principal vertex operator representations for toroidal lie algebras

Principal vertex operator representations for toroidal Lie algebrasYuly Billig Citation: Journal of Mathematical Physics 39, 3844 (1998); doi: 10.1063/1.532472 View online: http://dx.doi.org/10.1063/1.532472 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/39/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A Vertex Algebra Attached to the Flag Manifold and Lie Algebra Cohomology AIP Conf. Proc. 1243, 151 (2010); 10.1063/1.3460161 Indecomposable representations of the Diamond Lie algebra J. Math. Phys. 51, 033515 (2010); 10.1063/1.3316063 Finite growth representations of infinite Lie conformal algebras J. Math. Phys. 44, 754 (2003); 10.1063/1.1534890 Classical Lie subalgebras of the Lie algebra of matrix differential operators on the circle J. Math. Phys. 42, 3735 (2001); 10.1063/1.1380252 Indecomposable representations of the nonlinear Lie algebras J. Math. Phys. 41, 7839 (2000); 10.1063/1.1288246

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Principal vertex operator representations for toroidal Liealgebras

Yuly Billiga)

Department of Mathematics and Statistics, University of New Brunswick,Fredericton, New Brunswick E3B 5A3, Canada

~Received 30 June 1997; accepted for publication 30 March 1998!

We introduce the principal vertex operator representations for the toroidal Liealgebras generalizing the construction for the affine Kac–Moody algebras. We alsorepresent the derivations of the toroidal algebras and introduce analogs of theSugawara operators. ©1998 American Institute of Physics.@S0022-2488~98!02407-4#

I. INTRODUCTION

Vertex operators discovered by physicists in string theory have turned out to be importantobjects in mathematics. One can use vertex operators to construct various realizations of theirreducible highest weight representations for affine Kac–Moody algebras. Two of these, theprincipal and homogeneous realizations, are of particular interest. The principal vertex operatorconstruction for the affine algebraA1

(1) allows one to construct soliton solutions of theKorteweg–de Vries~KdV! hierarchy of partial differential equations. On the other hand, thehomogeneous realization is linked to the fundamental nonlinear Schro¨dinger equation.1

Eswara Rao and Moody2 studied the homogeneous vertex operator construction for toroidalLie algebras. The present paper is devoted to the principal realization. Here we construct theprincipal vertex operator representation for the toroidal Lie algebrag which is a universal centralextension ofg5 g^ C@ t0

6 ,...,tn6#, in which g is a simply laced simple finite-dimensional Lie

algebra overC. This generalizes the principal vertex operator realization of the basic representa-tions of affine Lie algebras.3,4

We add the Lie algebraD of vector fields on a torus to the toroidal Lie algebrag to form alarger algebrag. This is necessary in order to have a sufficiently large principal Heisenbergsubalgebra. To construct a representation ofg we consider the standard moduleF for the principalHeisenberg subalgebra and then extend the action to all ofg by means of the vertex operators. ThemoduleF is irreducible as a module overg, and it is reducible overg. It will be interesting to seehow this representation fits in the framework of the Verma modules over the toroidal Liealgebras.5

In a subsequent article6 we construct an extension of the KdV hierarchy using the represen-tations introduced in the present paper.

The vertex operator representations for the toroidal Lie algebras constructed here may also beuseful for quantum field theories in space–time of more than two dimensions.7,8

This paper is organized as follows. In Sec. II we recall the construction of the toroidal Liealgebra and in Sec. III we sketch the principal vertex operator realization of the basic representa-tions of affine Lie algebras. In Sec. IV we present the principal Heisenberg subalgebra ofg anddefine its standard module. We also introduce there vertex operators that will allow us to extendthis representation tog. At the end of Sec. IV we state the main theorem of the paper. The proofof the main theorem occupies Sec. V. In Sec. VI we construct the analogs of the Sugawaraoperators.

II. DEFINITIONS AND NOTATIONS

Let g be a simple finite-dimensional complex Lie algebra of typeAl , D l , or El ~i.e., simplylaced!. The Killing form ~•u•! is nondegenerate on the Cartan subalgebrah of g and induces themap n: h→ h* . Let D be the root system ofg. Since g is simply laced then (aua)52 for allnonzero rootsaPD.

a!Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 39, NUMBER 7 JULY 1998

38440022-2488/98/39(7)/3844/21/$15.00 © 1998 American Institute of Physics


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To construct a toroidal algebra let us consider a tensor product ofg with the algebra ofLaurent polynomials inn11 variable:

g5 g^ C@ t06 ,t1

6 ,...,tn6#.

We define the toroidal Lie algebra corresponding tog as the universal central extension ofg. Theexplicit construction of this extension9,10 can be given as follows. LetK be an(n11)-dimensional space with the basis$K0 ,K1 ,...,Kn%. Consider a derivationdp5tp(d/dtp) ofthe algebraC@ t0

6 ,t16 ,...,tn

6# which can be naturally extended on the tensor product

K 5K ^ C@ t06 ,t1

6 ,...,tn6#,

The space of the universal central extension ofg is

K 5 K /d K ,

where

d K 5H (p50

n

Kp^ dp~ f !U f PC@ t06 ,t1

6 ,...,tn6#J , K .

We will denote the image ofKp^ t0r 0tr in K by t0

r 0trKp , where r5(r 1 ,...,r n) and tr

5t1r 1...tn

r n. Note thatK has the defining relations

r 0t0r 0trK01r 1t0

r 0trK11¯1r nt0r 0trKn50.

The toroidal algebra is the Lie algebra

g5 g^ C@ t06 ,t1

6 ,...,tn6# % K

with the bracket

@g1^ f 1~ t0 ...tn!,g2^ f 2~ t0 ...tn!#5@g1 ,g2# ^ ~ f 1f 2!1~g1ug2! (p50

n

dp~ f 1! f 2Kp ~2.1!

and

@ g,K #50. ~2.2!

Finally, we shall add certain derivations tog. Specifically, letD be the Lie algebra of deri-vations ofC@ t0

6 ,t16 ,...,tn

6#:

D5H (p50

n

f p~ t0 ,...,tn!dpU f 0 ,...,f nPC@ t06 ,t1

6 ,...,tn6#J .

It is a general fact, that a derivation acting on a Lie algebra can be lifted in a unique way toa derivation of the universal central extension of this Lie algebra.11 Thus the natural action ofDon g,

f 1~ t0 ,...,tn!dp~g^ f 2~ t0 ,...,tn!!5g^ f 1dp~ f 2!, ~2.3!

has a unique extension tog. We shall denote the lift off (t0 ,...,tn)dp by f (t0 ,...,tn)Dp . Its actionon the subspaceg is unchanged, while the action onK is given by2

f 1Da~ f 2Kb!5 f 1da~ f 2!Kb1dab(p50

n

f 2dp~ f 1!Kp . ~2.4!

Consider a subalgebraD1 in D :

3845J. Math. Phys., Vol. 39, No. 7, July 1998 Yuly Billig


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D15H (p51

n

f p~ t0 ,...,tn!DpU f 1 ,...,f nPC@ t06 ,t1

6 ,...,tn6#J .

