From vertex operators to Calogero-Sutherland models

'~ ELSEVIER

NUCLEAR PHYSICS B Nuclear Physics B 476 (1996) 351-373

From vertex operators to Calogero-Sutherland models V. Marotta 1 A. S c i a r r i n o 2 Dipartimento di Scienze Fisiche, Universitd di Napoli "Federico H", and INFN, Sezione di Napoli, Mostra d'OItremare Pad. 19 1-80125 Naples, Italy

Received 26 April 1996; accepted 20 June 1996

Abstract The correlation function of the product of N generalized vertex operators satisfies an infinite set of Ward identifies, related to a Woo algebra, whose extension out of the mass shell gives rise to equations which can be considered as a generalization of the compactified Calogero-Sutherland (CS) Hamiltonians. In particular the wave function of the ground state of the compactified CS model is shown to be given by the value of the product of N vertex operators between the vacuum and an excited state and the Hamiltonian is identified with W02 generator. The role of the vertex algebra as underlying unifying structure is pointed out. PACS: 02.20.Tw; 03.65.Fd Keywords: Vertex operator; Woo algebra; Calogero-Sutherland model

1.

Introduction

Recently there has been a renewal of interest in the one-dimensional integrable models, in particular in Calogero-Sutherland ( C S ) models [ 1,2]. These models describe quantum mechanical systems of N one-dimensional particles interacting through specific two-body potentials. Although they have been proposed and completely solved in the beginning o f the seventy, they are at present object of intensive study for their unexpected connections with other fashionable models as the matrix models where the eigenvalues o f the matrix are identified with the momenta of the particles [3] and the c = 1 conformal field theory ( C F T ) models [4] and for their interesting and hidden 1E-mail: (Bitnet) [email protected];(Decnet) AXPNAI::VMAROTrA. 2 E-mail: (Bitnet) [email protected];(Decnet) AXPNAI::SCIARRINO. 0550-3213/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved. PII S0550-321 3 (96)00345-8

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V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

infinite symmetry structure. The connection of these models with the Lie algebras root systems has been clarified just a few years after they were proposed and their complete integrability was proved [5]. An alternative proof of the integrability as well as a new expressions of the quantum integrals of the Sutherland model is given in [6] using representation theory of the Lie algebra gl(N) and of its affine extension. Only recently the underlying infinite Kac-Moody symmetry has been understood and their invariance for a W~ discovered. In [7] it has been proven that the Calogero model is invariant for U( 1) Kac-Moody, the U( 1 ) current algebra appearing as the generating function of the quantum integrals of the model, and for a W1+~ algebra. In the case in which the interacting particles transform as the fundamental representation of a su(N) the symmetry is extended by an affine su(N) and the Wz+~ algebra becomes a coloured one. This symmetry has been further clarified in [8], where it is also shown that the Sutherland model is invariant only for the Yangian Y(su(N)). In [9] it has been proven that the ground state of the models satisfies the Knizhnik-Zamolodchikov equation. Moreover using the SN-extended Heisenberg algebra [ 10] an universal Hamiltonian for the CS model has been written as the anticommutator of one particle operators whose commutation relations reduce to the standard bosonic annihilation-creation operators in the limit of vanishing coupling constant [11]. In this approach therefore the potential is connected with the statistics of the particles. In this work we show that the very deep structure of these models, more exactly of their compactified version, i.e. the Sutherland model, is the vertex algebra structure, which constitutes the mathematically rigorous formulation of the algebraic origin of conformal field theory. We refer for an exhaustive discussion of the subject to [ 13] where references to the original papers can be found. For a pedagogical introduction for the physicist see Ref. [ 14]. The generalized vertex operators provide a well-known explicit construction of the vertex algebra and they are connected with the structure of Lorentzian algebras [ 15-17] which includes the indefinite Kac-Moody algebras [18] and the Borcherds algebras [19], the affine algebras being only a particular and peculiar subset of these algebras. Even if some of the topics we discuss here can be found scattered in the literature, we believe it is useful to present them here in a unified manner. In fact we believe that our approach permits a unified discussion of the algebraic structure, the connection with conformal field theory and the statistics of the particles. The paper is organized as follows: in Section 2 we write the product of N generalized vertex operators (GVO) in terms of an ordered product of the GVO times derivative of Jastrow-like functions and introduce a set of differential operators in terms of which we can get all the relevant results of the vertex algebra. In Section 3 we define the amplitude or correlation function for the GVOs product and show that it can be obtained by the action of differential operators applied to the amplitude of N standard vertex operators (VO). In Section 4 an (infinite) set of Ward identities is then derived for the amplitudes, and it is shown that the generators spanning a W~ algebra are a subset of the set of GVO symmetries.

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

353

In Section 5 we consider explicitly the particular case of the correlation functions for VOs and we show that the wave functions of the ground state of the CS model are obtained as the matrix element of the product of N VO between the vacuum and a suitable excited state, the Hamiltonian of the model appearing as a combination of differential operators appearing in the analogous of the Ward identities for off-shell amplitudes. Finally in Section 6 we briefly discuss further developments of the formalism previously developed which in the most general case gives rise to more general systems than the CS model. Let us remark that the vertex operators for the CS models already introduced in the literature [20] can be obtained by expansion of the GVOs.

2. Differential operators for GVOs In this section we introduce a set of differential operators that realize the vertex algebra in a complementary way to the usual vertex operators construction [ 13 ]. First we briefly recall the GVOs construction, which provides a well-known realization of vertex algebra [2l ] and then we show that it is possible to build up a set of differential operators by means of which to carry out the relevant operations on GVOs. Let us recall the particular choice of basis that we use in our construction that simplifies greatly the formal calculations. The standard (tachyonic) vertex operator (VO) is defined by U r ( z ) = : e ir'Q(z) : where Q (z) are standard Fubini-Veneziano fields Q ~ Z ( z ) = qjZ _ i p ~

inz + i ~

°t---~n~Z-n

(1)

g/

he0

on a d-dimensional Minkowskian toms with periodic boundary conditions given by a vector r in a lattice A. A general GVO is a product of Schur polynomials in the derivatives of Fubini fields times a standard vertex operator, b{r,ri} { ( n i ) ) [)Z/ ,

=:

rU r (z) :=: IIT~,~i(z) i

where

r t = r - ~i

ri

A{n,}

(zi)

IIuri(

zi) U r' (z):

,

(2)

i

and

d{"')(zi) = ]'-I lim

0n'

(3)

~ - z i ~ z ni!OZ n~ " i

At this point a few words of warning are necessary: in the above equation the label stands for a finite set of variables; to be rigorous we should put it in curly brackets. However, in order not to overweight the notation we use only the round brackets. We shall use this convention hereafter. Only for the GVO or VO the functional dependence is only from one variable. Moreover, in the above definition we should also write the explicit dependence on the z-point where the limit is taken at the end. For the same zi

354

1,~ Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

reason we shall omit here and in similar equations the dependence on the point(s) where the limit is (are) performed. Formal Laurent expansion of GVO is denoted as U {r'ri} /

{(,,,)}~z) ~ = / V_~~

A {r'r'}

{(~,)}., Z . . . . h

(4)

nl

r2

where h = ~ + ~ i ni is the conformal weight. The product of two VOs is simply Ur(z)

U ~ ( ( ) =: U r ( z )

US(£) : ( z - £)r.s

for ](I < [z[.

