Disponible à / Available at permalink :

- - -

- - -

Dépôt Institutionnel de l’Université libre de Bruxelles / Université libre de Bruxelles Institutional Repository

Thèse de doctorat/ PhD Thesis Citation APA:

Narganes-Quijano, F. J. (1990). Bosonization of symmetry current algebras in two-dimensional conformal field theory (Unpublished doctoral dissertation).

Université libre de Bruxelles, Faculté des sciences, Bruxelles.

Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/213143/3/27110e56-7050-4988-82dd-114104fa8ec2.txt

(English version below)

Cette thèse de doctorat a été numérisée par l’Université libre de Bruxelles. L’auteur qui s’opposerait à sa mise en ligne dans DI-fusion est invité à prendre contact avec l’Université (di-fusion@ulb.be).

Dans le cas où une version électronique native de la thèse existe, l’Université ne peut garantir que la présente version numérisée soit identique à la version électronique native, ni qu’elle soit la version officielle définitive de la thèse.

DI-fusion, le Dépôt Institutionnel de l’Université libre de Bruxelles, recueille la production scientifique de l’Université, mise à disposition en libre accès autant que possible. Les œuvres accessibles dans DI-fusion sont protégées par la législation belge relative aux droits d'auteur et aux droits voisins. Toute personne peut, sans avoir à demander l’autorisation de l’auteur ou de l’ayant-droit, à des fins d’usage privé ou à des fins d’illustration de l’enseignement ou de recherche scientifique, dans la mesure justifiée par le but non lucratif poursuivi, lire, télécharger ou reproduire sur papier ou sur tout autre support, les articles ou des fragments d’autres œuvres, disponibles dans DI-fusion, pour autant que :

Le nom des auteurs, le titre et la référence bibliographique complète soient cités;

L’identifiant unique attribué aux métadonnées dans DI-fusion (permalink) soit indiqué;

Le contenu ne soit pas modifié.

L’œuvre ne peut être stockée dans une autre base de données dans le but d’y donner accès ; l’identifiant unique (permalink) indiqué ci-dessus doit toujours être utilisé pour donner accès à l’œuvre. Toute autre utilisation non mentionnée ci-dessus nécessite l’autorisation de l’auteur de l’œuvre ou de l’ayant droit.

--- English Version ---

This Ph.D. thesis has been digitized by Université libre de Bruxelles. The author who would disagree on its online availability in DI-fusion is invited to contact the University (di-fusion@ulb.be).

If a native electronic version of the thesis exists, the University can guarantee neither that the present digitized version is identical to the native electronic version, nor that it is the definitive official version of the thesis.

DI-fusion is the Institutional Repository of Université libre de Bruxelles; it collects the research output of the University, available on open access as much as possible. The works included in DI-fusion are protected by the Belgian legislation relating to authors’ rights and neighbouring rights.

Any user may, without prior permission from the authors or copyright owners, for private usage or for educational or scientific research purposes, to the extent justified by the non-profit activity, read, download or reproduce on paper or on any other media, the articles or fragments of other works, available in DI-fusion, provided:

The authors, title and full bibliographic details are credited in any copy;

The unique identifier (permalink) for the original metadata page in DI-fusion is indicated;

The content is not changed in any way.

It is not permitted to store the work in another database in order to provide access to it; the unique identifier (permalink) indicated above must always be used to provide access to the work. Any other use not mentioned above requires the authors’ or copyright owners’ permission.

UNIVERSITE LIBRE DE BRUXELLES

FACULTE DES SCIENCES

L’épreuve publique pour l'obtention du grade légal de Docteur en Science de Monsieur NARGANES-QUI J AN0, Francisco, aura lieu le :

VENDREDI 23 NOVEMBRE 1990 A 14.30 HEURES en la Salle DF.2058, Forum, Campus de la Plaine,

Boulevard du Triomphe à 1050 - Bruxelles.

Monsieur NARGANES - QU IJAND présentera et défendra publiquement une dissertation originale intitulée :

"BosOnizatiOn of symmetry current algebras in two-dimensiona 1 conformai field theory”;

et une thèse annexe intitulée :

"Il est possible de résoudre algébriquement l'oscillateur quan­

tique anharmonique dans le cadre des algèbres universelles enve­

loppantes q-déformées".

Directeur de thèse : M. F. ENGLERT.

UNIVERSITÉ LIBRE DE BRUXELLES Faculté des Sciences

Service de Physique Théorique

BOSONIZATION OF

SYMMETRY CURRENT ALGEBRAS IN TWO-DIMENSIONAL

CONFORMAL FIELD THEORY.

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences.

(Grade Légal)

Francisco J. Narganes-Quijano

1990

Ithaque t’a donné un beau voyage.

Sans elle, tu ne te serais pas mis en route.

Elle n’a plus rien d’autre à te donner.

Même si tu la trouves pauvre, Ithaque ne t’a pas trompé.

Sage comme tu l’es devenu à la suite de tant d’expériences, tu as enfin compris

ce que signifient les Ithaques.