We will be working with the algebrag which is a deformation of the semidirect product ofg withD1 :

g5 g^ C@ t06 ,t1

6 ,...,tn6# % K % D1 .

Here multiplication ing is given by~2.1! and ~2.2!, the action ofD1 on g is given by~2.3! andon K by ~2.4!. The Lie bracket inD1 is the following:

@ t0r 0trDa ,t0

m0tmDb#5mat0r 01m0tr1mDb2r bt0

r 01m0tr1mDa2mar bH (p50

n

r pt0r 01m0tr1mKpJ .

The last term in this formula is a 2-cocycle with values in a nontrivialD-moduleK . This abelianextension of the Lie algebra of vector fields on a torus is a generalization of the Virasoro algebraand was introduced by Rao and Moody.2

III. PRINCIPAL REALIZATION OF THE BASIC REPRESENTATION OF AFFINE LIEALGEBRA

When n50 the algebrag constructed above yields the derived affine Kac–Moody algebra.We setn50 throughout this section. In this caseK is one-dimensional and is spanned byK0 ,while D1 is trivial, so

g5 g^ C@ t0 ,t021# % CK0 .

Lepowsky and Wilson3 constructed the principal realization of the basic representation ofaffine Kac–Moody algebraA1

(1), while the construction for a general affine algebra was given byKac et al.4 Let us recall this construction~see also Ref. 1 for details!. Its starting point is theprincipal Heisenberg subalgebra ing.

Let $a1 ,...,a l % be simple roots andu be the highest positive root inD. We define the heightfor a rootb5( i 51

l kia i as

ht~b!5(i 51

l

ki .

Note thatht(u)5h21, whereh is a Coxeter number ofg.12

Consider the principle gradation ofg defined by

deg~ ga1^ 1!5...5deg~ ga l ^ 1!5deg~ g2u^ t0!51,

deg~ g2a1^ 1!5...5deg~ g2a l ^ 1!5deg~ gu^ t0

21!521.

We choose nonzero root vectorse0P g2u^ t0 ,e1P ga1^ 1,...,el P ga l and form the elemente

5( i 50l eiP g. Sincee is of degree 1 then its centralizers in g is homogeneous with respect to the

gradation.Next, consider a projection

p: g^ C@ t0 ,t021#→ g by t0°1.

Note that deg(g2u^ t0)51 and deg(g2u

^ 1)5ht(2u)52h11, while p( g2u^ t0)5p( g2u

^ 1), and thus the principalZ-gradation ofg induces theZh-gradation ofg:

g5 (j PZh

gj .

3846 J. Math. Phys., Vol. 39, No. 7, July 1998 Yuly Billig


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The elementp(e) is regular ing, hence its centralizers5p~s! in g is a Cartan subalgebra,which implies that boths and s are abelian.

The preimages5 s% CK0 of s in g is an infinite-dimensional Heisenberg algebra which iscalled the principal Heisenberg subalgebra ofg. This subalgebra is the cornerstone of the con-struction of the vertex operator representation ofg. The scheme of this construction is the follow-ing: one starts with the standard irreducible representation ofs and then magically one can lift thisrepresentation to the whole ofg. Moreover the action ofg is prescribed and given by vertexoperators. Finally one has only to check in one way or another that this construction works.

The irreducible representation ofg constructed in this way is of the highest weightL0 , wherethe linear functionalL0 is given ~we consider the simply laced case! by

L0~n21~a i ! ^ t00!50, L0~K0!51.

This highest weight representation is called the basic representation ofg.The Cartan subalgebras is homogeneous with respect to the principleZh-gradation:

s5 (j PZh

sj .

Since (gi ugj )50 unlessi 1 j 50(modh) and~•u•! is nondegenerate ons then one can choosea basis ins: $T1 ,T2 ,...,Tl % such thatTiP gmi

, where 1<m1<m2<¯<ml ,h and

~Ti uTl 112 j !5hd i j .

The numbers$m1 ,...,ml % are called the exponents ofg.12 Note thatsù h5(0), andtherefore zerois not an exponent. Also,ml 112 i5h2mi .

It turns out that it is more convenient to work with a slightly different realization ofg basedon theZh-gradation ofg. Consider

gs5(j PZ

gj ^ sj% CK0 ,

with the Lie bracket

@g1^ si ,g2^ sj #5@g1 ,g2# ^ si 1 j1i

h~g1ug2!d i ,2 jK0 ,

@K0 ,gs#50.

The proof of the following lemma is straightforward, so we omit it.Lemma 1: The Lie algebrasg and gs are isomorphic and the isomorphism

c: g^ C@ t0 ,t021# % CK0→(

j PZgj ^ sj

% CK0

is given by

c~ea ^ t i !5ea ^ sht~a!1 ih,

c~n21~a! ^ t i !5n21~a! ^ sih1d i ,0

ht~a!

hK0 ,



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c~K0!5K0 .

We shall identifyg andgs using this isomorphism.In order to describe a basis in the principal Heisenberg subalgebrac~s! we need to introduce

the sequence$bi% i PZ such thatbi 1 j l 5mi1 jh for j PZ, i 51,...,l . Thenc~s! is spanned by$Ti

^ sbi,K0%, i PZ. Note thatb12 i52bi .Let Ds be the root system ofg with respect to the Cartan subalgebras. For aPDs choose a

root element

Aa5 (j PZh

Aja .

Sincep(e) is a regular element of the Cartan subalgebra then@p(e),Aa#5laAa with laÞ0.HenceAj

a5(adp(e)/la) jA0a and all the componentsAj

a are nonzero. We let the indicesi , j in Ti

andAja run overZ by settingTi 1

5Ti 2if i 1[ i 2(mod l ) andAj 1

a 5Aj 2

a if j 1[ j 2(mod h).

Define constantsl ia5a(Ti). Then@Ti ,Aa#5l i

aAa and @Ti ,Aja#5l i

aAj 1mi

a .

Consider an automorphisms of g such that

s5z j Id on gj ,

wherezPC is a primitiveh-root of 1. For a root elementAa5( j 51h Aj

a we have

@Ti ,s~Aa!#5s~@s21~Ti !,Aa#!5s~@z2miTi ,Aa#!5z2mil i

as~Aa!.

Thuss(Aa)5( j 51h z jAj

a is also a root element ands induces an automorphism ofDs . Since allAj

a are nonzero then the length of each orbit ofs in Ds is h.We may choosel rootsb1 ,...,b l PDs such thatA0

b1,...,A0b l spang0 . Note that if two root

elements belong to the same orbit under the action ofs then their zero components are propor-tional. Therefore every orbit ofs contains exactly one ofb1 ,...,b l . Consequently, we maychoose$s j (Ab i)%, i 51,...,l , j 51,...,h as our family of the root elements. The set$Aj

b i ^ sj ,Tj

^ sbj ,K0%, i 51,...,l , j PZ forms a basis ofgs .Remark:The properties of the automorphisms were studied by Kostant.12 He proved, in

particular, thats acts on the root systemDs as a Coxeter transformation. Remarkably, the complexcoefficientsl i

a could be interpreted as the orthogonal projections of the roots on a real planeinvariant under the Coxeter transformation. These projections were introduced by Coxeter in orderto visualize the regular polytopes of higher dimensions. In Chapters 12 and 13 of Ref. 13 and inSection 12.5 of Ref. 14 the case of the root system of typeH4 is treated, which also gives theanswer forE8 .