(5)

By means of locality properties the product is analytically extended to the whole C 2 space except the poles z = ( and z, ( = 0, o~. We can easily generalize this relation to N VOs product: N

N

I~grq(zq)=:

I-~grq(zq) :.~{NO'O'O'rrrl'}(Zl, Zl'),

q=l

q=l

(6)

where we have introduced a Jastrow-like function

.~{NO,O,O,r/'rI'}(Zl,

Zff ) = H ( Z l

(7)

-- Zl') rl'r'' •

l>l'

Using this result and the differential operators A{""nA(zi'scJ) = H i.j

lira lim

(8)

zi--*z ( j - , ( ni !3Z ni n . 13 y:nj .1" ~.i

we can compute any two GVOs product and get f {r'ri} ( z )

((.,)}

U(S,sJ} ( ( ) = A { n i ' n J } ( Z i , ( j )

{(.j)}

: u{r'ri}(z, zi)U{S'Sj}((,(j)

:

X • {aO ,aio,,toy,a(~,} ( Zi, ~ j ) ,

(9)

where u{r'ri}(z, zi)=: I-iuri(Zi)Ur'(z)

",

r'=r-

i

~-~ri,

(10)

i

U{S'SJ} ( s c , ~ : j ) = : I ~ u s J ( ( j ) U S ' ( ( ) : ,

st=s - ~-~sj

.i

(11)

i

and

F {aU'aiO'aOj'a°O)(Zi, (j) = H ( Z i -- ~j)ai)(Zi -- ()aiO(Z -- ~j)aoj (Z -- ()aoo ij

with aij = ri • s j , aoj = r t • s.j, aio = ri • s t e aoo = r t • s t. After some algebraic calculations we obtain

12)

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373 tli,nj

355

{ r,s,ri,$j }

U {r'r'} (.~ U {s'sj} [/::'~=

X{(kl)'(kJ)}

{(nl)}~''" {(nJ)}~'b: ~

--~)-r's+~iki+~jkj

(Z

x :

u{r,rl}

: .u{S,Sj}

{(n,_k,)}tz)

..~.

((nj-tp}t¢) :

(13)

with

{p~,pj} x{r,s,ri,sj }

×H(pi_li] ( aio

• .

t,J

~

aij "~ aoj li + lj J Q p j - - l j

J

(14)

"

Note that this relation is obtained in a non-local way without OPE expansion, so it holds also in the large distance limit and can be used to give non-perturbative results. The set of ate can be expressed in terms of upper triangular matrices in g l ( N ) , so relating our approach to the approach of Ref. [6]. To extend Eq. (9) we remark that only Jastrow-like functions appear in the GVOs product so we can define N

N

1-I"O"'%}(zq) =A(%}(z.,,) ~{(n,,q)} q

: Hu{r.,,%}(zq,

z.~) .

q

x.T:~ "''j`' ."ill' ,%, ,all,} ( Z~,, Zi,, ),

(15)

where

A{%}(z",, ) =

H lim 0% {q,Uq}Z"l----~Zq nv,, !8Z,",,:'q

(16)

is the generalization of operators in Eq. (8) and _ Z,I.)a,.,.(Z,

_Z,,)a,,I'

l>l' iLJl, X ( Zl -- Zj,, ) aljl' ( Zl -- Zl' ) all' , N • Hs{rq'rt'q}(zt'q'Zq) q

N :=: H H

grvq(Zvq)grq(zq)

: "

(17) (18)

q {Vq}

From these definitions we can introduce differential operators that, acting only on normal ordered products of the tachyonic vertex, give the expression for general GVOs: 'aill' 'alJl, "till' } zt Zi" ZJl' ) "f~{ailjF'aillt'alJll'allt} ( Zt~q) = /~{nt~q} ( Zl'q ) ~,']:'{ailJl, N ~(% }

If we expand these operators in terms of A t % } (Zt,q) we obtain

(19)

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

356

{n,,,}

tz~,,,) = ~

{,,q}

(zt, zt,) A{""~-k"qI(z,,q),

(20)

, zt ) = (A{~,.~t (z,,,,) "V,,{N""J,"%e'%"a"'} ( zi,,zj,,) ) . tzt,

(21)

"-'{k,.q}

{ k,.,,} where

S-'{"~,J,.%,','%,,",'} {k,,,}

These functions can be explicitly computed (see Appendix A) and have the following expression: ~-~OilJl/ ,aillt ,£11Jll ~111/} /

{k,,, }

~-

tz~, ze )

Z

{E,>~gq+E,,<,, ,'-

1-I } >t, (

X { ( kl; ),( k}t ' ) }ll' --rt.rt,+~.t ~;+~jt lctji' - zl, )

(22)

The explicit expansion of general N GVOs product becomes N

N

Il

ll{r q,rrq}} ( Zvq, Zq ) "="l'~{aqJt' v { (n,,q) ~{n,,q } 'aill' "aql' 'art' } ( Zt,q ) : I I u{rq'r"q } ( Zvq, Zq ) : q q {rt,r I, ,rit.rjV }

{Zt>q~.q+~v1' ( Z , N

x : II

u{rq,r,'q}

Zl') -rl'rF+/-''~il

" ~-'~

,

{(.,,q_k,,,,)}tZ,,q, Zq) " .

