Chiral Charged Fermions, One-Dimensional Quantum Field Theory and Vertex Algebras

Chiral Charged Fermions, One-DimensionalQuantum Field Theory and Vertex Algebras

FLORIN CONSTANTINESCU1 and GU« NTER SCHARF2

1 Fachbereich Mathematik, Johann Wolfgang Goethe-Universita« t Frankfurt,Robert-Mayer-Strasse 10, D-60054 Frankfurt am Main, Germany2 Institut fu« r Theoretische Physik, Universita« t Zu« rich, Winterthurerstr. 190, CH-8057 Zu« rich,Switzerland

(Received: 2 November 1999; revised version: 3 March 2000)

Abstract. We give an explicit L2-representation of chiral charged fermions using theHardy^Lebesgue octant decomposition. In the `pure'case such a representation has already beenused by M. Sato in holonomic ¢eld theory. We study both `pure' and `mixed' cases. In thecompact case, we rigorously de¢ne unsmeared chiral charged fermion operators inside the unitcircle. Using chiral fermions, we orient our ¢ndings towards a functional analytic study of vertexalgebras as one-dimensional quantum ¢eld theory.

Mathematical subject classi¢cations (2000): 81Txx, 81Rxx.

Key words: chiral charged fermions, vertex operators and algebras.

1. Introduction

Let Rn � [2na�1Ga be the octant decomposition of Rn by octants Ga considered withtheir parity. Each L2^function c�x�, x 2 Rn has a decomposition

c�x� �X2na�1

ca�x� iGa0�; i ��ÿ1p

; �1�

where ca are L2-boundary values in tubes Rn � iGa induced by the octantdecomposition in the conjugate Fourier variable. Remark that two different bound-ary values ca;jb, a 6� b of c;j 2 L2�Rn� are orthogonal, i.e.

Rca�x�jb�x� dx � 0.

We call (1) a Hardy^Lebesgue octant decomposition. Certainly, a compact versionin which R is replaced by the unit circle S1 is also possible by using the Cauchyindicatrix. In this case we have the classical Hardy decomposition. By means of thisdecomposition we introduce in Section 2 chiral charged fermions by an explicit rep-resentation in the antisymmetric Fock space F�L2�R��. This representation parallelsthe usual canonical anticommutation relations. Charged fermions polarize the

Letters in Mathematical Physics 52: 113^126, 2000. 113# 2000 Kluwer Academic Publishers. Printed in the Netherlands.

antisymmetric Fock space. Accordingly a graduation of the form

F�L2�R�� M1n�0F n�L2�R�

�M1n�0

Mn1�n2�n

F n1;n2 �L2�R� �M

n1;n2 X 0

F n1;n2�L2�R��2�

appears where n1; n2 X 0 refer to the octant decomposition and collect all Ga with n1positive and n2 negative components. The case n1 � n2 � 0 corresponds to thevacuum. We call spaces F n1;0, F 0;n2 `pure' and others `mixed'. The `pure' fermionicsectors were already used by M. Sato and coworkers [1] in holonomic ¢eld theory.We insist here on the `mixed' sectors. In the compact case, we rigorously de¢neanalytically continued chiral, charged fermionic operators inside the unit circle.

Our ¢ndings are analyzed from the point of view of vertex operator algebrasrealized as one-dimensional quantum ¢eld theory. There is no spectral conditionin momentum space. Translation invariance and especially locality play the centralrole. An interesting point is that our quantum ¢elds appear as boundary valuesof analytic (operator-valued) functions, a property which in the standard Wightmanquantum ¢eld theory is reserved to vacuum expectation values being a consequenceof the spectral condition (and Poincare invariance). In Section 3, in order to establishfull contact to vertex algebras, we have to pay a price by giving up complex con-jugation. This loss is fully compensated by rigorous operator product expansion.Formal algebraic machinery of vertex algebras is enriched by a functional analyticframework. The example of chiral-charged fermions and other examples workedout in this Letter can be extended to more general vertex algebras.