Now we shall construct the standard representation (w,F) of s. The space of this representa-tion ~called the Fock space! is the polynomial algebra in the infinitely many variables:

F5C@x1 ,x2 ,x3 ,...#.

For i>1, T12 i ^ s2bi is represented by a multiplication operator:

w~T12 i ^ s2bi !5bixi ,

Ti ^ sbi is represented by a differentiation operator:

w~Ti ^ sbi !5]

]xi

andw(K0)5Id.Indeed, the only relation to be checked is

[w~Ti ^ sbi !,w~T12 j ^ sb12 j !] 51

h~Ti uT12 j !bidbibj

w~K0!.



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But as it can be easily seen, both expressions are equal tobid i j Id.One can lift this representation ofs to the wholeg using vertex operators. We consider the

spaceg@@z,z21## of formal Laurent series in a variablez with coefficients ing and the spaceEnd(F)@@z,z21## of series with coefficients in End(F) ~see Chapter 2 in Ref. 15 for details!. Theadjoint action ofg on g@@z,z21## is well-defined.

Every representation w:g→End(F) defines a homomorphism w:g@@z,z21##→End(F)@@z,z21##.

The following theorem shows how to lift the standard representation of the principal Heisen-berg subalgebra tog:

Theorem 2: ~Ref. 1! There exists a representation

w:g→End~C@x1 ,x2 ,...# !

such that for iPN

w~T12 i ^ s2bi !5bixi , w~Ti ^ sbi !5]

]xi,

w~K0!5Id, (j PZ

w~Aja

^ sj !z2 j5Aa~z!,

where

Aa~z!5L0~A0a

^ s0!expS (i 51

`

l iazbixi D expS 2(

i 51

`

l12 ia z2bi

bi

]

]xiD .

SinceAja

^ sj together withTi ^ sbi andK0 spang then the above formulas completely deter-mine this representation.

We will use extensively the following Laurent series:

d~z!5(j PZ

zj and Dd~z!5(j PZ

jzj .

The first of these is the formal analog of the delta function.For an elementB5( j PZh

BjP g denote by B(z) the formal series( j PZ w(Bj ^ sj )z2 j

PEnd(F)@@z,z21##.Corollary 3: Let A5( j 51

h Aj , B5( j 51h BjP g and let Cj5@Aj ,B#P g. Then

@A~z1!,B~z2!#5(j 51

h S Cj~z2!1j

h~Aj uB2 j ! D S z2

z1D j

dS S z2

z1D hD1(

j 51

h

~Aj uB2 j !S z2

z1D j

Dd S S z2

z1D hD .

Proof of the Corollary:

@A~z1!,B~z2!#5(j PZ

(i PZ

@w~Aj ^ sj !,w~Bi ^ si !#z12 j z2

2 i

5(j PZ

(i PZ

H w~@Aj ,Bi # ^ sj 1 i !11

h~Aj uBi ! j d j ,2 iw~K0!J z1

2 j z22 i

5(j PZ

(k5 i 1 j PZ

w~@Aj ,Bk2 j # ^ sk!z12 j z2

j 2k11

h (j PZ

~Aj uB2 j ! j S z2

z1D j

5(j PZ

Cj~z2!S z2

z1D j

11

h (j 151

h

(j 2PZ

j 5 j 11h j2

~Aj 1uB2 j 1

!~ j 11h j2!S z2

z1D j 11h j2



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5 (j 151

h S Cj 1~z2!1j 1

h~Aj 1

uB2 j 1! D S z2

z1D j 1

dS S z2

z1D hD

1 (j 151

h

~Aj 1uB2 j 1

!S z2

z1D j 1

DdS S z2

z1D hD .

IV. CONSTRUCTION OF THE VERTEX OPERATOR REPRESENTATION FOR THETOROIDAL LIE ALGEBRA

In this section we construct a vertex operator representation for the toroidal Lie algebra thatgeneralizes the principal realization of the basic representation of affine Lie algebra. Again it willbe convenient to replaceg with

gs5(j PZ

gj ^ sjC@ t16 ,...,tn

6# % K % D1 ,

with the Lie bracket

@g1^ f 1~s,t!,g2^ f 2~s,t!#5@g1 ,g2# ^ ~ f 1f 2!1~g1ug2!H 1

h S sd

dsf 1D f 2K01 (

p51

n

dp~ f 1! f 2KpJ ,

@ f 1~s,t!Dp ,g^ f 2~s,t!#5g^ f 1~dpf 2!

while the multiplication inD1 and its action onK are the same as ing. Here and below we makeidentificationsshr0trKa5t0

r 0trKa andshr0trDb5t0r 0trDb , a50,1,...,n, b51,...,n.

The following lemma is an immediate generalization of Lemma 1:Lemma 4: The Lie algebrasg and gs are isomorphic and the isomorphism is given by

c~ea ^ t0r 0tr !5ea ^ sht~a!1hr0tr,

c~n21~a! ^ t0r 0tr !5n21~a! ^ shr0tr1

ht~a!

hshr0trK0 ,

c5Id on K % D1 .

We shall identifyg andgs using this isomorphism.The subalgebras with the basis$Ti ^ sbi,sihDp ,sihKp ,K0%, i PZ, p51,...,n is the principal

~degenerate! Heisenberg subalgebra ofg.Indeed,K0 is its central element and the multiplication ins is given by

@Ti ^ sbi,T12 j ^ s2bj #5bid i j K0 ,

@sihDa ,sjhKb#5dabis~ i 1 j !hK05 idabd i ,2 jK0 ,

@sjhDp ,Ti ^ sbi#50, @sihDa ,sjhDb#50,

@Ti ^ sbi,sjhKp#50, @sihKa ,sjhKb#50,

where i , j PZ,a,b,p51,...,n. This subalgebra is degenerate as a Heisenberg algebra since@Dp ,Kp#50.

The Heisenberg algebras can be represented on the space

F5C@qp6 ,xi ,upi ,vpi# i PN

p51,...,n

by differentiation and multiplication operators:



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w~Ti ^ sbi !5]

]xi, w~T12 i ^ s2bi !5bixi ,

w~sihDp!5]

]upi, w~s2 ihDp!5 ivpi ,

w~sihKp!5]

]vpi, w~s2 ihKp!5 iupi ,

w~Dp!5qp

]

]qp, w~Kp!50, w~K0!5Id,

where i>1 and p51,...,n. Our goal is to extend this representation ofs to g. Consider thefollowing elements ofg@@z,z21##:

(j PZ

sjhtrK0z2 jh, (j PZ

Aja

^ sj trz2 j .