% (23)

q

Locality properties for GVOs can be evaluated by means of the permutation group (notice that d{""o}(Z,,q) and ordered products are &v-symmetric so we do not indicate explicitly the symmetrization): N 1~

Hqq, 1 1 q

U {r''r'q }

{(nvq)} (Zq) Hqq,

= d{n"q}(Zvq)

N : I I u { r q ' r " q } ( Z q , Zvq) :Kqq, JT{N':titjl"aitF'aot''att'}(Zit,Zjv), q

(24)

where llqq, is an operator that exchanges the order of two GVOs while Kqq, exchanges the sets of indices {q, Vq} and {q', Uq, } in the functions ,1--{ ailJl t 'aillt 'LllJlt ,allt } tt Zi, , Zjr, ) = ~-Z,=0 qt --q 1 rq ~,.rq, ..i:.{aitj,, ,alt,, ,aot, ,an, } ( Kqq' J-N N Zi,, Zj,, ) ,

(25)

where e e irr. Hence we generalize this construction also to non-local cases (in particular rational values of rq . rq, give RCFT) where vertex algebras can be extended following Ref. [22]. =

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

357

3. GVOs eorrelators

By means of this formalism we calculate explicitly also correlation functions for GVOs using the property of normal ordered VOs: U{rq'rvq} ( Zq' Zvq ) :) = ( ~ , rq.O "

A{rq'r'q} ( Zvq, Zq ) = (: H

(26)

q SO by using Eq. (15) we simply have G{rq,r,.q} ( Z q ) = ( H g { r q , r v q } [ _

N{(n,.q)}

x\

{(nvq)} "~q)] =

~{aitjt''aitt"atJ,''att'}( .A{rq,rvq}( (n,.q} Zvq) St,q, Zq)

q

_~ %Jl, .%1,.%, .a,, }

= ,-.{%}

( zt. Zt, ) 6~-~q r,,.0 "

(27)

We can also verify the duality invariance by using the action of the symmetric group: K G {rq'rvq} ¢ H il N{(nrq)}kZq) = H T h l ( r i 14~i l~i

where

"rlil( r i •

. ~,._,{rq,r,,q} , 6:-r~G {rq'r''q) ¢, ~ " ¢l)LXN{(nvq))kZ q) = N{(n,q)}~.~qJ

rl) = e rrr' and neutrality implies ~ t ~ i rl . r i

V i, (28)

= --r 2.

A oN{(..,~)}tz q ) ~ { r ° ' r ' q. } " amplitude can be obtained also by means of the action of the d {~''~} ( z~.q) operators on a tachyonic amplitude:

G N{(n,,q)}~Zq {rq'r''q} " )" --- V {nvq} f%;'~'lll (Z~,~) t-'N+~, % tZ~.~. Zq)

(29)

where V [ aili; "ailll [ aid; 'aill [ -- 1 {n,.q) (Zvq) =A{%}(Zvq) "~N (Zil,Zt),

g",';'°i,l'(zi,,z,):l-XI-[(zi l

I --

Zi~ ) al,' ( Zil -- Zt ) a'll .

(30)

(31 )

iti~

In fact. applying this operator we obtain

V i,,,,,; ,a,,, ] {n,,q}

r.{""'r", } (Zvq) ".-'N+~nvq(Zvq,Zq)----

.~"l~l, '"ll' ,%, ,al,, } ( ~{nt,q} Zl, ZI') 8~qrq,O.

(32)

So all "massive" amplitude properties can be deduced simply from tachyonic ones. This set of operators gives relations between N and N + ~ nvo vertex amplitudes, so they appear as a non-linear realization of a larger algebra as pointed out by Witten [23] in the case of 2D string theory. By using a diverse factorization of functions: .]E.{ailjl, ,alll, ,atjl, ,aft, } ,Tz{ailJt, 'aill' 'a/Jl, 'all' --rrrl' } , x .r'.{O,O,O,rrrl, v N (Zil,ZI,,) = a N I. Zit,Zjt,)J-l~ }(Zl, ZI') ,

(33) we obtain also a relation between higher spin "excitations" and tachyonic amplitudes with the same momentum:

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

358

G{r,,,r,,,} _ V[¢4i~'~'trl t,.7{rq,r,,q} t. . N{(n,,q)}(Zq ) -- {n,:q} (Zu,,) u N ~,vq, ~,q)

= ~7 (~,,q) {a~r' ,%e ,%, ..,, ) (Z~,~j..'~U%,N ~ ,%r,,%, .~,, -r~'rl' } ( Zir, Zh , )

X .Jg'N{0'0'0'rr'rr' } ( Zl, ZI' ) a ~ r q ,0 (34)

"~{ ailJr' 'ailr' 'arjr' 'air'--rr'rr'} ( Zt,.r ) G {Nrq } ( Zq ) = ~{% I

An important point in this approach is that we can build all amplitudes by means of the action of the A{% } (z,,,) set of differential operators on the tachyonic correlation functions. Moreover, all information on the vertex algebra is contained in the contact S functions. Thus we can remove completely the GVOs from our construction and make use only of .]-{~litJrt ,airlJ ,£1lJrt,an~} / differential operators a n d . Jv t Zir, zj,, ) functions to study any amplitudes.

4.

Ward identity for correlation functions

At this point we can study the whole set of amplitudes in order to recover their symmetries. To do this we note that if the sum of roots vanishes (mass shell condition) ordered amplitudes .A{rq'r"q}(zt~q, Zq) are constant or, more generally, are symmetric functions, so they satisfy an infinity of differential equations: ( I ~ q zqmq+nq - t~n't )j4{rq'r"q}(Zvq, Zq).~O. ~("")" " A{r"'r'",}( Zq) a{mq} ~ Zq) Zv'r' = nq !tgZq '~

(35)

In the language of CFT these relations correspond to the insertion of quasi-primary fields [ 17]. In fact C~{Uq} (

{mq}kZq) A{rq'r"q}(Zcq,Zq)

]F~ .qlq-'-nq ~Uq u{rq,r,,q} z .. = (; I J Z q nq!OZq q {(.,,q)} tZq) :) q

(36)

is the ordered amplitude for descendent fields relative to the SL(2,I~) subalgebra. To these ordered amplitude symmetries we associate Ward identities for generic GVOs amplitudes, which, defining the operators 1~_{ailh , ,a4r, ,at,jr' ,art, }, { n,,q}

{ .,, }, { m,,}

.~-'{%Jr' 'a4r' 'alJl' 'art' } .

(Zq) = ~{ .,,, }

.,3{ nq } (

t Zt, Zt' ) v{ mq} "Zq )

~-~{tl,tjrf ,£lirtl,(Jrjr! title} -- I ×o{,,q} " ( z t , zt, ) ,

(37)

can be written in the following form:

£{'rJr,,%r','~%,,arr'},{%) i,_ ~ = 0 {nq},{mq} (Zq) GU,j,,q} N((n,,q)}~Zq)

for {nq 4= 0}

(38)

The operators introduced above satisfy the same commutation relations of the free differential algebra O{m,,} tZq):

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

[~__{

aith,.a#t'.at'jt,.a,' }.{n,'q)

{nq},{.,q}

=

~"v,, ,",,,' ,%, ,a,,, }(Zl, {n,,q}

359

~_{ait)tt,aitt',aFjl,,allt},{nvq} ( ] ( Zq) , {.~}.{m~} ~Zq)

co {""}" " o {";} [ {mqi[,~q) , {mtq}(Zq)

gl')

(39)

