], the covariant old operator formalism [ 3 ], where the orbital and ghost degrees of freedom circulate in the loops [ 4-9 ] a more hybrid technique where both the operator formalism and the path integral are used [ 10 ], a group theoretical approach based on overlap equations [ 11 ] and a new operator formalism based on the construction of the g-vacuum [ 12 ]. The basic object that has been considered in the approach of ref. [ 6 ], is the N-string g-loop vertex Vu.g which contains all the information of the perturbative string theory. It has the important property of reproducing the g-loop scattering amplitudes involving N arbitrary physical particles: these can be indeed simply obtained by saturating Vx,x with the N corresponding physical states. The starting point for obtaining the vertex Vu,g is the N-string vertex VN,o [ 5 ] which, saturated on N physical states, reproduces their corresponding tree scattering amplitudes. Vx.~, is indeed obtained from the ( N + 2g)-string vertex VN+ 2g,o through a sewing procedure of 2g of its legs by inserting a suitable propagator. The sewing procedure, which is explicitly performed with oscillators and coherent state techniques, yields straightforwardly to an expression o f VN.g as a function o f geometrical objects as the prime form, the period matrix and the abelian integrals. This formalism seems to be sufficiently general that it can be applied to an arbitrary conformal field theory. In this case the role o f the N-string vertex for string theories is played by an N-point vertex VN,O which has the property of reproducing the N-point correlation functions involving the primary fields of the theory at tree level, i.e., on the sphere, when it is saturated with the corresponding N highest weight states suitably defined. As in string theories, starting from Vx,o, the sewing procedure allows to construct Vu,g that generates the correlation functions with g loops, i.e. on a Riemann surface of genus g. O f course there are some differences between the case of string theories and the one o f conformal field theories. Indeed in the former the interest is devoted to scattering amplitudes so that an integration over K o b a Nielsen variables is needed; in the latter we are interested to reproduce correlation functions so that such an integration should not be performed. Furthermore, the conformal group in an arbitrary conformal field theory is not a gauge group as in the case o f string theories, hence no integration over the moduli of the Riemann surface must be performed. Supported in part by Fondazione A. Della Riccia. 544

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Volume 220, number 4

PHYSICS LETTERS B

13 April 1989

In this letter we apply these ideas to the simple case of conformal fields with dimension A = 1, corresponding to a theory of free fermions with spin ½. Starting from the ( N + 2g)-point vertex which reproduces the correlation functions for these fields on the sphere, and sewing together 2g external legs after the insertion o f a suitable twisted propagator, we obtain the N-point g-loop vertex Vu.g for free fermions on an arbitrary Riemann surface. It turns out to be a function of the Szeg6 kernel written in terms o f the P o i n c a r 6 0 series. Furthermore, V~.e reproduces the g-vacuum discussed in ref. [ 12 ]. We then perform the same construction for free bosons: it allows to check explicitly bosonization rules [ 13 ] of the free fermionic theory on an arbitrary Riemann surface, using the old operator formalism. The starting object is the vertex V:
Vx,o=

,(O[exp

(l

-~

i= I

b~°DS.~'/Z)(FV. 'Vj)b} j)

~

)

,

(1)

r,s= 1/2 i,j= I

where

D5,.'/2~( V(z) ) =

1 ( s - 1/2)!

O.~__,/2{V,(z)l/2[V(z)],_,/2} -

(2) _-=o

is an infinite dimensional representation of the projective group corresponding to a conformal weight A = ½ [ 15 ] and F ( z ) = 1/z. The V , ~(z) are projective transformations corresponding to a choice o f local coordinates vanishing at the Koba-Nielsen points zi. Although in the past the choice of Vy~(Z) found by Lovelace [ 16] has been widely used, our results do not depend on the particular choice for V,(z). The vertex ( 1 ) reproduces the correlation functions of the free fermionic theory on the sphere. Indeed if we want for instance to compute the fermion propagator on the sphere we have to saturate VV,o with the highest weight states associated to the primary fields of the theory ~,(z)'s; they are obtained by the relation

. {OVT'(z)'~ '/2

lm/ .... \

-az

) /

~'[Vy'(z)][0)-

1 [VT(O)],/2b~{/210),

(3)

corresponding to the local coordinate V y ~ ( z ) vanishing at the point z = zi. Here the following expression for the fermionic field is assumed: q/(z)=

~

b,.z - ' - t / 2

re I / 2 2

with b,.=b,?-. By saturating (1) with two states as in (3) we get the Green's function for free fermions on the sphere: 1

<~'(z)q/(y))- z-y"

(4)

In the same way one can obtain from ( 1 ) and (3) an arbitrary correlation function o f the free fermionic theory on the sphere. F The vertex Vx.~ is obtained by sewing together 2g legs of V Nv+ 2g.0 after the insertion of the following twisted propagator [ 3 ] : P(xj,) = ( - 1) ~1- t~''~/"~'xl,z'°Otl-xl,) w . . ,

/~=1, ..., g

(5)

on the (2~t- 1 )th leg, say, that is sewn to the next one. The twist operator t2 and Ware given by £2= e~-' ( - 1 )t-"-"'-/2W=Lo-Lj .