Note that

FTi ^ sbi,(j PZ

sjhtrK0z2 jhG50, FsihKp ,(j PZ

sjhtrK0z2 jhG50,

FsihDp ,(j PZ

sjhtrK0z2 jhG5(j PZ

r ps~ i 1 j !htrK0z2 jh5 (k5 i 1 j PZ

r pskhtrK0z2kh1 ih

5r pzih(j PZ

sjhtrK0z2 jh.

The theory of vertex operators suggests~see Lemma 14.5 in Ref. 1! that ( j PZ sjhtrK0z2 jh

should be represented by

K0~z,r !5qr expS (p51

n

r p(j >1

zjhup jD expS 2 (p51

n

r p(j >1

z2 jh

j

]

]vp jD .

Hereqr5q1r 1¯qn

r n.In a similar way the commutator relations

FTi ^ sbi,(j PZ

Aja

^ sj trz2 j G5(j PZ

l iaAj 1bi

a^ sj 1bitrz2 j5l i

a (k5 j 1biPZ

Aka

^ sktrz2k1bi

5l iazbi(

j PZAj

a^ sj trz2 j ,

FsihKp ,(j PZ

Aja

^ sj trz2 j G50,

FsihDp ,(j PZ

Aja

^ sj trz2 j G5(j PZ

r pAja

^ sj 1 ihtrz2 j

5r p (k5 j 1 ihPZ

Aja

^ sktrz2k1 ih5r pzih(j PZ

Aja

^ sj trz2 j

suggest that( j PZ Aja

^ sj trz2 j should be represented by



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Aa~z,r !5qrL0~A0a

^ s0!expS (i 51

`

l iazbixi D expS 2(

i 51

`

l12 ia z2bi

bi

]

]xiD expS (

p51

n

r p(j PZ

zjhup jD3expS 2 (

p51

n

r p(j PZ

z2 jh

j

]

]vp jD 5Aa~z!K0~z,r !.

The last formula hints that forBP g we may try to represent( j PZ Bj ^ sj trz2 j by

B~z,r !5B~z!K0~z,r !.

Note thatB(z) is known from the affine case~Theorem 2!.In fact, we shall see that the same approach is valid even forK andD1 , but for these we

should first discuss the concept of the normal ordering.Let X(z)5( i PZ Xiz

i , Y(z)5( j PZ YjzjPEnd(F)@@z,z21##. We say that the product

X(z)Y(z) exists if for everykPZ and for everyvPF the sum

(i PZ

XiYk2 iv

has finitely many nonzero terms. If this is the case then

X~z!Y~z!5(kPZ

S (i PZ

XiYk2 i D zk.

It is possible thatX(z)Y(z) exists whileY(z)X(z) does not. For example, this happens for

X~z!5(i>1

xizi , Y~z!5(

i>1

]

]xiz2 i .

To improve the situation we define the normal ordering for the product of operators onF. For adifferential operatorPPEnd(F) the normal ordering :xi P: is defined asxi P, while the normalordering of the product :(]/]xi)P: is defined asP(]/]xi). Note that forX(z) andY(z) from theabove example :X(z)Y(z):5:Y(z)X(z):5X(z)Y(z).

From the action of the principle Heisenberg subalgebra onF we see that the series( j PZ sjhKpz2 jh and( j PZ sjhDpz2 jh are represented by

Kp~z!5(i>1

iupizih1(

i>1

]

]vpiz2 ih

and

Dp~z!5(i>1

ivpizih1qp

]

]qp1(

i>1

]

]upiz2 ih.

Using the analogy withAa(z,r ) we shall represent( j PZ sjhtrKpz2 jh by

Kp~z,r !5Kp~z!K0~z,r !

and( j PZ sjhtrDpz2 jh by

Dp~z,r !5:Dp~z!K0~z,r !: .

Note that we use the normal ordering in the case ofDp(z,r ) in order for the product to exist.We summarize this discussion in our main theorem:Theorem 5: There exists a representationw of the toroidal Lie algebrags on the space

F5C@qp6 ,xi ,upi ,vpi# i PN

p51,...,n



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

such that for i51,...,l , p51,...,n, aPDs ,rPZn

(j PZ

w~sjhtrK0!z2 jh5K0~z,r !5qr expS (p51

n

r p(j >1


n

r p(j >1

z2 jh

j

]

]vp jD ,

~4.1!

(j PZ

w~Ti ^ smi1 jhtr !z2mi2 jh5Ti~z,r !5H (j >1

~ jh2mi !zjh2mixj l 112 i

1(j >0

z2 jh2mi]

]xj l 1 iJ K0~z,r !, ~4.2!

(j PZ

w~Aja

^ sj tr !z2 j5Aa~z,r !5L0~A0a

^ s0!expS (i 51

`

l iazbixi D

3expS 2(i 51

`

l12 ia z2bi

bi

]

]xiDK0~z,r !, ~4.3!

(j PZ

w~sjhtrKp!z2 jh5Kp~z,r !5H (i>1

iupizih1(

i>1

]

]vpiz2 ihJ K0~z,r !, ~4.4!

(j PZ

w~sjhtrDp!z2 jh5Dp~z,r !5:H (i>1

ivpizih1qp

]

]qp1(

i>1

]

]upiz2 ihJ K0~z,r !: . ~4.5!

Remark:By choosing a specializationqpPC\$0%, p51,...,n, we obtain a representation of thetoroidal algebrag on the Fock space

F5C@xi ,upi ,vpi# i PNp51,...,n .

V. PROOF OF THE MAIN THEOREM

First of all, let us check that formulas~4.1!–~4.5! define a linear mapw: g→End(F). Thelinear dependencies between the momenta at the left-hand side are of the form

w~Ajsk~a!

^ sj tr !5zk jw~Aja

^ sj tr !.

and

j w~sjhtrK0!1 (p51

n

r pw~sjhtrKp!50.

These dependencies extend to the following relations for the corresponding series:

(j PZ

w~Ajsk~a!

^ sj tr !z2 j5(j PZ

w~Aja

^ sj tr !~z2kz!2 j

and

21

hDz(

j PZw~sjhtrK0!z2 jh1 (

p51

n

r p(j PZ

w~sjhtrKp!z2 jh50,

whereDz5z(]/]z). We have to show that the same relations hold for the right-hand sides of

~4.1!–~4.5!. Indeed, noting thatl isk(a)5z2kmil i

a we obtain



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

Ask~a!~z,r !5L0~A0sk~a!

^ s0!expS (i 51

`

l iaz2kmizbixi D expS 2(

i 51

`

l12 ia zkmi

z2bi

bi

]

]xiDK0~z,r !

5L0~A0a

^ s0!expS (i 51

`

l ia~z2kz!bixi D expS 2(

i 51

`

l12 ia ~z2kz!2bi

bi

]

]xiDK0~z2kz,r !