.~,,J,, ,~v',%, ,",,' } -i ] ~{n,,q} (gl, Zl')

and the Jacobi identity. Note that the function.~{k,,~ ¢-'{a~°''} ,%l,,%, ,a., }-1 (zt, ze ) is a sum of Jastrow functions and is well defined everywhere except in the points zt = zt,. In these points, however, the singularity of the GVOs are regularized with normal ordering. Using the factorization given in Eq. (33) it is possible also to write a set of effective equations for the "excitations", which, introducing the operators ~__~{ OiljIt "~lillt,all Jl! ,all, -- r l.r l, }, { fl,,q}

{,,,,},{.,,,}

( Zq)

.T2.{O,O,O,rl.rlz} -- 1 jc{OilJl * 'ailll 'altj F 'Olll }'{nt'q } / \ ,.r-{O,O,O,rl'rF } , =" N (Zl, Zl') {nq},{mq} tZq)a"Al tZl, Z l ' ) ,

(40)

can be written as l~ {ail h, ,aill' ,al' j l, ,art,--rl'rl,},{n,,q} ( ,7 "~S ~ ailJl' ,aill' ,aljt, ,all'

{nq},{mq}

t'..q; {n,q}

--rl'rl'}(Z~,q) = 0

for {nq 4= 0}. (41)

The above equations are the analogues of Eq. (38) for the S functions and satisfy the same algebraic relations. The set of differential operators introduced in Eq. (38) can be considered as a generalization of Dunkl operators whose role is fundamental in the theory of integrable systems. In fact, from the property

S~,,,,j,, {°'} (zq) . ~ a;,j,, ,%,, ,~,j,, ,all' }--1 {%} ,a;,,,,%, ,,,l,, } ( z;, ze) a {m~} ~{.,,~ } (zt, ze) = 1-I Zq"+nq !.,(s~ai&"ail"aol"al"}({nvq} nq [ q

Zl..Zl')

(42)

°'}--O--~.~'5,,aitjl"aill"alh"aH'}-l'(zl,Zl') )';'" OZq {n,,q} '

using the identity S-~{ai,Jl, ,ai,t, ,alj,, ,all, } /

n,,~}

t Zt , zt, ) - -0 ,~-.{ailJ,, ,aiy ,ao:, ,all, } --1 ( Zl, Zl' ) OZq ~{ n,,,,}

-- __~{aiDtt'aitF'aul.''a#'}--l, " -- '.-'{&,q} tZl,Zt')

O.O_._,,5"~aitJl,,aill',ath,,~:ltt'}( C)Zq {n,,q} Zl, Z l ' ) ,

(43)

we can write a general differential operator in a simpler way: ~'{aqJl''aitl"al'jt''au'{nq},{mq}

}'{n"q} (zq ) = ( - 1 ) ~ q n q

mq+ nq I I -~/"{lq}Zqnq! 1"2"{ailh''aql''aejl''all'}'{n"q}"l'Zq))''~"q " q

(44) The generalized Dunkl operator is

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

360

{Iq}

(Zq)=-

{.qt.q}

'

c~ ~-~LlilJit.¢lill!,OtJit,~lllt

x -~zqS~.,,~ }

(Zl, Zlt)

(45)

} (zt. z~, ),

which in fact reduces to the usual Dunkl operator in the case of VOs: {O,O,O,au,} / {lq} • -

~ .r-{O,O,O,rt.rff}/ 0 ~{O,O,O,rt.rff} (Zq) =--J-lv ~Zl, Zl')~ZqJ-hl (Zl,ZI')

1~

3

~

= -- 3Z-~ +

alq

(46)

Zl - Zq "

Moreover, in this case we have N

~{l'/}?{0'0'0'a"}-l(Zq)G{Nrl}(Zi)= ((-~Zq -- rq .Q(') (Zq))Iffi

(Zi)) ,

(47)

where the singular product is understood as the limit z --+ Zq. So in this case we can identify the action of the Dunkl operator on the VOs amplitudes with the action of the operator ,9

--

CgZq

-- rq . Q(1)(Zq)

(48)

on the product of VOs, which is just a Miura transformation generating an explicit realization of W~ in terms of fields. In the general case of N GVOs, however, a Miura transformation cannot exist.

4.1.

Wo~ symmetries of amplitudes

In this section we describe some interesting subalgebras of the GVOs symmetries by using the correspondence between ordered amplitudes A{rq'r''}(Z,,~, Zq) and G{rq,r,,,,} U{(n,,,)}'( Zq) which allows us to single out the relevant symmetries.

It is obvious that the subalgebra of the ,~{nq} v{,,~} (zq) algebra spanned by the generators N

n = ~ - ~ z m + n on Wm ~ q OZ n q=l

(49)

q

satisfies a W~ algebra. The corresponding differential operators 1/V,", obtained by replacing 0 (m~} {'q} (Zq) with the operators of Eq. (40) are a realization of the same algebra. An important consequence of this is that all GVOs amplitudes are W~ invariant, as indeed the Wm ~ generators satisfy quantum commutation relations without central extensions, this is an obvious consequence of zero genus of the Riemann surface on which we take correlators, while anomalies arise on the genus one surface:

361

V Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

[ )/Vn,,, 1/~,,', ] = (n'm - n m ' ) 147~+"~-1 =2_~q

-t-...

l-lr. r .xmm'wn+nr-I

t'tl)n,~'

m+m' '

l=l

where q is a quantum deformation, which is q = 1 in this case, so

nt,m

C(1)n'"' =

[m' + n']t

__

l

and [a] b is the Pochhammer symbol. It should be possible to realize these generators by means of a particular combination of GVOs. In particular we compute the Virasoro (Witt) subalgebra generators []/vit ,1471,, ] = ( m -

m')VVl,+m , .

(51)

In the case of Virasoro algebra these vertices can be simply identified. In fact, in this case, it is well known that the tensor field T ( sc) is the generator of transformations that act on VOs in the following way:

+~(m+l)z

m Ur(z),

(52)

CLz

where the circuit C~,z includes the pole ~: = z. If we specialize these operators to the projective subalgebra we obtain wl_ 1~-~°'°'°'a''} (zt, zv ) = 0,

woo0 0J N

(Zl, Zl') =

l~{O00"all'}"tZl, Zl')

Wld'l~' '

=

(53)

~(~~alq) l~q

I ~q ( ~

1 000,

alq)Zq

tZl, Z l ' ) ,

(54)

~{O'O'O'all'} (Zl, ZI'),

(55)

~N

d-~

where ~ t # q alq = -r2q = -2hq. So the invariance is satisfied with a trivial comultiplica--

o-"{ailjt,,ailt, ,ath, ,an, } (

tion and we can use these relations to fix three variables in the o ( % }

zl, zv )

functions. In this case it is simple to realize these transformations by means of T(~) :