Extrait du poème Ithaque de K. Kavafis (pour en savoir plus, voir réf.[l]).

Un voyage peut en devenir un autre, une Odyssée ou un voyage au bout de la nuit;

je tiens à exprimer ici

MES VIFS REMERCIEMENTS à tous ceux qui ont contribué à ce que ce projet soit arrivé a bon port:

à Monsieur le Professeur François Englert, promoteur de cet entreprise, pour m’avoir tracé le chemin;

à Robertus Potting, Laurent Houart et José Gaite, pour les moments de discussion qu’ils m’ont consacrés et l’encouragement qu’ils m’ont prodigué;

à tous ceux qui m’ont aidé, à tous ceux qui m’ont accompagné tout au long de ce chemin: Christiane, Christine, Corinne, Jean, l’autre Jean, Jean François, Jean Louis, Jean Marie, Marc, Philippe, Pietro, Renaud et Robert;

à l’Institut Interuniversitaire des Sciences Nucléaires et au Gobierno Vasco/Eusko Jaurlaritza pour son aide financière;

à Walburga pour m’avoir initié (entre autres choses) à Kavafis, et à Kavafis pour

m’avoir appris ce que signifient les Ithaques.

UNIVERSITÉ LIBRE DE BRUXELLES Faculté des Sciences

Service de Physique Théorique

THÈSE ANNEXE;

Il est possible de résoudre algébriquement l’oscillateur quantique anharmonique

dans le cadre des algèbres universelles enveloppantes q-déformées.

Francisco J. Narganes-Quijano

Novembre 1990.

UNIVERSITÉ LIBRE DE BRUXELLES Service de Physique Théorique

BOSONIZATION OF SYMMETRY CURRENT ALGEBRAS IN TWO-DIMENSIONAL CONFORMAL FIELD THEORY.

by Francisco J. Narganes-Quijano

Director of the Thesis: Professeur François Englert November 1990

ABSTRACT

Following a coset-inspired construction, a représentation in tenus of free bosonic fields of the symmetry currents of a large number of two-dimensional extended conformai models is obtained. In particular, the Z;v, D3 and D4 paraferaiionic ciirrent algebras are bosonized, as well as other conformai cunent algebras closely related to them; the

= \ and Af = 2 superconformai algebras, the affine SU n {^) Kac-Moody algebra and, as a by-product, the cunent algebras defining the parafermionic models.

By making use of the corresponding représentation, it is shown that the Z a ' parafer­

mionic descendants of the identity define, up to null-currents, a BV-algebra. The hidden connection between the [Z;v] parafermionic models and the [W at ]^"^''’^ non-linearly extended conformai models is therefore exhibited in a manifest way. The essential features of this ‘parafermionic’ algebra are discussed, as well as the characterization of the parafermionic primaries as W^ a ^-invariant fields. Remarkably, this IV'A?-algebra can be represented in terms of only two free bosonic fields. This représentation tums out to interpolate between the second représentation of the IVa-algebra given by Fateev and Zamolodchikov, and the représentation of the PRS TToo-algebra obtained by Bakas and Kiritsis.

Within the same framework, two inequivalent bosonic représentations of the BRST sym­

metry current associated to the NSR superstring are obtained, and the respective local BRST cohomologies are analyzed. While the ‘covariant’ bosonization gives rise to a doubling of the BRST cohomology in the extended bosonized field space, the ‘standard’

bosonization enhances such a cohomology to a trivial one. The picture changing opera­

tion in the BRST cohomology of the minimal field space emerges as a residual effect of

such a triviality.

CONTENTS:

Acknowledgements...ii

Abstract... iii

1. Introduction...1

2. Conformai Field Theory... 9

2.1. The Conformai Group ... 9

2.2. Conformally Invariant Field Théories... 11

2.3. Radial Quantization... 14

2.4. Ward Identities and Operator Product Expansions... 16

2.5. The Virasoro Algebra... 18

2.6. Null Vectors and Minimal Models ... 25

2.7. The Free Boson... 27

2.8. The Dotsenko-Fateev Approach... 30

3. Extended Conformai Field Theory... 34

3.1. Conformai Models with Extended Symmetry ... 34

3.2. Extended Conformai Current Algebras... 35

3.3. Affine Kac-Moody Algebras... 44

3.4. The Coset or GKO Construction... 47

4. Bosonic Représentation of Conformai Current Algebras... 52

4.1. A Coset-Inspired Bosonic Construction... 52

4.2. Our Starting Bosonic System... 54

5. The Parafermionic Conformai Models 59

5.1. Introduction... 59

5.2. The Z;v Parafermionic Current Algebra... 60

5.3. Current Algebras Related to Z;v Parafermions... 69

5.4. The D/v Parafermionic Current Algebra... 73

5.5. Spécial Cases of the Construction ... 78

6. The Conformai iy;\/-Algebras ... 80

6.1. Introduction... 80

6.2. H'^/vf-Algebras and Non-linearly Extended Conformai Models... 82

6.3. The Parafermionic Algebra... 87

6.4. The Conformai Models ... 97

6.5. The N —* oc Limit of the Parafermionic W’Ar-Algebra... 100

7. The (6c,/?7)-Systems and the Superstring BRST Current...106

7.1. The First-order Superconformai or (6c,/?7)-Systems ... 106

7.2. The BRST Symmetry Current in the NSR Superstring...113

7.3. BRST Cohomology in Superstring Theory... 121

7.4. The PCO as an “Adjunction of Variable” Mechanism... 128

7.5. The Covaxiant Bosonization...135

8. Conclusions and Comments...140

Appendix A: Basic Conformai Calculus ... 144

Appendix B: Wick’s Theorems...150

References...155

1. INTRODUCTION.

There are essentially two facts that make two dimensions very spécial in physics.

One is the “bursting” of symmetry due to the fact that the local conformai group in two dimensions is infinité dimensional [2]. Another is the complété équivalence of bosons cUîd fermions [3]. These two features hâve motivated in the last few years a lot of interest and intensive work in the development of two-dimensional conformai field theory (CFT), in view of its applications in the characterization of two-dimensional statistical Systems and string theory. Another reason of this interest certainly is the beautiful mathematical structure exhibited by two-dimensional CFT, which shows once more the deep connections between mathematics and physics.

Conformally invariant quantum field théories describe the long range behavior of Systems which are undergoing second-order phase transitions [4], the standard example being the Ising model. In an arbitrary number of dimensions the conformai invariance of the critical theory does not turn out to give much more information than ordinary scale invariance, and it is not particularly useful for studying the relations between physical quantities. In two dimensions, however, the situation is quite different. Since any cinalytic function mapping the complex plane onto itself is conformai, in two dimensions the group of conformai transformations is exceptionally larger than in any other number of dimensions, and its représentation theory restricts considerably the possible types of two-dimensional critical Systems. Conformally invariant two-dimensional quantum field theory then is the theory underlying two-dimensional second order phase transitions, and it has been argued that CFT could ultimately give a classification of ail possible critical phenomena in two dimensions.

Two-dimensional conformally invariant field théories are also essential in string the­

ory [5-6], which has been considered in the last few years as a good candidate for what heis been called the “Theory of Everything”. It appears that the parameter space de- scribing the world history of a string (the ‘world sheet’) is two-dimensional, and the dynamical variable of string theory constitutes a two-dimensional quantum field theory.

The physical content of this field theory has to be independent of the the metric cho-

sen in the description of the world sheet. There is, however, a conformai factor which

usually affects the physics. Requiring that this effect vanishes is simply the statement

that the quantum field theory is Weyl invariant, t.e. it is a conformally invariant the-

ory. ConformaJ invariance puts restrictions on the allowed dimensions and the internai degrees of freedom. The classification of conformally invariant two-dimensional quan­

tum field théories would moreover provide useful information on the classicad solution space of string theory eind could lead to a deeper imderstanding of the nonperturbative aspects of strings [7-8].

To achieve the program of ‘solving’ CFT eind classifying its solutions, Belavin, Polyahov and Zamolodchikov (BPZ) hâve revived the ‘conformai bootstrap’ program of Polyakov [9] in the two-dimensional case. Indeed, the idea of constructing solutions of CFT by combining the requirement of conformai invariance with the Ansatz that the operator fields form em algebra tums out to be feasible in two-dimensions [2j. This is due to the fact that the operator field algebra organizes into représentations of two commuting copies of the Virasoro algebra Vir — the infini te-dimensional Lie algebra defined by the two-dimensional conformai group, and which admits a central exten­

sion characterized by a free parameter c. In Ref.[2], BPZ succeeded in constructing an infinité set of exactly solvable ‘minimal’ models (related to strongly degenerated repré­

sentations of the Virasoro algebra) in which the space of operator fields organizes into a finite number of irreducible représentations. Such minimal models are described by a central charge

c = 1 — 6

(m + l)(m -I- 2) (1.1)

for rational values of m > 1. Unitarity, which puts severe constraints on the allowed physical Systems, restricts further m to take integer values [10-11].

The property of finite décomposition makes the study of the corresponding field thé­

ories much easier. Among the tools used in carrying out the analysis of these minimal models, the most powerful is the Coulomb-gas approach. The basic idea is to repre- sent the irreducible représentations of Vir via free boson fields. Feigin and Fuchs [12]

showed how to reconstruct the Vir minimal models (1.1) with such a représentation,

and Dotsenko and Fateev [13] then developed the Coulomb-gas approach, introducing

screening operators to compute multipoint correlators on the pleine. This approach has

enabled in particular the characterization of the operator field content, as well as the

computation of corrélation functions and the structure constants of the operator alge-

bra. The method présents moreover the advantage of allowing a generalization to higher genus surfaces [14-15].