5Aa~z2kz,r !.

The verification of the second relation is also straightforward:

DzK0~z,r !5Dzqr expS (

p51

n

r p(j >1


n

r p(j >1

z2 jh

j

]

]vp jD

5 (p51

n

r pH (j >1

zjh jhup j1(j >1

z2 jhh]

]vp jJ K0~z,r !5h(

p51

n

r pKp~z,r !.

Observing that the momenta of the series on the left-hand sides of~4.1!–~4.5! spang weconclude that the linear mapw: g→End(F) is well-defined. We need to show now thatw is ahomomorphism of Lie algebras.

The following Lemma and its Corollary which we state without proof are very useful for thecomputations with formal series.

Lemma 6:~cf. Proposition 2.2.2 in Ref. 15!. Let

d~z!5(kPZ

zk, Dd~z!5Dzd~z!5(kPZ

kzk.

If the products in the left-hand sides exist then the following equalities hold:

~ i ! X~z1!dS z2

z1D5X~z2!dS z2

z1D ,

~ i i ! X~z1!DdS z2

z1D5X~z2!DdS z2

z1D1Dz2

~X~z2!!dS z2

z1D .

Corollary 7: Let X(z)5Y(zh). The following equalities hold provided the products in theleft-hand sides exist:

~ i ! X~z1!dS S z2

z1D hD5X~z2!dS S z2

z1D hD ,

~ i i ! X~z1!DdS S z2

z1D hD5X~z2!DdS S z2

z1D hD1

1

hDz2

~X~z2!!dS S z2

z1D hD .

Note thatAa(z,r )5Aa(z)K0(z,r ) andTi(z,r )5Ti(z)K0(z,r ), hence for everyBP g we haveB(z,r )5B(z)K0(z,r ).

In order to verify the commutator relations between the elements of( j PZgj ^ sj

3@C@ t16 ,...,tn

6# we have to show that for everyA,BP g,

(j ,i PZ

w~@Aj ^ sj tr,Bi ^ si tm# !z12 j z2

2 i5 (j ,i PZ

@w~Aj ^ sj tr !,w~Bi ^ si tm!#z12 j z2

2 i . ~5.1!

Indeed, letCj5@Aj ,B#. Then



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

(j ,i PZ

w~@Aj ^ sj tr,Bi ^ si tm# !z12 j z2

2 i

5 (j ,i PZ

w~@Aj ,Bi # ^ sj 1 i tr1m!z12 j z2

2 i1 (j ,i PZ

~Aj uBi !wS sj 1 i tr1mS j

hK01 (

p51

n

r pKpD D5(

j PZ(

k5 j 1 i PZw~@Aj ,Bk2 j # ^ sktr1m!z1

2 j z2j 2k

1 (j 151

h

(j 2PZ

j 5 j 11h j2

(i 2PZ

i 52 j 11hi2

~Aj 1uB2 j 1

!j 11h j2

hw~sh~ j 21 i 2!tr1mK0!z1

2 j 12h j2z2j 12hi2

1 (j 151

h

(j 2PZ

j 5 j 11h j2

(i 2PZ

i 52 j 11hi2

~Aj 1uB2 j 1

! (p51

n

r pw~sh~ j 21 i 2!tr1mKp!z12 j 12h j2z2

j 12hi2

5 (j 151

h

(j 2PZ

(kPZ

w~Ckj 1^ sktr1m!z2

2kS z2

z1D j 1S z2

z1D h j2

1 (j 151

h

(j 2PZ

(k5 j 21 i 2PZ

~Aj 1uB2 j 1

!j 11h j2

hw~shktr1mK0!z1

2 j 12h j2z2j 11h j22hk

1 (j 151

h

(j 2PZ

(k5 j 21 i 2PZ

~Aj 1uB2 j 1

! (p51

n

r pw~shktr1mKp!z12 j 12h j2z2

j 11h j22hk

5 (j 151

h

Cj 1~z2 ,r1m!S z2

z1D j 1

dS S z2

z1D hD1

1

h (j 151

h

j 1~Aj 1uB2 j 1

!K0~z2 ,r1m!

3S z2

z1D j 1

dS S z2

z1D hD1 (

p51

n

r p (j 151

h

~Aj 1uB2 j 1

!Kp~z2 ,r1m!S z2

z1D j 1

dS S z2

z1D hD

1 (j 151

h

~Aj 1uB2 j 1

!K0~z2 ,r1m!S z2

z1D j 1

DdS S z2

z1D hD .

For the right-hand side of~5.1! we use Corollary 3, Corollary 7, and the fact that

K0~z,r !K0~z,m!5K0~z,r1m!:

(j ,i PZ

@w~Aj ^ sj tr !,w~Bi ^ si tm!#z12 j z2

2 i

5@A~z1 ,r !,B~z2 ,m!#

5@A~z1!K0~z1 ,r !,B~z2!K0~z2 ,m!#

5@A~z1!,B~z2!#K0~z1 ,r !K0~z2 ,m!

5(j 51

h

Cj~z2!S z2

z1D j

dS S z2

z1D hDK0~z1 ,r !K0~z2 ,m!

11

h (j 51

h

j ~Aj uB2 j !S z2

z1D j

dS S z2


1(j 51

h

~Aj uB2 j !S z2

z1D j

DdS S z2




130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

5(j 51

h

Cj~z2!S z2

z1D j

dS S z2


11

h (j 51

h

j ~Aj uB2 j !S z2

z1D j

dS S z2


1(j 51

h

~Aj uB2 j !S z2

z1D j

DdS S z2


1(j 51

h

~Aj uB2 j !S z2

z1D j

dS S z2

z1D hD 1

hDz2

~K0~z2 ,r !!K0~z2 ,m!

5(j 51

h

Cj~z2!S z2

z1D j

dS S z2

z1D hDK0~z2 ,r1m!1

1

h (j 51

h

j ~Aj uB2 j !

3S z2

z1D j

dS S z2

z1D hDK0~z2 ,r1m!1(

j 51

h

~Aj uB2 j !S z2

z1D j

DdS S z2

z1D hDK0~z2 ,r1m!

1(j 51

h

~Aj uB2 j !S z2

z1D j

dS S z2

z1D hD (

p51

n

r pKp~z2!K0~z2 ,r !K0~z2 ,m!

5(j 51

h

Cj~z2!S z2

z1D j

dS S z2

z1D hDK0~z2 ,r1m!1

1

h (j 51

h

j ~Aj uB2 j !

3S z2

z1D j

dS S z2

z1D hDK0~z2 ,r1m!1 (

p51

n

r p(j 51

h

~Aj uB2 j !S z2

z1D j

dS S z2

z1D hD

3Kp~z2 ,r1m!1(j 51

h

~Aj uB2 j !S z2

z1D j

DdS S z2

z1D hDK0~z2 ,r1m!.