2. Fock Space Representations of Charged Fermions

In order to explain what we are going to do, let us consider the well-known rep-resentation of canonical anticommutation relations in the antisymmetric Fock spaceF�L2�R��:�

A� f �c�n�x1; . . . ; xn� � �n� 1�1=2

Zdxf �x�cn�1�x; x1; . . . ; xn�; �3�

�A�� f �c

�n�x1; . . . ; xn�

� nÿ1=2Xni�1�ÿ1�iÿ1f �xi�cnÿ1�x1; . . . ; xi; . . . ; xn� �4�

114 FLORIN CONSTANTINESCU AND GU« NTER SCHARF

and, in the unsmeared form,

�A�x�c

�n�x1; . . . ; xn� � �n� 1�1=2cn�1�x; x1; . . . ; xn�; �5�

�A��x�c

�n�x1; . . . ; xn�

� nÿ1=2Xni�1�ÿ1�iÿ1d�xÿ xi�cnÿ1�x1; . . . ; xi; . . . xn� �6�

Here c � �c0;c1; . . . ;cn; . . .� where cn � cn�x1; x2; . . . ; xn� 2 L2�Rn� is an elementof the antisymmetric Fock space, c0 is the vacuum, and the hat denotes the missingvariable. On the vacuum we have

A�f �c0 � 0; f 2 L2�R�:

The smeared A�f � and A��f � are densely de¢ned, closed operators withA��f � � A�f ��being the adjoint of A�f �. The unsmeared A�x� is an operator, whereasA��x� is only abilinear form. The maps

f 7! A�f �; f 7! A��f �

are antilinear and linear, respectively, and

A�f � �Z

dxf �x�A�x�; A��f � �Z

dxf �x�A��x�: �7�

The operators A�f � and A��f � satisfy the canonical anticommutation relations

fA�f �;A�g�g � fA��f �;A��g�g � 0 �8�

fA�f �;A��g�g � �f ; g�; �9�

where �f ; g� is the scalar product in L2�R�. The operators A�f �, A��f � are boundedwith norm

kA�f �k � kA��f �k � kf k2: �10�

Now we introduce charged fermions a�f �; a��f �; f 2 L2�R� by the following de¢ning

CHIRAL CHARGED FERMIONS 115

relations in antisymmetric Fock space:�a�f �c

�n�x1; . . . ; xn�

� �n� 1�1=2Z

f��x�cn�1�x; x1; . . . ; xn� dx�

�nÿ1=2Xnj�1�ÿ1�jÿ1fÿ�xj�cnÿ1�x1; . . . ; xj; . . . ; xn�; �11�

�a��f �c�n�x1; . . . ; xn�

� �n� 1�1=2Z

fÿ�x�cn�1�x; x1; . . . ; xn� dx�

�nÿ1=2Xnj�1�ÿ1�jÿ1f��xj�cnÿ1�x1; . . . ; xj; . . . ; xn�; �12�

where f � f� � fÿ is the Hardy^Lebesgue decomposition of f .In compact form the relations (11) and (12) can be given with the help of A#, (A#

stays for A or A�) as

a�f � � A�f�� A��fÿ�; �13�

a��f � � A�fÿ� � A��f��; �14�

where A��f�� A�f��. Remark the similarity of (13) and (14) to two-dimensional(time-independent) Dirac fermions which are also de¢ned in the (antisymmetric)Fock space over the direct sum H� �Hÿ. The difference is that for Dirac fermionswe start in momentum space and both H� and Hÿ are copies of the same L2 Hilbertspace. In addition, the natural conjugation in H� and Hÿ accounts for the chargeconjugation (see, for instance, [2]). Writing in this case f � �f�; fÿ�,f� 2 H� � L2�R� the formal difference is that for Dirac fermions the second termin a�f � has fÿ instead of fÿ, a��f � being the adjoint of a�f �. In this way no mixedlinear/antilinear dependence on f appear, which is typical for (13) and (14) (seelater). On the other hand, it is well known that by putting together chiral fermionsof opposite chirality one obtains the massless Dirac fermion.


Time-zero Dirac fermion operators satisfy CAR relations, whereas chiral chargedfermions a�f �; a��f � de¢ned above satisfy anticommunitation relations of the form

fa�f �; a�g�g � fa��f �; a��g�g � 0; �15�

fa�f �; a��g�g � hf ; gi; �16�where in contradistinction to (9) hf ; gi is no longer the scalar product in L2�R� butit is

hf ; gi �Z

f��x�g��x� dx�Z

gÿ�x�fÿ�x� dx: �17�

We call the attention of the reader to the fact that in our case it is not possible to takefÿ instead of fÿ in (13) as in the case of Dirac fermions, which would turn the innerproduct h�; �i into the usual scalar product