Jill' a{Nrq}(zq)= ~ i ~i'wi~m+2(Url(Zl)""T(~)urq(zq)"''ur^'(ZN)):0" q C(,zq

(56) In the general case additional terms are needed:

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

362

all,( m+z m+t Czt, zv ) . I .i ~ >.l Z l .- - Z l.' ) J ' f i Zl -- ZI'

.F{O,O,O,a.,}.

w~,, --N

.,-{0,0,0,a,,, },(Zl, Zl')

= Zau'H(Zt--e2k/("'+i)Zt,)U{NO'O'O'~"'}(Zl, Zl,). (57) l>F

k=l

In fact, as the vacuum state 10) is invariant only for projective transformations, in this case it is necessary to consider also the contribution of the action of T ( ( ) on the vacuum:

~i~l'G{Nrq}(Zq):Zi d~GT"i~m+2(uri(zl)'"T(~)urq(zq)'"UrN(ZN)) q C~,:q

+ C£,0 f

<:Z,:,

-r- i 2-~'ITi~m÷2(T(~)url(zl)"'UrN(ZN))=O"

(58)

The interpretation of this result is very simple: the Wo~ symmetry of GVOs in the quantum case is realized by taking into account the anomalous transformations of the vacuum, so the symmetry is restored also in non-critical dimensions.

4.2. Additional equations In the previous section it is shown that a Wo~ algebras exists for any GVOs correlator and the explicit realization can be given in terms of generalized Dunkl operators. This is only a subalgebra of the full symmetry algebra that can be realized; in this section we want to understand the role of the remaining differential operators. As recently pointed out [24], canonical quantization of two-dimensional identical particles give unusual interesting results. We consider GVO correlators as multiparticle form factors and the generalized Dunkl operators as one-particles operators. In this framework the W~ generators become the completely symmetric single-particle operators that give the observables of the N-particle system. We can also construct operators not SN invariant that must be related to the differential equations not in the W~ algebra. Two cases can arise: in the first the particles are distinguishable and these operators give observable results; in the second case the particles are identical and they cannot be observed to avoid the possibility of identifying the particle. In this framework the existence of this additional symmetry has a very simple physical interpretation in terms of breaking the Hilbert space of the particles in N! sectors that cannot be mixed by Hamiltonian evolution of the system. So all non-symmetric operators must be unobservable. This aspect is very interesting because it implies that many null

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

363

states for W~ representations must exist corresponding to these unobservable operators; this is just the case of quasi-finite representations that are well studied in Ref. [25]. Notice that degenerate representations of W-algebras arise also in the classification of hierarchies in the quantum Hall effect [26] without any explicit requirement of the generalized exclusion principle. In the following section we explore in more detail this aspect in the case of the CS model. These further equations relate amplitudes in which there are quasi-secondary fields to pure quasi-primary ones. Therefore, by means of these equations we reduce the vertex operators space to quasi-primary states only. In this way we remove not only the dependence on quasi-null states but also on all quasi-secondary as a signal of enhanced symmetry. This invariance has a simple interpretation in terms of the universal enveloping algebra of projective symmetry and does not depend on the full Virasoro constraints. As pointed out in [ 17] this is very important for the consistency of the GVO construction of Lorentzian algebras. Besides the considered linear symmetry algebra of GVOs which commutes with the N operator (number of GVOs), we remark that there is another set of symmetry obtained by applying the ~ operator of Section 3 which relates the product of GVOs with different values of N, see the end of Section 5.

5. Relationship with Sutherland model In the first part of this paper we have constructed the differential operators that generate the symmetries for all GVO amplitudes. In what follows we describe the connection with the CS one-dimensional integrable systems. The compactification of external space implies that periodic boundary conditions for the momentum of particles must be imposed, so the model that we describe is of the Sutherland form that describes a system of non-relativistic particles on a circle interacting with an inverse square potential. This model is completely integrable and gives eigenfunctions expressed in terms of Jack polynomials J ~ ) ( Z q ) that are indexed by the Young diagrams that can be interpreted as the distribution of the momentum of pseudo-particles (holes). The Young diagram is parameterized by the N numbers {t} = ( q . . . . . t N ) , tl >~ tz . . . >1 tN >~ 0 with total number of boxes denoted by It] = ~q~l tq. To make a reduction of GVO space we have to impose that the number of particles L--j

be a constant, so we consider only the set of G~"~r~'qt~(Zq) with a fixed value of N. t , Vq.j The correspondence with the CS model is obtained by the identification of the states and of the Hamiltonian, which can be done using the projective invariance to fix two states corresponding to the in-vacuum and out-excited state: {s,sj.rq,rL,q,r,ri}

.

.

N

{rq,rvq}

~((.j)(.,,q)(.,)}Nt Zq) = (s, s j ( n j ) I H U{(.,,q)} ( Zq)Ir, ri(ni) ) . q=l

(59)

364

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

Specializing these wave functions to the Sutherland model we consider the ground state (rq =

r, r~,,~= 0 V q) N

I/l{Nr'h} (Zq) =

N

(Nr + h I H Ur(zq)la) = H (zt - z t ' ) r 2 H Zqr,A • q=l

(60)

q=l

I>F

The wave function vanishes for zt - zt, ~ 0 if r 2 > 0, which indeed is the condition ensuring the absence of poles in the product of two VOs and assures the normalizability of the wave functions. It is very interesting to notice that we could consider also the case r 2 < 0 where bound states should exist. When expanding VOs in terms of symmetric Jack polynomials we obtain the wave functions of excited states [ 14] :

~-~(--1 "~[t[l(r2)~{,) [, ,Zq) j /

:Hur(zq)'l/~)=HZq

q:l

q=l

-,/2l{t},gr+A),

(61)

{t}

where (j;2) -I/2 is a normalization factor. All information on the symmetry of states are encoded in the ground-state wave functions. This allows us to understand the importance of GVOs in one-dimensional integrable models. GVOs can be factorized in a Jastrow-like function and a symmetric term that depends only on ordered products. Statistical properties are related only to ffUN{a,tjlt ,aid, ,atjff ,alff } ( Zit, Z.h, ) functions. By orthogonality properties of the Jack polynomials we obtain N

({t),gr+AlIIUr(zq)l~)=IX(zt-zz,) q=l

- 1 ) Irl "{t} (Zq)

q=l

l>l t

Jt (62)

The completeness of Jack polynomials in the space of c = l CFT at R = ~ implies that there exists an out-state for each state of the Sutherland model, so Woo symmetry projects the functions defined in R u onto the single-particle space. In this way a full set of wave functions can be obtained acting only on out-state (collective state). In this case the first two generators of Woo algebra are

wo' -- II(z,

-

-

I>l'

0 H(zt_zt,) q

W 2 = H(Z, l>l'

q

=~-""~Z' q Z2q q ~022..~ -

~

3

q~zq

q,l,lt 4:q

2N(N-I)