Thus ~5.1! holds.It is easy to see that operatorsw(sjht rKp), p50,1,...,n commute with w(( j PZ gj ^ sj

3C@ t16 ,...,tn

6# % K ). This follows from the obvious equalities:

@Kp~z1 ,r !,Aa~z2 ,m!#5@Kp~z1 ,r !,Ti~z2 ,m!#50,

@Ka~z1 ,r !,Kb~z2 ,m!#50, p,a,b50,1,...,n.

In order to check the action ofD1 on ( j PZ gj ^ sjC@ t16 ,...,tn

6# it is sufficient to verify thefollowing equality for the generating series:

(j ,i PZ

w~@sjhtrDp ,si tmAia#!z1

2 jhz22 i5 (

j ,i PZ@w~sjhtrDp!,w~Ai

a^ si tm!#z1

2 jhz22 i . ~5.2!

The corresponding equalities forTi ^ sbitm andsihtmKp will follow since sjhtrDp acts as a deri-vation andsi tmAi

a generate( j PZ gj ^ sjC@ t16 ,...,tn

6# % K .Let us prove that~5.2! holds.

(j ,i PZ

w~@sjhtrDp ,si tmAia#!z1

2 jhz22 i5 (

j ,i PZmpw~si 1 jhtr1mAi

a!z12 jhz2

2 i

5(j PZ

(k5 i 1 jhPZ

mpw~sktr1mAka!z1

2 jhz22k1 jh

5mpAa~z2 ,r1m!dS S z2

z1D hD .



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

Next, we will show that the right-hand side of~5.2! reduces to the same expression. In thefollowing computations it will be convenient to denotew(sjhDp)5]/]up j by gp( j ), w(s2 jhDp)5 j vp j by gp(2 j ) and w(Dp)5qp(]/]qp) by gp(0). Note that @gp( j ),K0(z,m)#5mpzjhK0(z,m) for j PZ.

(j ,i PZ

@w~sjhtrDp!,w~Aia

^ si tm!#z12 jhz2

2 i5@Dp~z1 ,r !,Aa~z2 ,m!#

5F :H (j PZ

gp~ j !z12 jhJ K0~z1 ,r !: , Aa~z2!K0~z2 ,m!G

5Aa~z2!F :H (j PZ

gp~ j !z12 jhJ K0~z1 ,r !: , K0~z2 ,m!G

5Aa~z2!F H (j ,0

gp~ j !z12 jhJ K0~z1,r !, K0~z2 ,m!G

1Aa~z2!FK0~z1 ,r !H (j >0

gp~ j !z12 jhJ , K0~z2 ,m!G

5Aa~z2!F(j ,0

gp~ j !z12 jh , K0~z2 ,m!GK0~z1 ,r !

1Aa~z2!K0~z1 ,r !F(j >0

gp~ j !z12 jh , K0~z2 ,m!G

5Aa~z2!(j ,0

z12 jhz2

jhmpK0~z2 ,m!K0~z1 ,r !

1Aa~z2!K0~z1 ,r !(j >0

z12 jhz2

jhmpK0~z2 ,m!

5mpAa~z2!K0~z1 ,r !K0~z2 ,m!dS S z2

z1D hD

5mpAa~z2!K0~z2 ,r !K0~z2 ,m!dS S z2

z1D hD5mpAa~z2 ,r1m!dS S z2

z1D hD .

From the above computation we can see that

@Dp~z1 ,r !,K0~z2 ,m!#5mpK0~z2 ,r1m!dS S z2

z1D hD .

To complete the proof of Theorem 5 we need to compute the commutators@w(sjhtrDa),w(sihtmDb)#. We will check the following equality for the generating series:

(j ,i PZ

w~@sjhtrDa , sihtmDb# !z12 jhz2

2 ih5 (j ,i PZ

@w~sjhtrDa!, w~sihtmDb!#z12 jhz2

2 ih . ~5.3!

To compute the left-hand side we use multiplication ing:



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

(j ,i PZ

w~@sjhtrDa ,sihtmDb# !z12 jhz2

2 ih

5 (j ,i PZ

w~mas~ j 1 i !htr1mDb2r bs~ j 1 i !htr1mDa!z12 jhz2

2 ih

2mar bwS s~ j 1 i !htr1mH jK 01 (p51

n

r pKpJ D z12 jhz2

2 ih

5(j PZ

(k5 j 1 i PZ

~maw~skhtr1mDb!2r bw~skhtr1mDa!!z12 jhz2

2kh1 jh

2(j PZ

(k5 j 1 i PZ

mar bwS skhtr1mH jK 01 (p51

n

r pKpJ D D z12 jhz2

2kh1 jh

5~maDb~z2 ,r1m!2r bDa~z2 ,r1m!!dS S z2

z1D hD

2mar b(p51

n

r pKp~z2 ,r1m!dS S z2

z1D hD2mar bK0~z2 ,r1m!DdS S z2

z1D hD .

Observe that

@Da~z,r !,gb~ i !#5F :H (j PZ

ga~ j !z2 jhJ K0~z,r !:,gb~ i !G5F(

j ,0ga~ j !z2 jhK0~z,r !,gb~ i !G1FK0~z,r !(

j >0ga~ j !z2 jh,gb~ i !G

5(j ,0

ga~ j !z2 jh@K0~z,r !,gb~ i !#1@K0~z,r !,gb~ i !#(j >0

ga~ j !z2 jh

52r bzih(j ,0

ga~ j !z2 jhK0~z,r !2r bzihK0~z,r !(j >0

ga~ j !z2 jh

52r bzihDa~z,r !.

We use the last equality to compute the right-hand side of~5.3!:

(j ,i PZ

@w~sjhtrDa!,w~sihtmDb!#z12 jhz2

2 ih

5@Da~z1 ,r !,Db~z2 ,m!#

5FDa~z1 ,r !,(i ,0

gb~ i !z22 ihK0~z2 ,m!G1FDa~z1 ,r !,K0~z2 ,m!(

i>0gb~ i !z2

2 ihG5(

i ,0@Da~z1 ,r !,gb~ i !#z2

2 ihK0~z2 ,m!1(i ,0

gb~ i !z22 ih@Da~z1 ,r !,K0~z2 ,m!#

1@Da~z1 ,r !,K0~z2 ,m!#(i>0

gb~ i !z22 ih1K0~z2 ,m!(

i>0@Da~z1 ,r !,gb~ i !#z2

2 ih

52r b(i ,0

Da~z1 ,r !z1ihz2

2 ihK0~z2 ,m!1(i ,0

gb~ i !z22 ihmaK0~z2 ,r1m!dS S z2

z1D hD

1maK0~z2 ,r1m!dS S z2

z1D hD(

i>0gb~ i !z2

2 ih2r bK0~z2 ,m!(i>0

Da~z1 ,r !z1ihz2

2 ih



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

5maDb~z2 ,r1m!dS S z2

z1D hD2r b(

i ,0S z1

z2D ih

(j ,0

ga~ j !z12 jhK0~z1 ,r !K0~z2 ,m!