�f ; g� �Z

f��x�g��x� dx�Z

fÿ�x�gÿ�x� dx �Z

f �x�g�x� dx;

because the relations (15) are no longer true ((16) remains valid). In spite of not beinga scalar product, h�; �i is positive de¢nite. Indeed

hf ; f i �Z

f��x�f��x� �Z

fÿ�x�fÿ�x� dx

� �f ; f � � kf k22 �18�

becauseZf��x�gÿ�x� dx �

Zfÿ�x�g��x� dx � 0 for arbitrary f ; g 2 L2�R�:

As remarked in the introduction the antisymmetric Fock space decomposesaccording to

F�L2�R�� M

n1;n2 X 0

F n1;n2 �M1n�0

Mn1�n2�n

F n1;n2 �M1n�ÿ1

Mn1ÿn2�n

F n1;n2 �16�

giving rise to the `pure' and `mixed' states already mentioned. Here n1; n2 are thenumbers of the a� and a operators, respectively, applied to the vacuum.

Neither a�f � nor a��f � annihilates the vacuum as in the neutral case. Vacuumexpectation values of a#�f � satisfy neutrality condition (i.e. they do vanish if thenumber of a-operators is not equal to the number of a�-operators) and can be givenin closed form. They are Gram determinants in the pure cases and a kind of gen-eralized Gram `determinants' in the mixed case. Moreover, a#�f � are bounded


operators:

ka��f �k � kf k2: �20�It is interesting to remark that the proof of (20) doesn't follow directly from CARbecause h�; �i in (16) is not a scalar product. Nevertheless, the reader can checkwithout dif¢culty that the usual boundedness proof for fermions [3] can be easilyadapted because it only uses positivity of h�; �i.

The explicit Fock space realization of chiral charged fermions (11) and (12) can betaken over to the compact case S1.

It is somewhat unpleasant that in the non-compact as well as in the compactcase a�f � and a��f � do not show sharp linear or anti^linear dependence on f suchthat the unsmeared a�z�, a��z�, z 2 R or S1 cannot be looked at asoperator-valued `kernels' for a�f �, a��f �. In the compact case this point willbe discussed (and improved) in the next section. This is the main point of thisLetter.

Coming to the end of this section, let us remark that the notation a�f � for the chiralfermion is not fortunate. Indeed, we want to look at a�f � as a one-dimensional ¢eldoperator and as such a better notation would beF�f �. The reason we choose a insteadof F is explained in the next section where the unsmeared a�z� will be interpreted as amember of a vertex algebra.

3. Towards Vertex Algebras as One-Dimensional Quantum Field Theories

Vertex algebras have been related to the two-dimensional Wightman theory [4]. Theresults of the preceding section, together with results obtained in [5], show that anatural variant is to start with a one-dimensional quantum ¢eld theory. Certainlywe cannot expect bona ¢de Wightman ¢elds but the ¢elds we are suggesting havebetter behaviours.

Indeed, in Wightman theory, as a consequence of spectral condition togetherwith Poincare invariance, vacuum expectation values show up as certain bound-ary values of analytic functions. The ¢elds by themselves do not appear asboundary values.

In contradistinction to that, our one-dimensional ¢elds are boundary values ofoperator-valued analytic functions. But in order to achieve full contact with vertexalgebras, we still have to do some work which we accomplish within the frameof our example of chiral fermions. Some other examples follow later. In orderto start let us remark that strong tools in quantum ¢eld theory and string theorylike operator product expansion (OPE) have to be formulated in the unsmearedform, which is not well de¢ned from a functional analytic point of view. In conformalquantum ¢eld theory, a rigorous way out are vertex algebras in which OPE is for-mulated within the frame of formal power series, formal Laurent expansionsand formal distribution theory.


What we claim in this Letter is that the algebraic framework of vertex algebras canbe realized in functional analysis and this implies a search for a kernel calculus. Wedecide to give up complex conjugation and to de¢ne all our ¢elds as operatorsin a given complex domain which will be the unit disc jzj < 1. Even in the caseof chiral fermions, it is not at all clear that this is possible. The following explicitrepresentation of a�z� and b�z� (instead of a��z�) as operators inspired from the for-mal unsmeared version of (11) and (12) solves the problem:�

a�z�c�n�z1; . . . ; zn�

� �n� 1�1=2cn�1��zÿ1; z1; . . . ; zn

��

�nÿ1=2Xnj�1�ÿ1� j 1

zÿ zjcnÿ1�z1; . . . ; zj; . . . ; zn� �21�

�b�z�c

�n�z1; . . . ; zn�

� �n� 1�1=2cn�1ÿ �z; z1; . . . ; zn�

�nÿ1=2Xnj�1�ÿ1� j 1

zÿ zÿ1jcnÿ1�z1; . . . ; zj; . . . ; zn�; �22�

where zj 2 S1, jzj < 1. We are now going to show that a�z�; b�z� are densely de¢nedoperators in fermionic Fock space with a domain of the de¢nition presented below.In some sense, complex conjugation z! z is simulated in (21) and (22) by theinversion z! zÿ1. Although these relations are similar to the unsmeared versionof (11) and (12), they show particularities not present in (11), (12) which will bemade clear by progressing in this section.

We use the notations cn�1� �z; z1; . . . ; zn� in order to denote Hardy components of

cn�1 in the ¢rst variable. For instance, for the pure case

c�z1; . . . ; zn� � det�ji�zj��i;j�1;...;n; ji�z� 2 L2�S1�;we have

c��z1; . . . ; zn� �j1��z1� . . .jn

��z1�j1�z2� . . .jn�z2�j1�zn� . . .jn�zn�

��:

Further, cn��zÿ1; . . .� means `wrong boundary value', i.e. we ¢rst take cn

��z . . .� andthen replace z by zÿ1. In order to make sense of this type of boundary value, wewill restrict cn to Laurent polynomials instead of Laurent series. This restrictionenables us to rigorously de¢ne a�z� and b�z� as Fock space operators. Indeed, we


identify F�L2�S1�� with the corresponding space of analytic functions inside andoutside the unit circle obtained by the Hardy decomposition (which are Laurentseries) and de¢ne a�z�; b�z� through (21), (22) on the dense domain in F�L2�S1��obtained by cutting the Laurent series to Laurent polynomials. In order to de¢neproducts of such operators (beside introducing the normal ordering), we have toextend the de¢nition to some semi-in¢nite Laurent series. This will be discussed lateron in this section.

The reader can check anticommutation relations of a and b, in particular

fa�z�; b�w�g � d�zÿ w�; �23�where now

fa�z�; b�w�g � a�z�b�w�jjzj>jwj � b�w�a�z�jjwj>jzj

� 1zÿ w

��jzj>jwj� 1wÿ z

��jwj>jzj� d�zÿ w� �24�

and d�zÿ w� replaces the formal d-function [4]:

d�zÿ w� � zÿ1Xn2Z

�wz

�n� wÿ1

Xn2Z

� zw

�n� d�wÿ z�: �25�

Some hints are given below. This is exactly the state of affairs within the vertexalgebra frame where, by now, a�z�, b�z� are genuine densely de¢ned operators inFock space for jzj < 1.

Some remarks are in order concerning (23). Let us introduce ¢rst some notationssimilar to those in Section 2:

a�z� � a1�z� � a2�z�; �26�

b�z� � b1�z� � b2�z�; �27�where�

a1�z�c�n�z1; . . . ; zn� � �n� 1�1=2cn�1

��zÿ1; z1; . . . ; zn

�; �28�

�a2�z�c

�n�z1; . . . ; zn� � nÿ1=2

Xnj�1�ÿ1�j 1

zÿ zjcnÿ1�z1; . . . ; zj; . . . ; zn� �29�

and similar relations for b1�z� and b2�z�. The operators ai�z�; bi�z�, i � 1; 2; jzj < 1 arede¢ned on Laurent polynomials cn but in order to write down (23) we need productsof ai and bi. Such products are, a-priori, not de¢ned. Let us discuss the case a1�z�b2�w�with jzj > jwj. Here we have to extend the domain of de¢nition of a1�z� from Laurentpolynomials to some semi-in¢nite Laurent series. Indeed, for the ¢rst critical term


ÿ1=�wÿ zÿ11 � in b2�w� we have

a1�z� ÿ 1wÿ zÿ11

� �

� a1�z� z11ÿ wz1

� �

� a1�z�X1n�0

wnzn�11

!�X1n�0

wna1�z��zn�11 �

�X1n�0

wnzÿnÿ1 � 1zÿ w

for 1 > jzj > jwj and this is the correct result. The other term �wÿ z�ÿ1b1�z� with1 > jwj > jzj results from b1�w�a2�z� after extending the domain of de¢nition ofb1�z�. We have used jzj; jwj < jz1j; jz1jÿ1.