-

,

(63)

q - Zl')

(64)

r"

2 a +2r2(r2+l)Z _2r2~--~ Z20 t-~zj--Zl'~zff

-

r2 -

I>l'

l>l'

Zl -- Zl t

2

+r4

g

=Z

l>l" 02 Zq2--~Z2 H ( Z ,

-- z F ) r 2 Z

--r2

Zq + (Zl-- Zq)(Zq-- Zl')

ZlZl,

I>l'

r 2 ( r 2 + 1)

2

N(N-

(z17~,)2 l)

"

(65)

V. Marotta,A. Sciarrino/NuclearPhysicsB 476 (1996)351-373

365

The first generator corresponds, up to a constant, to the total momentum P:

P=~-~ZqoOzq-- ,

(66)

q

while the first two terms of the second generator are the differential operators whose eigenvectors are, according to Stanley's theorem [ 30], the Jack polynomials, while the fourth term is a constant. This suggests that there is a close relationship between this generator and the CS Hamiltonian, which in our notation is written as ( 0 )

ztzr ~-~ ( zl ~ zr ) 2 .

2

H = ~-~ -~I Zq -~zq

_ r2 ( r 2 1 )

q

l>F

(67)

Moreover, by construction on the ground state we have W'N

(Zq)=

WO2'I'{r'A} v'N (zq)

P--

= [2 H +

N(N-1) ~

N(N 2

~{Nr"l)(Zq)=p~z{Nr"l}(Zq),

(68)

-- 1)] ~{r'A}(Zq) =2E~l{Nr'h}(Zq),

(69)

where p = Nr • ,~ and E = ½ N ( r . ,~)2 are the eigenvalues of relative operators. In order to be able to actually identify the CS Hamiltonian we have to modify our approach introducing the symmetric hermitian exchange operators Ku,, so we can discuss the more general case in which the exchange operator appears in the CS Hamiltonian. In the following we give more general results than is possible to obtain by the VO construction if rt ¢ r t , . The CS wave function 1/"(z u) can be written in a factorized form as the product of the ground-state wave function g,{Nr"'~)(Zq) times the wave function of the collective CS Hamiltonian, i.e. for art, = r 2 a Jack polynomial ¢(Zq). If the particles are identical (indistinguishable) the wave function has to be invariant under the action of the operator Ku,, so we must impose, for any couple IIr

( l -- "Qll,gll,)~O'(Zq) -- ~t{N%'a} (Z.q) ( 1 - KiF)q~(Zq) -----0 ,

(70)

where rill, is the phase, computed in Section 2, produced by the action of Kit on ~t{Nr"'a}(Zq). The phase rhl, takes in account the fact that the statistics depends on the length of the rq roots. Let use remark that the invariance of the wave function should require identical particles, i.e. all rq equal or the introduction of mutual exclusion statistics [32], whose connection with the CS model has been discussed in Refs. [33,34]. The VO approach naturally leads to models with this type of statistics. We make use of the results in Section 4.2 that allow us to replace the differential operator of Eq. (46) by the Dunkl operator,

ozqC) ~

dq = : - - +

aql

- - ( 1 Zq

Z/

Kqt).

(71)

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

366

When aq! = r 2 the dq satisfy the relations of an affine Hecke algebra, but now we have

aqq, aq,l

[a,,,

de ] = ~

aqlalq,

(zq - zq, ) (z+, - zl) -

l #:q,q'

(z,, - zl) (Zq, - z,)

aqq'aql ) -- (Zq -- Zq')(Zq -- Zl) gqq,(gql - Kq,l) ,

(72)

[zq, Zq,] = 0 ,

(73)

[dq'zq']=(~qq' ( l + ~--2~aqlKql)

(74)

so the Woo algebra structure is now lost. In the order to see whether and how this symmetry can be recovered we note that the first two Woo generators are

W2,(d)=~--~+z,,

,,,+~

o

q

l>l'

m+2

W~'2,(d)=~Zq

zl "'+j - zF, '+1

~zq+Zau,

q

(1 - Ku,) ,

Zl -- ~l'

02

zm+2 O

OZ2q + 2 Z a n '

l

I>l'

(75)

+m+2 c?

~ - Zl' gl -- Zl'

Zff'+2 + Zfit+2 - Z all, ~Zt -- -Zt-~ ( 1 - Ku, ) l>l' a -

_m+2

lqUql'Zq --

~

q,l~l'#=q

( l -- Klq) (1 -- Kql, )

(76)

( Zl -- Zq ) ( Zq -- Zl' )

where we have used

2

__

dq = 0z 2 +

~

aql Zq--- zl

~zt +

+ ~zTzt (1 - Kql)

(77)

-- Z aql aqlaql, ( 1 -- Klq ) ( 1 -- Kql, ). ( Z q T Z l ) 2 (1 -- Kql) ~- Z (Zq-- Zl)(Zq-- ZF) I~q q,l+F4=q If all all, = r 2 they still close a We+ algebra on the states that we have used to realize them on which the exchange operators Kqq, becomes a c-number,

(d),W'nn;Cd)

~ ( Z q ) --

nT,W~,,

qbCzq) ,

(78)

where W,", are the W,~,(d) with Kit, = 1. In particular W2o(d) becomes the effective CS Hamiltonian when it is applied to the symmetric functions. If we use the freedom o f adding any antisymmetric term that cannot give observable results, we restore the Woo structure that is realized only modulo the transformations generated by the operators o f Section 4.2. If we define the operator Lq as

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

L

367

,r-(O,O,O,bnt}," _ ",.I .'r'(O,O,O,bu'}-1 q = "/-lq 1,Zit, Z.jt, )UqJ-N (Zit, Zjr, )

= 0 OZq

~ ~q

1 Zq

Zl

[ bql

aql ( 1 -- rI (bql)

Kql) ]

the Lq satisfy the same algebra as the Dunkl operator we have

(79)

dq with

Kn, ~ ~u, (bu,) Kll'. Now

"1/~)1, = f'{O'O'O'ba'}" W 1 t A ~ "ic{O'O'O'bu'}- 1 N t Z i l ' Z J F ) " m ~'~1~ N (Zil'ZJ, , ) =~

Zql+l t~ZqO q

Z z/m+lzl -- Zff zlm+l [bit' -- all,(1 -- 7](bll,)Kll,) ] I>F

(80) (81)

and )/~n2

"r'{O,O'O'bltt} . IM2 { d'~ .7£'{O,O,O,btt'}-1 = "]'lq [,Zit, Zj,, ) ,,n,,,,'~]..," N (Zi,, Zj,, )

_m+2 ,9 = ~,'-'_.,+2 02 ~q Lq C)Zq2 + ~ l>F +

2 Z ( bu, - all, ) ~1 l>l'

(82)

_m+2 O

~l Zl' Zl -- Zl'

z[ "+2 + zt",'+2 ~ t --- z t - ~ [ (btt, - art,) b t t , - btt, (an, - 1 ) - art, ( 1 - ~7( btt,) Ktt,) ]

Z q,l+Fv~q

zm+2 q )< [blq(bql' - aql,) q- bql,(blq - alq) (Zl -- Zq)(Zq -- Zl')

--blqaql, -- bql, alq -- alqaql,( 1 - rl( bql) glq ) ( 1 - 71( bql, ) gql, ) ] .