2r b(i ,0

S z1

z2D ih

K0~z1 ,r !(j >0

ga~ j !z12 jhK0~z2 ,m!2r bK0~z2 ,m!(

i>0(j ,0

ga~ j !z12 jhK0~z1 ,r !

3S z1

z2D ih

2r bK0~z2 ,m!(i>0

K0~z1 ,r !S z1

z2D ih

(j >0

ga~ j !z12 jh


z1D hD2r b(

i ,0(j ,0

S z1

z2D ih

z12 jhga~ j !K0~z1 ,r !K0~z2 ,m!

2r b(i ,0

(j >0

S z1

z2D ih

z12 jhK0~z1 ,r !@ga~ j !,K0~z2 ,m!#

2r b(i ,0

(j >0

S z1

z2D ih

z12 jhK0~z1 ,r !K0~z2 ,m!ga~ j !

2r b(i>0

(j ,0

z12 jhS z1

z2D ih

@K0~z2 ,m!,ga~ j !#K0~z1 ,r !

2r b(i>0

(j ,0

z12 jhS z1

z2D ih

ga~ j !K0~z2 ,m!K0~z1 ,r !

2r b(i>0

(j >0

S z1

z2D ih

z12 jhK0~z2 ,m!K0~z1 ,r !ga~ j !


z1D hD2r bdS S z2

z1D hD :H (

j PZga~ j !z2

2 jhJ K0~z2 ,r !K0~z2 ,m!:

2mar b(i .0

(j >0

S z2

z1D ~ i 1 j !h

K0~z1 ,r !K0~z2 ,m!1mar b(i<0

(j ,0

S z2

z1D ~ i 1 j !h

K0~z1 ,r !K0~z2 ,m!


z1D hD2r bDa~z2 ,r1m!dS S z2

z1D hD

2mar b (k5 i 1 j .0

(j 50

k21 S z2

z1D kh

K0~z1 ,r !K0~z2 ,m!

1mar b (k5 i 1 j ,0

(j 5k

21 S z2

z1D kh

K0~z1 ,r !K0~z2 ,m!


z1D hD2mar b(

kPZkS z2

z1D kh

K0~z1 ,r !K0~z2 ,m!


z1D hD2mar bK0~z1 ,r !K0~z2 ,m!DdS S z2

z1D hD


z1D hD

21

hmar bDz2

~K0~z2 ,r !!K0~z2 ,m!dS S z2

z1D hD2mar bK0~z2 ,r !K0~z2 ,m!DdS S z2

z1D hD


z1D hD2mar b(

p51

n

r pKp~z2 ,r1m!dS S z2

z1D hD



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

2mar bK0~z2 ,r1m!DdS S z2

z1D hD .

This establishes~5.3! and completes the proof of Theorem 5.

VI. SUGAWARA OPERATORS

In the representation we constructed the operators fromD1 act as derivations on the algebrag5 g^ C@ t0

6 ,t16 ,...,tn

6# % K . It is possible to extend this action onD , that is to representt0j trD0

by operators onF. For the affine case these are the Sugawara operators.We will work with the toroidal algebra realized as( j PZ gj ^ sjC@ t1

6 ,...,tn6# % K . For this

realization

c~ t0j trD0!5

1

hsjhtrDs2

1

hn21~r! ^ sjhtr ~6.1!

whererP h* with (rua i)51 for every simple roota iPD andDs5s(d/ds).Since foraPD the elementsAa

^ t0i tm generateg then we have to check that

c~@ t0j trD0 ,Aa

^ t0i tm# !5@c~ t0

j trD0!,c~Aa^ t0

i tm!#.

Note thatAaP ght(a) and (rua)5ht(a).We have

c~@ t0j trD0 ,Aa

^ t0i tm# !5 ic~Aa

^ t0i 1 j tr1m!5 iAa

^ sht~a!1h~ i 1 j !tr1m,

while

@c~ t0j trD0!,c~Aa

^ t i0tm!#5F1

hsjhtrDs2

1

hn21~r! ^ sjhtr,Aa ^ sht~a!1 ihtmG

51

h~ht~a!1 ih2~rua!!Aa

^ sht~a!1h~ i 1 j !tr1m

5 iAa^ sht~a!1h~ i 1 j !tr1m.

This establishes~6.1!. It is then sufficient to construct operators onF corresponding tosjhtrDs .It will be convenient to denote in this sectionw(Ti ^ sbi) by t( i ), w(sjhDp) by gp( j ), and

w(sjhKp) by kp( j ).In these notations

K0~z,r !5qr expS (p51

n

r p(j >1

kp~2 j !

jzjhD expS 2 (

p51

n

r p(j >1

kp~ j !

jz2 jhD ,

Aa~z!5L0~A0a

^ s0!expS (i>1

l ia t~12 i !

bizbi D expS 2(

i>1l12 i

a t~ i !

biz2bi D .

Nontrivial commutators in the principal Heisenberg subalgebra are

@t~ i !,t~12 i !#5bi and @gp~ j !,kp~2 j !#5 j .

The action of the Heisenberg subalgebra on the vertex operators is determined by

@t~ i !,Aa~z!#5l iazbiAa~z!,

@gp~ j !,K0~z,r !#5r pzjhK0~z,r !.

Consider the following operators on the spaceF:



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

Lt~ j !51

2 (i PZ

:t~12 i !t~ i 1 j l !:,

Lgk~ j !5 (p51

n

(i PZ

:gp~2 i !kp~ i 1 j !: .

These are the analogs of the Sugawara operators.Construct the formal generating series for these:

Ds~z!52(j PZ

~Lt~ j !1hLgk~ j !!z2 jh.

Proposition 8: Formula (4.5) together with

(j PZ

w~sihtrDs!z2 ih5Ds~z,r !5:Ds~z!K0~z,r !:

defines the representation of the Lie algebraD on the space

w~ g!5wS (j PZ

gj ^ sjC@ t16 ,...,tn

6# % K D .

Proof: We have an action ofD on g which is a unique extension of its natural action ong.Thus we need to prove thatw(@D,B#)5@w(D),w(B)# for everyDPD andBP g. For subalgebraD1 this was proved in the course of Theorem 5, hence we need to consider onlyD5sihtrDs .SincesihtrDs act ong as derivations andg is generated as an algebra by elementsAj

a^ sj tm then

it is sufficient to prove that

w~@sihtrDs ,Aja

^ sj tm# !5@w~sihtrDs!,w~Aja

^ sj tm!#.

Again we shall replace this with an equivalent identity for the corresponding series:

wS F(i PZ

sihtrDsz12 ih ,(

j PZAj

a^ sj tmz2

2 j G D 5F(i PZ

w~sihtrDs!z12 ih ,(

j PZw~Aj

a^ sj tm!z2

2 j G .~6.2!