The reader is asked to check the usual axioms of vertex algebras for our Fock spacerepresentation (21, 22). In particular, the translation operator T (which coincideswith the Virasoro generator Lÿ1) and the state ¢eld correspondence is inducedby a�z�; b�z� as strongly generating set of ¢elds ([4], chapter 4) together with relations

a�z�j0ijz�0 � ÿzÿ11 ; b�z�j0ijz�0 � z1; �30�

where j0i is the vacuum and z1; zÿ11 appear as Laurent monomials interpreted asstates in our Fock space. Here the vacuum is de¢ned to be the constant Laurentpolynomial. Derivatives andWick products of a�z�; b�z�, which are de¢ned algebraic-ally in vertex algebras, allow straightforward interpretation as Fock space operators.The operators a�z�; b�z� de¢ned in (21, 22) also satisfy

fa�z�; a�w�g � 0 � fb�z�; b�w�g �31�

independent of the relative position of z and w inside the unit circle. We give somedetails.

The proof of (31) is similar to the proof of (23) and is based on the equation

1zÿ z1

�� 0;

where + means `wrong boundary value' as de¢ned above. Ordering conditions on zand w are not necessary here as it should be on symmetry reasons. We mention thatthe relations (28), (29) can be expressed by using the Cauchy kernel in order to gen-erate c� from c.


The translation operator T can be identi¢ed as

Tcn�z1; . . . ; zn� �Xni�1

@

@zicn�z1; . . . ; zn� �32�

and satis¢es [4]

�T ;Y �a; z�� @

@zY �a; z�; �33�

where Y �a; z� is the general element of the vertex algebra generated by a�z� and b�z�and a is the Fock space vector

Y �a; z�j0ijz�0 � a:

In particular, using (30) we have

Y �ÿzÿ11 ; z� � a�z�; Y �z1; z� � b�z�: �34�

A more precise de¢nition of Y �a; z� will be given below after introducing the Wickproduct and the smearing out operation on a�z� and b�z�. The Wick product: a�z�b�z�: is de¢ned as

: a�z�b�z�: � a2�z�b�z� � b�z�a1�z� �35�

and it is again a densely de¢ned operator for jzj < 1. From the explicit relations (21),(22) it is clear why a1�z� cannot stand in front of b2�z� (and b1�z� in front of a2�z�).Formulas giving the Wick product through contour integrals, which are well knownin conformal quantum ¢eld theory and string theory are within our frame, rigorousequalities between densely de¢ned operators, instead of being used formally as usual.

Let us remark that the relations (26), (27) suggest the interpretation of a�z� and b�z�as densely de¢ned operator-valued analytic functionals (hyperfunctions). This prop-erty persists in all other examples of vertex algebras and is very natural if one remem-bers that elements of vertex algebras are usually de¢ned as formal Laurent serieswith operator coef¢cients. It suggests a one-dimensional Wightman quantum ¢eldtheory (cf. Section 4). In the particular case of chiral charged fermions, the denselyde¢ned smeared operators a�f �; b�f �, f 2 L2�R� can be shown to be bounded, as thiswas the case with chiral fermions in Section 2 but we refrain from giving detailsbecause this property is incidental here and is not true for the other examples thatfollow. Instead, let us discuss the nature of the singularity in a�z�; b�z� when z passesfrom the interior to the boundary of the unit circle. This problem is related tothe operator product expansion which was claimed to be under rigorous functional


analytic control. To see this, the reader can write down for z � reiy; r < 1,

a�f � � limr!1ÿ

12p

Z 2p

0a�z�f �eiy�eiydy; �36�

b�f � � limr!1ÿ

12p

Z 2p

0b�z�f �eiy�eiydy; �37�

where a�z�; b�z� are explicitly given by (21), (22) and f 2 L2�S1�. Formally butsuggestive, we write instead of (36), (37):