This procedure of ‘bosonization’ constitutes certainly a powerful method in two- dimensional CFT. Besides providing a method of computing corrélation functions and anaJyzing partition functions, it allows for an explicit représentation of the operator field algebra defining the corresponding conformai model in terms of vertex operators as simple functions of free bosons. Since it was noticed [3] that soliton operators of a bosonic theory behave as fermions, such a bosonization procedure has become a general technique in two-dimensional théories. In particular it has been applied to fermion théories on any Riemann surface [16-17] and to Wess-Zumino-Witten models [18] at level one [19-20], and it has proved extremely useful for understanding the covariant description of superstrings [7] and in proving the équivalence of the Green-Schwarz and the Neveu-Schwarz-Ramond (NSR) superstrings [21].

The minimal sériés (1.1) certainly does not exhaust the class of ail solutions of CFT in two dimensions. Conformai models described by values of c > 1 are unitary [10].

They involve, however, an infinité number of Vir représentations [22], and the model is not tractable with the BP Z method. It has been argued that, in order to provide minimal conformai models associated to values of the central charge c > 1, the conformai symmetry has to be extended with additional symmetries. The combination of an additional symmetry with the conformai one gives rise to a larger infinité dimensional algebra. The Virasoro algebra Vir then is promoted to a bigger Ext algebra which contains Vir as a subalgebra. It happens that the infinité number of représentations of Vir of some conformally invariant models having c > 1 can be classified into a finite number of représentations of some extended algebra Ext

Vir. Such Ext-invariant models then become minimal with respect to Ext and therefore exactly solvable.

Early examples of such Ext aJgebras are just the affine Kac-Moody algebras [23]

and the Af = 1 and M = 2 superconformai algebras [24-26] which provide respectively conformally invariant solutions of the WZW models [18] and exactly solvable models with superconformai invariance [27-28].

The research for Ext-conformally invariant models is also motivated by the coset

construction of Goddard, Kent and Olive (GKO) [il]. It is possible to associate to every

Lie group G an affine Kac-Moody algebra which generates unitary représentations of Vir via the Sugawara construction. The coset space construction amounts to define a new conformai model associated to the coset GfH, H being a subgroup of G. This procedure gives rise to a systematic construction of a huge class of unitary models with c > 1 which 2ire exactly solvable. Since the method is not very practical for calculating corrélation functions and representing the space of invariant operator fields [29], one needs to identify the Ext symmetry algebra underlying such solvable models.

Consequently, attention has moved in the last few years to extensions of the con­

formai symmetry, i.e. to constructing and characterizing infinite-dimensional algebreis conteûning the Virasoro algebra

a subalgebra. There axe different approaches to characterize new extensions of the conformai symmetry. The most relevant one is the method of Zamolodchikov [30], who constructs symmetry current algebras obeying two ba^ic consistency requirements: closure of the current algebra under the operator prod- uct and associativity of the operator product — equivaJently, Crossing symmetry of the corrélation functions involving the symmetry currents [2].

Two of the new Ext current algebras hâve attracted spécial attention. On the one hand, the parafermionic current algebras with an internai symmetry, which give rise to models generalizing the critical [Z2] Ising model [31-33]; on the other hand, the Wyv-algebras, which constitute the symmetry current algebras defining the ‘non-linearly’

extended conformai models [W’ at ] [34-35].

There axe essentially two kinds of parafermionic current algebras, usually denoted by Z N and D/v. Both are constituted by nonlocal currents with fractional spins, and they generate indeed Vir as a subalgebra. The most relevant one is the Z^' current algebra, which turn out to be closely related to the N = 2 superconformai algebra [33]

and the SU{2) Kac-Moody algebra at level N [31]. In fact, it constitutes a sort of building block for larger classes of interesting théories. By combining Zj^ parafermions with boson fields with suitable background charge, most known rational conformally invariant field théories are generated [36].

The TV;v-algebra originates from a straightforward generalization of Vir. The (right

and left) Virasoro algebra is essentially the algebra generated by the modes of the

(holomorphic and antiholomorphic components of the) stress tensor {T{z),T{z)), a

tensor of conformai dimension two. The Ty^f-algebra is the symmetry current aJgebra generated by a set of currents conformai dimension n, where W2(z) = T{z). The closure of the operator product algebra turns ont to be realized in a non-linear sense, i.e. it generates normeil ordered products of the currents and their dérivatives.

(In terms of the modes of the currents, the commutator algebra closes in the universal enveloping aJgebra.) The représentation theory of the ly^v-algebra is very similar to that of Vir. In spite of the complexity of such algebras, the conformai models invariant under such symmetry current algebra hâve been constructed and analyzed, and they exhibit a link with the SU (N) weight lattice.

Apart from its intrinsic mathematical interest, the relevance of these Ty>/-algebras goes beyond its applications to CFT. Their rich and peculiar algebraic structure hâve inspired a lot of work in the last two years in many areas of physics: they serve as pro­

totypes of gauge théories for non-linear algebras (‘IF-gravity’) [37-38], their structure might elucidate the géométrie interprétation of higher spin field théories (‘VT-geometry’) [39], and the limit case W^o exhibits connections with the area-preserving diffeomor- phism algebra [40] and membrane théories [41].