(6) 545

Volume 220, number 4

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Even if we use for convenience the twisted propagator, our results are largely independent from the specific choice of the propagator as it will appear from subsequent calculations. Furthermore, B~, in (5) can be regarded as the/.tth component of a g-dimensional vector, which can take the values 0 or 1, corresponding to the possible insertion in P(x~,) of the fermion parity operator ( - 1 )F. The quantities B~, specify the boundary conditions fixed for the fermionic fields around the g homology cycles commonly denoted by b of a compact Riemann surface of genus g: hence the vector B, so defined, takes into account of the spin structures. The sewing procedure, explained in detail in ref. [ 17 ], leads to the following automorphized expression for the vertex: N V x.~.-F _

1)NiCkn-l/2 ] H [i<0[]

l~' fi [1--( o~

It= I

i= 1

× e x p ( - ~1 ~

t,l=

b,,,D,,/2)(FV., ) ~. (_I)N[{D,,,/2,(Tc.)D,.,.~(2)(~)b,a)) .

_~

(7)

o~e.'/

I r.s= 1/2

where the sum runs over the elements 7", of the discontinuous Schottky group 5f whose generators are given by

S,=FV~,IP~,~,_I,

/z= 1, ..., g .

(8)

Z' denotes that in the terms where i=j the identity must be left out and, furthermore, a sum over the indices l and m from ½to m is understood; N~ is defined as follows:

N~= L B,,N~,,

(9)

It= ]

where N~, corresponds to the number of times the generator S~, appears in the element 7", of the Schottky group. Furthermore I-[~ in (7) denotes a product over all prime classes each characterized by the multiplier k,. Using the relation

D~/,/e)(FVT~ ) ~ (_I~N(~n(,/2~(T~)D~,If2)(Vj) = 1 I f - - a r1- l / 2)! s uh,, ..... (r-~). (s - 1

a:~(,-t/2G( V l ( z ) ,

177~l(y) ) ~= i,= o

(lO) where

G(V~(z), Vj(y))= ~ ( - 1 ) N~ [V'~(z)T"(~(Y))]'/2 V,(z) - T. ( ~ ( y ) )

'

(11)

it is possible to rewrite the vertex (7) in the following elegant form:

[l--(--1)N[{k~-'/2] VvN,O

v..F..~= l--[' f i (~

It =

I

( 11L 2~ b~)' , r _ , , ~

×exp -

/2

t,j=

1 r,.s =

:

~ ~1

t,s_/~,~ : ~ 1 ~0) 5I

- -1 / 2

O~.-I/2(~[Vt(z), ~ ( y ) ] _-=v=o bj i) ) ,

(12)

I

where ~[V,(z),

Vj(y)I=G[V,(Z), ~ ( y ) ] -

[ v', (z) v>(y) ] '/2 V,(z)- ~(y)

(13)

The g-loop vertex ( 12 ) consists of three pieces. The first one is just the g-loop partition function for a Majorana fermion: 546

Volume 220, number 4

Z~ e ) = l - I ' ot

[1--(--1)Ngk'~-'/2] •

fi n=

PHYSICS LETTERS B

13 April 1989

(14)

I

The second one is the vertex for the sphere while the last term is the contribution coming from the automorphization procedure needed to go from the sphere to an arbitrary genus-g Riemann surface. If we saturate ~,,e with two states as in (3) we get the fermion correlation function on a genus-g Riemann surface:

(~,(z)~(y) ).4 = G(z, y),

15 )

with G(z, y) defined in ( 11 ). This function is called the Szeg6 kernel in the literature. Actually the Szeg6 kernel is usually defined in a different form in terms of the O-functions and the prime form:

G(z,y)=O(~;)(riog)/E(z,y)O(~;)(zlO),

(16,

),

where ( o~~, % ) define an even spin structure on the Riemann surface. In ( 16 ) r and E (z, y) denote respectively the period matrix and the prime form of a Riemann surface of genus g. Since the two expressions ( 11 ) and (16) have the same singularities and periodicity properties around the various cycles they must be identical. It is interesting to check the equivalence between ( 11 ) and (16) on the t o m s where ( 11 ) gives the following expression for the spin structures Z = ( - + ) and Z = ( - - ), corresponding to B = 0 and B = 1, respectively, +~

( + 1 ),,k,,/2

G_±(z,y)=,,=~_~

z-k"y

'