a�f � � 12pi

Za�z�f �z� dz; �38�

b�f � � 12pi

Zb�z�f �z� dz �39�

with integrals on the unit circle. These operator relations are understood as beingapplied to Laurent polynomials or even on semi-in¢nite Laurent series accordingto the de¢nition of the operators a�z�; b�z�; jzj < 1 and the extension of their domainof de¢nition given above. The complex conjugation of test functions which wasessential for the linear/anti-linear nature of (11), (12) was completely ignored here.This might spoil some special operator properties (like boundedness) but, asremarked above, this is not the point. What counts is a rigorous natural (linear)interpretation of a�z�; b�z� as operator boundary values (operator-valuedhyperfunctions). This interpretation is particularly rewarding when one introducesoperator product expansions. Indeed, the rigorous de¢nition of OPE takes placeinside the unit circle and segregates the expected singularity. It is in perfect agree-ment with formal physical work in string and conformal ¢eld theory. As an example,we can look at products of the form a�z�b�w� or b�z�a�w� with jzj > jwj which can beanalyzed exactly as above within the context of verifying (23). This is in contrastto ordinary quantum ¢eld theory where even in the free case, the correspondingde¢nition looks rather heavy. Indeed, in order to formulate a rigorous OPE (andde¢ne Wick products) of free ¢elds one ¢rst writes down formal expressions andonly in a second step gives a smeared out de¢nition over the Fourier transform (seefor instance, [6]).

Specializing in (38), (39) to monomial test functions we obtain operators satisfyinganticommutation relations. They have interesting representations in our Fock spacewhich can be traced back to the reproducing property of the Cauchy kernel. We usethem to give a direct de¢nition of Y �a; z� which also works for the general caseof vertex algebras by constructing ¢rst the Fock space vector a to which Y �a; z�is associated; a! Y �a; z�. Let us start with the formal expression

a�z� �Xn2Z

anzÿnÿ1 �40�


common in vertex algebras considered as formal Laurent series with operatorcoef¢cients. Within our setting, the coef¢cients an are

an � a�zn�; n 2 Z; �41�

where (41) is the particular case of (38) with f �z� � zn. By a similar formula, we de¢nethe coef¢cients bn; n 2 Z. It is interesting to remark that both an and bn; n 2 Z are notonly densely de¢ned as was the case with a�z�; b�z� but, in addition, the set of Laurentpolynomials is an invariant domain of de¢nition with respect to forming products(this was not the case with a�z�; b�w�; remember the necessity of the argumentordering). Now in order to make a long story short, in the general case we haveto consider monomial smearing (41) of the entire generating set of the vertex algebraunder consideration [4]. Arbitrary products of such operators with n < 0 applied tothe vacuum generate the Fock vector a and the correspondence a! Y �a; z� is givenby the formula (4.4.5) of [4] by means of Wick products.

Let us ¢nally remark that the operator properties of a�z�, b�z� are similar to thoseof vertex operators V�1�z� and their smeared out counterparts obtained in [5], whichwere introduced by different methods within a different framework. Indeed, bothV�1�z� exist as densely de¢ned operators for jzj < 1 but do not have a meaningfor jzjX 1. As far as the smeared versions of a; b (and V�1) is concerned [5], weare allowed to approach the unit circle from the interior. We want to stress thatEquations (21), (22) show, in a clear way, that the smeared operators a�f �, b�f �,f 2 L2�S1� involve both f� and fÿ from the Hardy decomposition f � f� � fÿ in spiteof the fact that a�z� and b�z� are de¢ned only inside the unit circle. This is a centralpoint of our approach. Certainly, the similarity between a�z� and b�z� on the oneside and V�1�z�, Vÿ1�z� on the other side is a consequence of the correspondencebetween bosons and fermions in two dimensions. It appears here in conjunction with[5] at the true operator level.

After the example of chiral fermionic vertex algebra as one-dimensional quantum¢eld theory, we proceed to other examples. The simplest one is the ~u�1� theory gen-erated by currents

J�z� � a�z�b�z�; �42�

where we have left out the Wick dots. The precise relations giving J�z� as denselyde¢ned operator in Fock space can be assembled from (21) and (22). It is clear thatthe typical square of the Cauchy kernel makes its appearance. The central statementis the locality of J�z�:

fJ�z�; J�w�g � d0z�zÿ w� � ÿd0w�zÿ w� �43�

and can be veri¢eld in the sense of operators by a direct computation. It also followsby twice applying the Dong lemma (cf. Section 4) to a�z�, b�z� and J�z�. Otherexamples include non-Abelian generalization of ~u�1� like current algebras with


currents (see, for instance, [7])

Ja�z� �XNi;j�1

ai�z�taijbj�z�; �44�

where ai; bj are independent chiral fermions and taij are transformation matrices in thede¢ning representation of su�N� such that

Trtatb � dab;Xa

taijtbkl � dildjk ÿ 1

Ndijdkl;

�ta; tb� �Xc

fabctc;Xa;b

fabcfabd � 2Ndcd

with fabc being the structure constants. The (mutual) locality of the currents Ja�z� canagain be veri¢ed by direct computation.