(83)

When ate = a and bu, = b it is possible to prove along the lines of the proof of Ref. [29] that the commutators of any symmetrized operators do not contain any extra terms depending on KLr, i.e. the abstract algebra does not depend on the statistics of the particles. Moreover, by a translation of the qth Dunkl operator by Zq, we can introduce an harmonic oscillator potential in the Hamiltonian. Notice that if we consider the case bu, = an, the generator Wo2 becomes the multispecies Hamiltonian of the CS model with a three-body potential plus terms that contain the projectors ( 1 - r/(bqt) Klq) ( 1 - ~7( bql' ) Kql' ) vanishing on the relative wave functions. Now we are in a position to discuss the CS model with spin. We introduce an internal space (spin space of dimension 2s + 1) so that the true wave function is the tensor product of the q~(Zq) with a spin wave function X. We can simulate the action of the spin by introducing a (2s + l)-dimension "vacuum" such that Ptt' Ix) = r/u, ( s ) I X ) ,

(84)

where now we write the exchange operator as the product of an operator Ktt, acting on the labels ll' of the excited states and the operator Ptt, acting on the spin vacuum. Now the equation that the eigenstates of the collective Hamiltonian have to satisfy becomes

(P., - K . , ) ¢ ' ( z q ) I X )

= 0,

(85)

368

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

where ~b'(zq) can be related to the symmetric function ~(Zq) by a Jastrow-like function that gives the same phases of the spin vacuum: (86)

O~' ( Zq) = ~h( Zq)5~'~ ( zt, zl, ) . On this space the Dunkl operator is written as c9 dq - c~zq

~

(87)

aq__l (Pq/ - Kql) zq - zl

The above formula is a particular case (/z = p) of the general case considered in [31 ]. The general case can be reproduced in our approach by a translation of any root r q ( r 2 = p) by a quantity tq such that/z = (rq + tq) 2 and by defining the generator Lq with the Jastrow function which depends only o n / . z - v = tq(tq-q-rq), i.e. the generators of the differential algebra are written with a factor which does not take the statistics (case bll, ~ art,) into account completely. It is interesting to look more in detail at this interpretation of the statistics in terms of the shift of the roots. The free case (no interaction) is obtained when r 2 = 0, i.e. when r is a light-like vector. So the lattice A can be considered in a Minkowskian space. If we take h in the dual lattice: a=a+n~K++n

K_,

n~Z+,

(88)

where K+, K_ are the two light-like vectors such that K+ • K_ = 1 and a is a vector orthogonal to K+ and K_. The interaction with statistics u can be introduced by translating the light-like roots (let us say rq = r = nK+ for any q) by t a vector which, requiring the invariance of the total momentum (~-~wr q . /~), c a n be written as t. a = 0.

(89)

The introduction of the spin requires the extension of the algebra of the observables to a charged W-algebra, Notice that the ground-state function has an expression very similar to the measure of the Selberg correlation integrals, whose relevance for the CS models has been discussed in [27]. Many relations between the CS model and other topics, for instance KdV equations, can simply be understood in this framework. It is an interesting point to study the action of the operators of Section 3, which give the amplitudes of GVOs, in the case of the CS model. We restrict the discussion to the case of VOs and apply the operator to obtain the CS ground state of N - 1 particles from the N-particle one: //t'[r¢'t'A}~ N--1

, =~{rN.r,,~. ~Zq)

= where

lim

ZN~ZN I

I}

, ,{r,,,h}/

(ZN, ZN-1)qIN

,

[Sq)

(ZN -- ZlV-1) -~N~N ~{Nr'~'a}(Zq) ,

(90)

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373 rlq=rq

369

V q= l ..... N-2,

!

rN_ l = rN + r N - l •

(91)

We hope that this formulation can be applied to second quantization of the CS model, where the ~7 operator can be identified with the annihilation operator.

6. Conclusions

We have shown that by a similarity transformation ("dressing") of the free differential operators generating the W~ we still preserve the algebraic structure and that with a particular choice of the "dress" we can identify the CS Hamiltonian with the generator The choice of "dressing" depends on two aspects. The first is related to the existence of non-unitary similarity transformations that give the W~ algebra for any GVO that is discussed in Sections 2 and 3. The second gives a sector-independent basis by using the permutation operator Kn, that takes in account the multiply connected configuration space for the CS model. In terms of vertex algebra this is just the duality invariance property of amplitudes. The dressing function is given by the correlation function of the product of N VOs, computed out of mass shell between an arbitrary in-state and an out-state fixed by the choice of the in-state and by the value of the roots. The W algebra is thus related with the Ward identities for VO amplitudes, identities always satisfied also by correlation functions for the product of generic GVOs. Although we have not yet explicitly computed the correlation function for product of any GVO, we have presented here the whole formalism to emphasize the role, we believe fundamental, of the vertex algebra. In the case of the product of N VOs with the same root, corresponding to the CS model, we find all the results of [28], but our formalism allows us to get more general equations with a potential whose coupling constant is not necessarily equal for all the particles. The system of N VO's appears as a system of N particles, of equal mass, with internal quantum numbers specified by the roots r i. The particles are identical if their roots are equal. The observables of a system of identical particles must be symmetric operators, so operators not belonging to the diagonal W~ must be not observable. For instance we have ( £ i are the operators given by Eq. (46)) ( zl El ( zl ) - Z2 Ej ( Z2)~bN( Zq) ) = ( rl -- r2) • A ~ N ( Zq)

(92)

and the r.h.s, vanishes if rl = r2. More generally, as the momenta defined on the circle have to be quantized, the product rq. A must be an integer, which implies that A has to belong to the dual lattice of the lattice A of the roots. So translating A by an element of the lattice does not change the value of the relative momenta (up to an integer), so the translation of a can be interpreted as a Galilean boost of the system. So the algebra of observables is the whole algebra of differential operators. One may expect that a classification of all the possible vertex operators may give a classification of

V Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

370

all the possible integrable models. Let us emphasize once more that we have shown that the invariance for a W~ algebra, which in the literature has been established only for particular models, is a general feature, a consequence of the algebra of differential operators for correlation functions. Moreover, the connection between integrable 14-1 models as CS and vertex algebra is given by the intrinsic structure of GVOs that are an explicit realization of the vertex algebra. We conjecture that "integrability conditions" as the Yang-Baxter equation can be deduced by properties of vertex algebra as it happens in the simplest case of the CS model. A very interesting question to understand is if there are physical models corresponding to correlation functions of the product of arbitrary GVOs. Independently of any physical interpretation or interest it is natural to raise the question if such models are integrable. We believe that the answer is negative, at least if integrability is meant in the usual sense. In fact integrability requires that r 2 in Eq. (56) be non-negative. This condition is not guaranteed either for the roots or for the weight for a generic (not affine) Kac-Moody algebra. It is interesting to discuss the connections and the differences between the structure of the integrable non-relativistic models and the structure of string theory, where the Lorentzian algebras play an essential role. For instance many of the above considerations still hold replacing the term particle with the term string. However, a thorough discussion of this topic requires further analysis and it will be eventually carried out elsewhere. Finally let us remark that Woo algebras are related to area preserving diffeomorphisms and these structures arise as a property of vertex algebra independently of any physical models.

Appendix A To give an explicit expression of amplitudes and GVOs products we need to compute any arbitrary S~{k,.~ a~ljl'} ,%t, ,al~I' ,all, } ( zt, zt')

function. This can be done in the following way.

By definition S~t'lilJlt ,tlillt ,at)it ,all; } ,.

~,,t}

.~{aitJll ,°lilt ,at)it ,till!

tzt, zt,) = {z,.,,--.z,,} lim ~{k,q }

} (zt, zt,)

(A.1)

and

S{k,,q} { a,%t't l,a,)j l,, ,a,,, , }. ~ Zl, Zl t ).

. .Tc{¢',tJl, ,atlt' ,a~j,. ,,,., }.

= ( o { k " q } ( Zuq)

N

[. Zit' ZJt '

)) .

(A.2)

Factorizing .7" in terms of the q-variable .~.N{att& ,ail l, ,alYt, ,a u' }

( zit, zj,, ) =

1-I

7c{ ail)l, ,airt, ,at& ,all, } (

- n,

zil, zjl, )

l>l':l,l ~ 4=q

( Z~,, Zt,, ) 1-I ",.t~.{ qt'avqjit ,avqt' ,aqj It .aqt, } ( Z~,,,, Z•j,, ) ' X ]--t I t " ~-{~t,,,,,atiq,,t,,q,~t,} lq l>q

l'
(A.3)

371

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

(we indicate with k~q and kvq r the number of derivatives that act, respectively, on terms and Uql') these numbers must satisfy the identity

Ytq

~-'~ k~q-]- ~--~ klt[q = kVq I>q l'
(A,4)

~ {q, Uq}.

•

.

{ailJlt ,aillt ,aljff ,all I }

At this point it is possible to give a factorlzatmn of S{k,,~}

(zl, z,, ) in terms

of two-point functions: .~ { OilJlt ,ailff ,alt Jlt ,allt }

`5 { ailj t, ,ai d ' ,atj I, ,air, } { kt,q } ( Zl, Zl' ) =

{g,>q~ +E,,. <=~,~}

( Zl, Zl' ) ,

,>,'

(A.5) where the indices l, I' include also the renamed q, and, by means of the substitutions / rl ~ r, kl, ~ ki V l,1 I, (A.6) rij --+ ri, r j~, ~

S.j,

rl, ~ s,

k JiI.I ~ kj

V l, l'

(A.7)

a general ,5 is aO0} [,Zi,~j, ~ ki,k, `5{aij,aio,aoj,

= (c){ti'k'}(Zi, ej)~{aq'aio'ao''aoO}(Zi,:~j) ) .

(a.8)

In this way it is necessary to compute only this class of functions. By the factorization of 5r{~q'a~°'a°j'a°°} (Zi,/j j) it is possible give an explicit expression for its derivatives: 03ki(zi)c)kj(~j).~{aq'ai°'a°j'a°°}(Zi,~j)

=

{p~,vj} ~

~ {~iPi=k,,~jpj=kj

(A,9)

} {li,l,}

× [ I [o',(z,)o',(¢~)( z,- ej) °,,] [on,-,,
which can be written in a more compact form introducing the coefficients

X {a~''a~°'a°j}

{(l,+lj).(p,-ll),(pj-lj)}

= ( - 1 ) ~jlj

V[ ( a i--o li) ( a i lij )nt-( lja ° J )pj

----.• \Pi t,j

-- lj

'(a'lO)

s ki,kj{ a q,aoo}, a[\ Zi,i o~j ,) a= j,~{aij o j ,ai0,a0j,aoo} ( Zi, ~:j ) {v,,pj} ×

Z

{Z,p,=k,,~, ~,_-k,} {,,,,,}

,¥ { aij,alo,aoj } { ( li+lj ),(pi-li),(pj-lj ) } ( Zi -- ~j)li+lJ( zi -- ~)li( z -- ~j)lj

(A.11 To give a final formula for $~ak'if~"%"'%' ,a,,} (Zt, Zt' ) it is necessary to take the limits

zi -~ z and ~j ~ ~ for all i,j:

V. Marotta, A. Sciarrino/Nuclear Physics B 476 (1996) 351-373

372

~ aij ,aio,aoj ,aoo} ( /: ,~ i,kj t Z i , g j , -~ l i m

.~ { aij,aio,aoj ,aoo}

lim ~ki,kj

(Zi,(j)

=(A{ki'kj}(zi,~j).~{ao'ai°'a°j'a°°}(Zi,~j)

)

x{r,s,ri,sj} {(ki),(kj) }

(z - ~)-r"+~'

(A.12)

k,+~j ~j '

where {Pi,pj}

x{r;'ri'~ }

)( { aij,aio,ao) ) { ( li4-lj ),(pi--li ),(pj--l j ) } "

(A.13)

Replacing this formula in Eq. (A.4) with the obvious renaming of indices ri ---+ rit, s j --+ r h, ,

r ~

rt,

I'

ki ---+ k ~

s - ~ rl, ,

(A.14)

V l, 1t

k j --~ k (.lit

(A.15)

V l, l ~

we obtain the final formula --ds) ait& ,airt, ,at& ,art, } ( Zl , Zl' ) =

Z

{~,.q }

{ ~,>q k(,q • ~,t
×

17"

X

11 z>r ( zt -

zt, )

{rl,rlt ,ri I ,rj F } l' {(k,t ),(k~¢ )}It' -- r l "r t' + ~'~ i, ~ii; + ~-]~j l' k t.& "

[]

(A.16)

References [I] [21 [3] 14] 15] [6] [7] [8] [9] [ 10] 111 ] 112] [ 13] 114] [ 15] 116] [ 17] [ 18]

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