(17)

while from ( 16 ) one gets 1

r~

( 1-k'')2

G_+(z,y)= z--~y, 12, (1-T-k"-'/2) 2

(1-T-k"-'/2Y/Z)(1-T-k"-'/2z/Y) (1-k"y/z)(1-k"z/y)

(18)

In deriving (18) we have used the following explicit expression of the prime form:

E(z,y)=(z-y)

fi (z-k"y)(y-k"z) ,,=l z Y ( 1 - k " ) 2

(19)

It is possible to check that the two expressions are identical [ 18 ] using the identities R~ 8 and R2 ~ of ref. [ 19 ]. It is also easy to see that they have the same expansion around k = 0. It is interesting to obtain from our formalism the g-vacuum, introduced in ref. [ 12 ]. It is given by the vertex V~.~.. By choosing the local coordinates such that V~ (z) = z we get for V~.gthe following expression:

(1

V ~,=Z~') (O I exp - ~

~

)

b,.B,.,bs ,

(20)

r,s=l/2

with B,.,=

(r_½)------~V.(s_½)~v.O"_--'/201~,-'/2 G ( z , y ) -

1

_-=o

that reproduces the definition of the g-vacuum for fermions of spin ½ given in ref. [ 12 ]. The previous construction of VFN,,~can be extended to the case where we have many fermion fields. In partic547

Volume 220. number 4

PHYSICS gETTERSB

13 April 1989

ular, in the case of two Majorana fermions the theory is equivalent to the one with a free real scalar field. In the following we want to construct V~,e for this theory and check how bosonizations works in our formalism. We assume the following expansion for a scalar field ~(z):

~(z) = q - i a o l o g z + i ~ a,, z -'' , he0

(22)

H

where ao= -iN,

(23)

with Nbeing an operator that has integer eigenvalues; the oscillators a~ are related to conventionally normalized oscillator operators by a,, = x/na,, for n > 0 and a_,, = x / ~ l a _+, for n < 0. B The vertex Vx.~. for scalar fields can be computed by means of the same procedure used for free fermions. It is given by

1

( 1 x

B

V~,~,= [ I ' f i ( 1 - k S ) V N , o e x p ~ 2 n= I

~

110,,

\z. i,.j= I n,m=O

a},')n!m! : O : 7 1 o g E [ V , ( z ) , ~ ( y ) ]

_-=,,=o

a}~'

)

I',(z)

xoO~, r2 ~ a~,')1 ~ 0'_'\ o ¢ 2 / \ ,=l ,,=o

f)

o9 ,

(24)

20

where the co are the g first abelian differentials. The vertex Vx, Bo appearing in (24) is the following:

Vx.o=

,(n,,O.I

I

5

1

/

N, exp - ~

icj

a,,") D,,,,,(FVi-'V/)a};{ ) ,

(25)

n,m=O

D,,, being the infinite matrix representation of the projective group with conformal weight A-,0. If we bosonize free complex fermions they are expressed in the bosonic theory by ~'(z)--,exp[0(z)] , q)(z)--+exp[-0(z)] .

(26)

In order to compute the fermion Green's function in this theory we must saturate

lim(OV:'(z)~'/2

_-.=

Oz

,/

exp[O(z)]lO),-

(OV~:(z)~ '/2

lira \

2 *22

0z

]

V~..~ with the states

1 [r,(o)],/2 In~=l,O,~)~,

exp[-~(z)]10)2--

1 [vi(0)l,/21n2=-1,0,~)2.

(27)

So one obtains the following two-point function: ~¢(Zl)@(7-2)~:O(~:)(~';(J))/E(zI,Z2)O(~:)(7~IO),

(28)

21

that has been normalized dividing by the bosonic partition function

1(::)

Z~ ~''= ~I' ,,=fi, ( 1 - k ~ , ) O

(rl0).

(29)

Bosonization implies that (Z~)2=Z~. 548

(30)

Volume 220, number 4

PHYSICS LETTERS B

13 April 1989

This equation can be explicitly checked in the case o f the torus for those spin structures described by both vertices ( 12 ) and (24). In fact from ( 12 ) we get

Z~L~= fi [1-(-1)nk"-~/2], n:

(31)

I

while from (24) we get Z[~)=

O

~_, kn2/2[exp(2rfia2)k'~']nk'~/2exp(2~rioqa2)

(rl0)=

(32)

So (30) is simply checked using the Jacobi identity

E

k"~-/2z"= f i ( 1 - k " ) ( l + k " - ' / 2 z ) ( l + k n - ' / 2 / z ) ,

II - - - - ¢~S)

I1=

(33)

|

with the identification 1 -B=2a2. By saturating the vertex the more general correlation function

V~.g (24) with the states In,, 0 ) j we can compute

(lq

O o~1 ( r l 0 ) ,

exp[n,~(zi)]