Finally, we remark that we can obtain from the charged (complex fermions (21),(22)) explicit representations of real fermions which, in turn, can be used for gen-erating explicit representations of bso�N� current algebras at level one or even higherlevels (see [7]).

4. Remarks and Conclusions

The aim of this Letter is twofold. First, we gave an explicit representation of chiralcharged fermions in Fock space insisting on what we called `mixed states'. Second,and more important, we modi¢ed the above (unsmeared) representation in orderto have full contact with de¢nitions, methods, and techniques in vertex algebras[4]. This is the main point of the Letter. Although we have presented only simpleexamples, it is clear that the present results can be generalized. Summarizing, wehave shown how functional analysis penetrates vertex algebras formulated asone-dimensional quantum ¢eld theory.

It would be interesting to develop our ¢ndings into a general axiomatic approachto vertex algebras. Within the above mentioned framework of one-dimensionalquantum ¢eld theory, one should start with a locally convex algebra of distributionaltest functions on the circle (Borchers algebra) on which vertex algebra elements arede¢ned as (unsmeared) operators inside the unit disc. If positivity is expected, thenthe factorization and Hilbert space completion common in Wightman theory boildown to a symmetrization property of the Borchers algebra consistent with sym-metry properties of the given vacuum expectation values. The Cauchy indicatrix,and its variants present in our examples discussed above, has to be replaced by somereproducing kernels characterizing the given vertex algebra. This remembers pro-posals in [8], Section 5, and [9], Section 8.

We want to stress the very appealing idea of a one-dimensional quantum ¢eldtheory with ¢elds being operator boundary values (operator-valued hyperfunctions),


as opposed to the standard Wightman theory where this property is reserved to vac-uum expectation values. This might have drastic consequences. For example, wemention the Dong lemma of vertex algebra which within our framework turnsout to follow from the transitivity of (mutual) locality via weak locality (i.e. com-mutativity inside expectation values). It is a simple example of the celebratedBorchers transitivity of local quantum ¢eld theory.

Another remark concerns the quality of our operators representing elements ofvertex algebras. They are densely de¢ned but might be ugly, for instance, concerningclosability (cf. [5]). But nobody would expect more from them as long as they can bemultiplied. The examples presented in this Letter, i.e. chiral charged fermions andcurrent algebras are easily understood within this framework. In fact, the programcan be extended to vertex algebras with a fermionic representation. We do not knowhow to obtain explicit representations (if any) of the type (21, 22) for general latticevertex algebras, although our feeling is that such representations could exist.

Last but not least, an explicit operator realization of the type (21), (22) and itssmeared counterpart can be used to study the C�^content of vertex algebras. Thisseems to be useful in the context of recent developement at the interface betweenstrings and non-commutative geometry; see, for instance, [10].

Acknowledgement

We thank A. Hoffmann for discussions on the subject of functional analytic study ofvertex algebras.

References

1. Sato, M., Miwa, T. and Jimbo, M.: Aspects of holonomic quantum ¢elds, In:D. Iagolnitzer (ed.), Complex Analysis, Microlocal Calculus and Relativistic QuantumTheory, Lecture Notes in Phys. 126, Springer, New York, 1980.

2. Carey, A. L. and Ruijsenaars, S. N. M.: Acta Appl.Math. 10 (1987), 1.3. Araki, H. and Wyss, W.: Helv. Phys. Acta 37 (1964), 136.4. Kac, V.: Vertex Algebras for Beginners, 2nd edn, University Lecture Series 10, Amer.

Math. Soc., Providence, RI, 1998.5. Constantinescu, F. and Scharf, G.: Comm.Math. Phys. 200 (1999), 275.6. Wightman, A. S. and GÔrding, L.: Ark. Fysik 28 (1964), 129.7. di Francesco, P. Mathieu, P. and Senechal, D.: Conformal Field Theory, Springer, New

York, 1997.8. Gaberdiel, M.R. and Goddard, P.: Axiomatic conformal ¢eld theory, hep^th/9810019.9. Borcherds, R. E.: Vertex algebras, q^alg/9706008.10. Lizzi, F. Noncommutative geometry, strings and duality, hep-th/9906122.


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Chiral Charged Fermions, One-Dimensional Quantum Field Theory and Vertex Algebras