Motivated by the strong calculational power of the Dotsenko-Fateev approach, it would be useful to describe different classes of models with extended conformai in­

variance within a unified Coulomb-gas représentation. By making use of the fact that suitable combinations of free bosonic fields can describe explicitly a large class of non- trivial conformai models, we présent in this thesis a systematic bosonization of a large class of Ext symmetiy' current algebras. By bosonization we mean here a représentation in terms of free bosonic fields, i.e. fields whose oscillator modes satisfy a Heisenberg- type algebra and that define a conformai System with stress tensor at most quadratic in the fields.

To this end, we introduce a particular bosonic System constituted of three free boson fields, and characterized by a unique free parameter u — the background cheirge of one of the bosonic components. In spite of this restriction, and following a method inspired by the coset construction, we eventually succeed in representing a large class of symmetry current algebras in terms of this bosonic System or some of its bosonic subsystems.

In particular we obtain a vertex représentation of the 2 in parafermionic algebra

for generic values of iV, as well as of the D3 and D4 parafermionic aJgebras. This last limitation is due to the fact that the number of bosonic fields présent in om initial bosonic System is not enough to consider the aJgebras D^r for N > 4, zmd it illustrâtes a simple ‘boson counting rule’ that we dérivé in Section 4.2. Such bosonic realizations will provide us with représentations not only of the symmetry current £ilgebras intimately related to the Z n and D4 parafermions (the SU{2) n Kac-Moody aJgebra, the Af = l and 2 superconformaJ aJgebras), but aJso of other spécial cases (the current algebras of the models and realized respectively as the parafermionic

subeilgebra md the x Z a / algebra).

The bosonic représentation of the Z/v parafermionic current algebra allow us to show explicitly the hidden connection between the [Z a '] parafermionic models and the non-linearly extended conformai models. To this end, we show that the Z A/ parafermionic descendants of the identity can be identified with a set of currents defining a hona. fide W'A'-algebra. Such an identification will be carried out modulo a family of null-currents which constitutes an idéal in the space of fields of the theory, and consequently can be consistently removed from the space of fields. We discuss the essentiel features of this ‘parafermionic’ H^/v-algebra, as well as the chciracterization of the parafermionic space of fields as WAf-invariant fields. Remarkably, this W/v-aJgebra is represented, by construction and for generic values of N, in terms of only two free bosonic fields. Contrarily to other realizations of the W a ?-algebra, the structure of our two-boson realization will reveal itself more appropiate in analyzing the N 00 behavior of the WA^-algebra. To make contact with other représentations known in the literature, we will show that our représentation reproduces, as particular cases, the second représentation of the Wa-aJgebra given by Fateev and Zamolodchikov in Ref.[34], emd the représentation of the PRS Woo-algebra [42] obtained by Beikas and Kiritsis in Ref.[43j.

Our method aJso provide us with two different représentations of the first-order superconformai Systems, or (6c,/S7)-systems. These two représentations, which turn out to correspond to the well-known standard bosonization [7] and to the ‘covariant’

bosonization [44], give rise to two different représentations of the BRST symmetry cur­

rent associated to the covaxiant formulation of the NSR superstring. Though équivalent

in describing the BRST cohomology of the superstring, we show that both bosoniza-

tions behave in a quite different but complementary way. In particular, the standeu-d bosonization of the BRST current give rise to an enlaxged field space in which the local BRST cohomology defined by the BRST current becomes trivial. The picture changing operation (PCO) in the initial field space will emerge in a quite natural way as a residual effect of this triviality. The covariant bosonization, however, will induce a ‘doubling’ of the BRST cohomology in the enlarged space of fields.

In this Work we restrict our attention to the vertex représentation of the symmetry currents and the invariant operator fields. We will not of course be able to give a com­

plété coverage of ail the important topics related to the conformai models considered here. In particular, we will not touch on the important topic of modular invariance.

We give here the Dotsenko-Fateev screening operators, but we omit here examples of the computation of corrélation fimctions on the sphere that can be carried out straight- forwardly following the standard methods (see Refs. [13,45]). Since CFT splits into the holomorphic and the antiholomorphic sectors that are isomorphic, we restrict our con­

sidérations, as it is usual, to the holomorphic sector.

The thesis is organized as follows: In Chapter 2, after considering briefly the confor­

mai group in D dimensions, we shall consider the constraints imposed by the conformai invariance in two-dimensions. For the sahe of completeness, we review the fundamen- ted features characterizing two-dimensional CFT as well as the spécifie techniques used in analyzing the conformai models. In Chapter 3, after discussing the motivations in extending the conformai symmetry, we give an overview on Ext-symmetry current al- gebras and the characterization of the corresponding extended conformai models. (The standard reference is [2j; some review articles are [23,46-50].)