We have

wS F(i PZ

sihtrDsz12 ih ,(

j PZAj

a^ sj tmz2

2 j G D5(

i PZ(j PZ

j w~Aja

^ sj 1 ihtr1m!z12 ihz2

2 j

5(i PZ

(k5 j 1 ihPZ

~k2 ih !w~Aka

^ sktr1m!z12 ihz2

2k1 ih

5(kPZ

kw~Aka

^ sktr1m!z22kdS S z2

z1D hD2h(

kPZw~Ak

a^ sktr1m!z2

2kDdS S z2

z1D hD

52~Dz2Aa~z2 ,r1m!!dS S z2

z1D hD2Aa~z2 ,r1m!S Dz2

dS S z2

z1D hD D

52Dz2S Aa~z2 ,r1m!dS S z2

z1D hD D ,

whereDz25z2(]/]z2).



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

In order to compute the right-hand side of~6.2!, we will need two lemmas.Lemma 9:

~ i ! @Lt~ j !,Aa~z!#5zjh~Dz1 jh !Aa~z!,

~ i i ! @Lt~z1!,Aa~z2!#5Dz2S Aa~z2!dS S z2

z1D hD D .

Proof: We will prove ~i! assuming thatj >0. The casej ,0 is analogous.

@Lt~ j !,Aa~z!#51

2 F(i PZ

:t~12 i !t~ i 1 j l !:,Aa~z!G5

1

2 F(i .0

t~12 i !t~ i 1 j l !,Aa~z!G11

2 F(i<0

t~ i 1 j l !t~12 i !,Aa~z!G5

1

2 (i .0

t~12 i !@t~ i 1 j l !,Aa~z!#11

2 (i .0

@t~12 i !,Aa~z!#t~ i 1 j l !

11

2 (i<0

t~ i 1 j l !@t~12 i !,Aa~z!#11

2 (i<0

@t~ i 1 j l !,Aa~z!#t~12 i !

51

2 (i .0

t~12 i !l iazbi1 jhAa~z!1

1

2 (i .0

l12 ia z2biAa~z!t~ i 1 j l !

11

2 (i<0

t~ i 1 j l !l12 ia z2biAa~z!1

1

2 (i<0

l iazbi1 jhAa~z!t~12 i !

51

2 (i .0

t~12 i !l iazbi1 jhAa~z!1

1

2 (i .0

l12 ia z2biAa~z!t~ i 1 j l !

11

2 (i<2 j l

t~ i 1 j l !l12 ia z2biAa~z!1

1

2 (2 j l , i<0

l12 ia z2bi@t~ i 1 j l !,Aa~z!#

11

2 (2 j l , i<0

l12 ia z2biAa~z!t~ i 1 j l !1

1

2 (i<0

l iazbi1 jhAa~z!t~12 i !

51

2 (k5 i .0

t~12k!lkazbk1 jhAa~z!1

1

2 (k5 i 1 j l . j l

l12ka z2bk1 jhAa~z!t~k!

11

2 (k512 i 2 j l .0

lkazbk1 jht~12k!Aa~z!1

1

2 (2 j l , i<0

l12 ia z2bil i

azbi1 jhAa~z!

11

2 (0,k5 i 1 j l < j l

l12ka z2bk1 jhAa~z!t~k!1

1

2 (k512 i .0

l12ka z2bk1 jhAa~z!t~k!

5zjhDzAa~z!1

j

2zjhS (

k51

l

lkal12k

a DAa~z!5zjh~Dz1h j !Aa~z!.

At the last step we used the equality

1

2 (k51

l

lkal12k

a 51

2 (k51

l

~Tkun21~a!!~T12kun21~a!!5h

2~n21~a!un21~a!!5

h

2~aua!5h,

which follows from the fact that$1/h T12k% is the dual basis for$Tk%.Part ~ii ! is an immediate consequence of~i!:



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

@Lt~z1!,Aa~z2!#5(j PZ

z12 jh@Lt~ j !,Aa~z2!#

5(j PZ

z12 jhz2

jh~Dz21h j !Aa~z2!

5~Dz2Aa~z2!!dS S z2

z1D hD1Aa~z2!S Dz2

dS S z2

z1D hD D5Dz2S Aa~z2!dS S z2

z1D hD D .

Lemma 10:

@ :Lgk~z1!K0~z1 ,r !:,K0~z2 ,m!#51

hdS S z2

z1D hDDz2

~K0~z2 ,m!!K0~z1 ,r !.

Proof:

@ :Lgk~z1!K0~z1 ,r !:,K0~z2 ,m!#

5(j PZ

z12 jh (

p51

n

(i .0

@gp~2 i !kp~ i 1 j !K0~z1 ,r !,K0~z2 ,m!#

1(j PZ

z12 jh (

p51

n

(i<0

@kp~ i 1 j !K0~z1 ,r !gp~2 i !,K0~z2 ,m!#

5(j PZ

z12 jh (

p51

n

(i .0

@gp~2 i !,K0~z2 ,m!#kp~ i 1 j !K0~z1 ,r !

1(j PZ

z12 jh (

p51

n

(i<0

kp~ i 1 j !K0~z1 ,r !@gp~2 i !,K0~z2 ,m!#

5(j PZ

z12 jh (

p51

n

mp(i .0

z22 ihK0~z2 ,m!kp~ i 1 j !K0~z1 ,r !

1(j PZ

z12 jh (

p51

n

mp(i<0

kp~ i 1 j !K0~z1 ,r !z22 ihK0~z2 ,m!

5(j PZ

(k5 i 1 j PZ

(p51

n

mpz12 jhz2

2kh1 jhkp~k!K0~z2 ,m!K0~z1 ,r !

51

hdS S z2

z1D hDDz2

~K0~z2 ,m!!K0~z1 ,r !.

Using these two lemmas we can handle the right-hand side of~6.2!.

F(i PZ

w~sihtrDs!z12 ih , (

j PZw~Aj

a^ sj tm!z2

2 j G5@Ds~z1 ,r !, Aa~z2 ,m!#

5@ :Ds~z1!K0~z1 ,r !:,Aa~z2!K0~z2 ,m!#

52@Lt~z1!K0~z1 ,r !,Aa~z2!K0~z2 ,m!#2h@ :Lgk~z1!K0~z1 ,r !:,Aa~z2!K0~z2 ,m!#

52@Lt~z1!,Aa~z2!#K0~z1 ,r !K0~z2 ,m!2hAa~z2!@ :Lgk~z1!K0~z1 ,r !:,K0~z2 ,m!#

52Dz2S Aa~z2!dS S z2

z1D hD DK0~z2 ,m!K0~z1 ,r !2Aa~z2!dS S z2

z1D hDDz2

~K0~z2 ,m!!K0~z1 ,r !



130.18.123.11 On: Mon, 22 Dec 2014 02:09:49


z1D hDK0~z2 ,m!K0~z1 ,r ! D


z1D hDK0~z2 ,m!K0~z2 ,r ! D52Dz2S Aa~z2 ,r1m!dS S z2

z1D hD D .

ACKNOWLEDGMENT

This work is supported by the Natural Sciences and Engineering Research Council of Canada.

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130.18.123.11 On: Mon, 22 Dec 2014 02:09:49

principal vertex operator representations for toroidal lie algebras

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