= ]-[ [E(z,,z,)]O i
zi ,

"

(34)

Og2 =o

that reproduces the result obtained in ref. [ 20 ] with other techniques. In particular, if the vertex (24) is saturated with M s t a t e s I n~= 1, 0•) ( i = 1..... M ) and with M s t a t e s I n i = - 1, 0 , } (3'= 1..... M ) , then from the " a d d i t i o n - t h e o r e m " for abelian functions [21 ], it is possible to write the corresponding correlation function (34) as follows:

/,H" exp[~(zD] exp[-O(yj)]} =det((~,(zDq)(yj))),

(35)

i=1

that is equal to the correlation function o f M fields ~U(z,) and ~ ( y j ) . In conclusion, we have seen that the sewing procedure used in the old operator formalism extended to free bosonic and fermionic conformal theories provides a very powerful tool for computing correlation functions on an arbitrary R i e m a n n surface starting from those on the sphere and the propagator ( 5 ) and for showing bosonization. I a m greatly indebted to Paolo Di Vecchia for his assistance throughout this work; I would like to thank also A. Lerda, F. N i c o d e m i and M.H. S a r m a d i for helpful discussions and N o r d i t a for their kind hospitality.

References [ 1] E. D'Hoker and P.H. Phong, Rev. Mod. Phys. 60 (1988) 917. [2] S. Mandelstam, Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986). [3] V. Alessandrini, D. Amati, M. Le Bellac and D. Olive, Phys. Rep. 6 ( 1971 ) 269. [4] A. Neveu and P. West, Nucl. Phys. B 278 (1986) 601. [5] P. Di Vecchia, R. Nakayama, J.L. Petersen and S. Sciuto, Nucl. Phys. B 282 (1987) 103; P. Di Vecchia, R. Nakayama, J.L. Petersen, J. Sidenius and S. Sciuto, Phys. Lett. B 182 (1986) 164. [6] P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Nucl. Phys. B 298 (1988) 526; Phys. Lett. B 199 (1987) 49; P. Di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, Phys. Lett. B 211 (1988) 301. [7] G. Cristofano, F. Nicodemi and R. Pettorino, Phys. Lett. B 200 (1988) 292; G. Cristofano, R. Musto, F. Nicodemi and R. Pettorino, Universith di Napoli preprint (1988), Intern. J. Mod. Phys. A, to be published. 549

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[8] U. Carow-Watamura and S. Watamura, Nucl. Phys. B 288 (1987) 500; Nucl. Phys. B 301 (1988) 132; B 302 (1988) 149. [9] A. LeClair, Nucl. Phys. B 297 (1988) 603; B 303 (1988) 189. [ 10] J.L. Petersen and J. Sidenius, Nucl. Phys. B 301 (1988) 247; J.L. Petersen, J. Sidenius and A.K. Tollst6n, NBI preprints NBI-HE-88-29, NBI-HE-88-30 (1988). [ 11 ] A. Neveu and P. West, Phys. Lett. B 193 (1987) 187; B 194 ( 1987 ) 200; B 200 (1988) 275; Commun. Math. Phys. 114 (1988) 613; CERN preprint TH-4903/87; M.D. Freeman and P. West, Phys. Lett. B 205 (1988) 30; P. West, Phys. Lett. 205 B (1988) 38. [ 12] N. Ishibashi, Y. Matsuo and H. Ooguri, Mod. Phys. Lett. 2 (1987) 119; L. Alvarez-Gaum6, C. Gomez and C. Reina, Phys. Lett. B 190 (1987) 55; C. Vafa, Phys. Lett. B 190 (1987) 47; L. Alvarez-Gaum6, C. Gomez, P. Moore and C. Vafa, Nucl. Phys. B 303 ( 1988 ) 455; S. Mukhi and S. Panda, Phys. Lett. B 203 (1988) 387. [ 13 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93; E. Martinec, Nucl. Phys. B 281 (1987) 157. [ 14] E. Corrigan and C. Montonen, Nucl. Phys. B 36 (1972) 58. [ 15 ] A. Clarizia and F. Pezzella, Nucl. Phys. B 298 ( 1988 ) 636; A. Neveu and P. West, Phys. Lett. B 200 (1988) 275. [16] C. Lovelace, Phys. Lett. B 32 (1970) 490. [ 17] C. Montonen, Nuovo Cimento 19A (1974) 69. [ 18 ] M.H. Sarmadi, private communications. [ 19 ] D. Mumford, Tara Lectures on Theta ( Birkh~iuser, Basel, 1983 ). [20] E. Verlinde and H. Verlinde, Nucl. Phys. B 288 (1987) 357. [ 21 ] J.D. Fay, Theta functions on Riemann surfaces, Springer Notes in Mathematics, No. 352 (Springer. Berlin, 1973).

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