In Chapter 4 we characterize our basic bosonic System and its space of vertex fields,

£ind we introduce a coset-inspired bosonization procedure that we are going to follow

eilong this work. After an introduction to the [Z;v] and [D a >] paxafermionic conformai

models, in Chapter 5 we bosonize the parafermionic currents algebras D3 and D4,

as well as other related conformai algebras. For the sake of completeness and to illustrate

the method, we rederive some of the results that are known in the literature. We give

also the bosonic représentation of the ZAr-primary fields, as well as the Dotsenko-Fateev

screening operators corresponding to our représentation.

In Chapter 6, we begin by introducing the essentiaJ features characterizing the W/vr- algebras and the non-linearly extended conformai models We introduce the concept of nuU-currents in a Wyv-algebra, and we shall show that the Z;v parafermionic currents generate a set currents that defines, up to null-currents, a weU-defined aJgebra. We shall also characterize the space of fields of this parafermionic WA/-algebra in what we shall call here the parafermionic basis, and consider the correspondence between the invariant field spaces of the models [Z a ?] and Particular at­

tention will be payed to the N ex limit of our W;v-algebra, which will reproduce the essential properties of the PRS Wœ-algebra.

In Chapter 7 we introduce the general features of the (6c, ^7)-systems, and in par­

ticular of the BRST superconformai ghosts associated to the formulation of the NSR superstring. We shall analyze the structure of the local BRST cohomology associated to the BRST current in the two different bosonization prescriptions, shedding new light into the compréhension of the peculiarities associated to the superconformai ghosts. To this end, we shall use here the powerful tools of algebraic cohomology, and incidentally we shall illustrate some of its fundamental theorems.

In Chapter 8 we présent our conclusions together with some comments.

Finally, in Appendix A we give a small overview on the technicalities of the conformai calculus, and in Appendix B we dérivé a useful “Wick’s theorem” which will constitute the basic ingrédient in characterizing the parafermionic WAf-algebra. Such a theorem will also provide us with Jaeobi-like identifies. These prove useful in the study of the constraints imposed by the associativity condition of the operator product expansion.

We shall use them extensively along this work.

2. CONFORMAL FIELD THEORY.

2.1. The Conformai Group.

Let ds^ = g^^dx^dx'' be the line élément of the space with fiat metric

of signature (p,q)- Under a change of coordinates x*‘ -* x'^{x‘'), transforms els

9nv —* = {dx°/dx'>^){dx^/dx'^') ga0{x). The conformai group is defined to be the subgroup of coordinate transformations that leave the metric invariant up to a scale change:

9'^À=^') = 9tiM (2-1)

and consequently they preserve the angle a ■ between two vectors a and b.

Taking infinitésimal coordinate transformations + e^(x‘'), it is easy to see that (2.1) is satisfied if

+ = ^(de)r}^^. (2.2)

Thus fl(x) = 1 + (2/£>)5^e'^(x), and from (2.2) it follows that

[v^uid ■d) + {d- 2)d^d,] (a • e) = 0. (2.3)

In any number of dimensions D > 2, équations (2.2)-(2.3) require the function e(x") to be at most quadratic in x. For e ~ (x)° we hâve the translations (e'' = a^); for e ~ (x)^ we hâve the rotations antisymmetric) and the scale transformations (e^ = Ax**); and for c ~ (x)^ we hâve the spécial transformations (e^ = b^x^ -2x^{b-x)) which axe équivalent to the product of two inversions and a translation. The generators eire a''5^, A(x • d) and 6''[x^a^ — 2x^^(x • a)] respectively, and they constitute an algebra locally isomorphie to50(p + l,g + l).

The fini te conformai transformations contain the Poincaré group (composed of trans­

lations X —> x' = X -f a and Lorentz transformations x —> x' = A • x, G SO{p,q), for which n = 1), the dilatations (x —> x' = Ax, for which fl = A~^) and the spécial confor­

mai transformations (x —> x' = (x-f6x^)/(l-t-2fex-|-6^x^), with fl(x) = [1 -|-26x-|-6^x^]^).

In twodimensional Euclidean space, (2.2) translates into the Cauchy-Füemann con­

ditions

d\t\ = 82^2 , d\t2 = —d2^\- (2-4)

In complex coordinates (2,2) = ± ix^ (2.4) implies that the combinations (e,ë) =

± dépend only on 2 and z respectively. Thus, the two dimensional conformai transformations coincide with the analytic coordinate transformations 2 —> f{z), z —>

/(f), the aJgebra of which is infinité dimensional. In this case ds^ = dz dz Qdzdz, where Q, =[ df /dz p. For infinitésimal transformations, it is convenient to take the basis

z z' = z — , Z ^ z' = Z — ïnZ'^'^^ (n £ Z) (2.5) where are in general complex parameters. The corresponding generators are

^jn + ë„/„, where

= = ( 2 . 6 )

which satisfy the algebra

[^ni ^m] — (^ + n i — (^ ^)^m-|-n i [^ni^m] 0- (2-7)

Since the /n’s commute with the /m’s, the local conformai algebra is the direct sum of two isomorphic subalgebras. To take advantage of this fact, it is useful to consider 2 and 2 as independent coordinates (in other words, to take (x^,x^) 6 instead of R^). This will aJlow us to study the holomorphic and the antiholomorphic sectors independently. The Euclidean plane and the Minkowski space-time are just real sections of this extended complex space, and to reconstruct the original theory it will be enough to impose the

‘physical’ condition 2 = 2* by combining in an appropiate way both sectors.

Note that the unique generators {InJn} of the local conformai algebra that are globally well-defined on the Riemann sphere 5^ = C Uoo are those for which n = 0, ±1.

Such operators define the global conformai group in two dimensions**: the operators

h The distinction between local and global conformai groups only emerges in two dimensions, for

in higher dimensions there exists only the global one.

(/_i, /_i), lo + /o) *(^o ~ ^o) aJîd (/i, 11) are respectively the generators of translations, di­

latations, rotations and spécial conformai transformations. They are the only invertible conformai transformations on the z-pleme and they generate the group SL(2,C)/Z2 of projective conformai transformations

az + b cz + d

az b

cz + d (a, b,c,d£ C, ad — bc = 1). (2.8) This group will chaxacterize some of the physical properties of States in a quantum realization of (2.7). Note that, since / q -|- / q and i{lo — I q ) hâve been identified with the generators of dilatations and rotations respectively, the (/ q , ?o)-eigenvalues {h, h) of a State \h,h') of scale dimension A and spin s will satisfy the relations A = h + h, s = h — h.

2.2. Conformally Invariant Field Théories.

In an arbitrary number of dimensions D, the — symmetric and in general divergence- free — stress-energy tensor of a quantum field theory can be defined as the response of the System to an external gravitational field:

6S

6gi^‘'{x) (2.9)

For an arbitrary local coordinate transformation + V^(x'^), gives us the associated generator current J^IV^]

j da^ = J da>^ ^{( 2 . 10 )}

and is conserved if = 0. In particular, the generators of translations and rotations are

pu = /.-xT«^ , = (2,„)

respectively. In this context, the conformai group introduced in the last section can be

equivalently defined as the subgroup of coordinate transformations which are conserved

if the stress tensor is traceless. Indeed, the conservation of the dilatation current 7^ =

implies = 0, and the rest of currents (with V'' satisfying

(2.2)) are also conserved if is traceless.

In two dimensions, and using complex coordinates = gu = 0, = g-^^ = 1/2), we hâve (Tjj,Tiî) = j(Too T 2iTio - Tu) and T^i = = -j(roo + Tu) = \T>^. From the conservation law g°^daTpi, = 0 and the traceless condition it follows that

d-J,,=d,Tu = 0 , r,, = 0 (2.12)

which implies factorization into holomorphic and anti-holomorphic sectors. Now, any aneilytic and antianalytic Vj constitute a solution to (2.4), and we recover the resuit of the previous section. Thus, in two dimensions every local quantum field theory with a traceless tensor has an infinité number of conformai symmetries.

In order to desbribe a field theory with conformai invariance, we assume the foUowing natural and immédiate properties:

— There is a vacuum |0) invariant under the global conformai group.

— There is a (in general infinité) set of fields {^4^}, labeled by (a set of) indices j, which contains the identity operator 1 and the dérivatives of ail the fields involved.

— In {Aj] there is a subfamily of fields called ‘quasi-primary’, which transform as

(2.13) under global conformai transformations x —» i', where Q =\dx'/dx and Aj is the (anomalous scale) dimension of (f>y The theory is said to be covariant under (2.13) if the corrélation functions (or vacuum expectation values of time-ordered fields) satisfy

... (2.14)

— Every field in {Aj} can be expressed in terms of the quasi-primary fields {(f),} and their dérivatives, in a way that will become clear in the following.

Conformai invariance imposes constraints on the A^-point functions of a generic

quantum field theory in D dimensions. In particular, the covariant property (2.14)

implies that the TV-point functions dépend on invariants of the conformai group in D

dimensions [4]. Thus, translation invariance implies that a generic N-point function dépends only on the D{N — 1) quantities (i, — Xj), rotational invariance implies (for spinless objects) dependence on the N{N — l)/2 quantities r,j =| i, — xj |, and scale invariance implies dependence only on the ratios rijfrid. It can be shown that spécial conformai transformations constrain further the A^-point fimction to dépend only on the N {N — 3)/2 [46] independent cross-ratios rijrki/r,kT"ji. As a conséquence of such state- ments, it follows that 2,3 and 4-point functions of quasi-primary fields in a conformai field theory are given by

{<f>l{Xi)<f>

{x

)) = (5 a ,,A, (2.15)

12

{(f>l{Xi)(l>2{x2)(l>3{X3)) = Cl 23

Ai-fA2—As^Aj-fAs —Ai^As-fA,—Aj

23

12

' 31

— rf

^12^34 ^12^34 \ TT )-t-^ .A,/3

^13^24 r23r4i ; n «O

where the coefRcients C12 and C123 dépend on the normalization of the fields and F{a, b) is an unfixed function of two independent cross-ratios. Thus, for TV > 4 conformai invariance is not enough to fully détermine the dependence of the A^-point functions of a conformally invariant field theory in D > 3 dimensions.

In two dimensions, the transformation law (2.13) can be written as

(2.16)

which States that <j}j{z,z)dz"dz^ is in\ariant, i.e. ^,(2, 2) is a tensor with h lower 2-indices and h lower f-indices. The field <f>j{z,z) is said to be a primary field of conformai weight {h, h) — and it is automatically a quasi-primary (or 5L(2, C) primary) field. Under infinitésimal transformations z z e(z), z z ë{z)

6((f>j{z,z)= (^{hde + ed) + {JiBê + êd)^(i>j{z,z) (2-17)

where h, h are real valued and d = di.

Now the A^-point functions Zi) = (rin=i 4>n{^ni^n)) are supposed to satisfy the infinitésimal form of (2.14). This gives differential équations, the solutions of which are

G^^\zi,z,) =

C\2

-2h^2h Zj2 Zi2

(2.18)

U23

hi+h^ — hs h^ + hz — hi /13+/11 — Aj-hi + Aj —/13 -hj + hs — hi ^^hs+hi—h^

'12 =•23

‘31

\213224 Z iz Z2 a J

where 2,j = Zi — zj. Taking A, = /i,- + hi, h{ = hi (spinless partiales) and r,y =| 2,j |, we recover (2.15). The difFerential équations do not fixed, for the time being, the function F(a,b). We shall see, however, that under some conditions the function F{a,b), and in general every two-dimensional A^-point function, can be completely determined (at least in principle) using the techniques of two-dimensional conformai field theory.

2.3. Radial Quantization.

The most useful operator interprétation for a two dimensional conformally invariant quantum field theory is ‘radial quantization’, in which the rôle of time ordering is played by radial ordering. Such a procedure for defining a quantum theory on the plane will allow us the use of the powerful complex calculus to analyze two dimensional CFT.

Consider fiat Euclidean space and time coordinates a and r. The light-cone coordi- nates in Minkowski space would be t ± ct , and the cinalogs in Euclidean space are r±icr.

The left- and right-moving character of massless fields in two dimensional Minkowski space become purely holomorphic and antiholomorphic dependence on the coordinates in the Euclidean context. Compactification of the space coordinate a = a 2n defines a cylinder in (cr, r) coordinates. Now, the transformation z = maps conformally the cylinder into the complex 2-plane: equal time is now equal radius, the infinité past (future) becomes 2 = 0 (2 = 00), time reversai r —♦ —r becomes 2 —* \/z* and intégrais over equal time planes become contour intégrais around circles of fixed radius. Prop­

agation in T (time translations) becomes dilatation of 2, and time ordering becomes

radial ordering. Note also that in this description the dilatation generator on the plane

(^0

+ ^o) plays the rôle of Hamiltonian of the System, and the Hilbert space is built up on surfaces of constant radius.

The infinitésimal symmetry variation of a field <p is given by 6i4> = where Q = J dx jo is the intégrai over a fixed-time slice of the 0-component of the associated symmetry current j^. Now, T{z) = Tzz and T{z) = Ta are the generators of the anaJytic (z —> /(z)) and antianalytic (2 f{z)) coordinate transformations, respec- tively. Henceforth, the infinitésimal local conformai transformation of a field 4>{w,w) is the equal-time (constant-radius) commutator with the charge corresponding to the energy-momentum tensor*”

6^4>{w, w) = ^ dz e{z)[T{z), (f>{w, tD)] + dz ë(f)[T(f), li))] (2.19)

where the counter-clockwise line intégral is performed over some circle of fixed radius.

Since the product of two fields A{z)B{w) in Euclidean space radial quantization are only defined for | 2 |>| [, it is useful to introduce the radial ordering operator R\

R(A(z)B(w)) A{z)B{w) if I 2 |>| w I

eB{w)A{z) if I 2 |<| w \ (2.20) where e = +/— depending on the bosonic/fermionic character of the fields. Taking into account now the former prescriptions, equal time commutator of an operator A with the spatial intégral of an operator B translates into the contour intégral of their radially ordered product;

J ^dxB,A

If B is an analytic operator, then the contour can be distorted at will. In a corrélation function, however, we hâve to pick up the singularities when the position of B coincides with that of some other operator. To see how (2.21) works, note the contour intégral équivalence lbl>l«i’l ~ §Co^^ l|‘l<l“>l~ §C equal-time commutator (2.19)

t) The line intégrais contain implicitly the factor \j2vi.

pmia_î timp

j dz R(^B{z)A{w)y (2.21)

becomes now 6((f>{w, w)

U|>|u,| U|<|u;|

= ^ {dz t{z)R{T{z)<p{w, w)) + dz e{z)R{T{z)(j)[w, • (2.22)

Comparing (2.22) with (2.17), it follows that the sort distance behavior of T{z) and a primary field (f>(w,w) is given by

R{T{z)(f){w,w)) = R{f{z)4>{w,w)) =

{z — wy [z — w) Hjiv^w) ^ dü,4>{iv,w) ^

{z — wy (z — w)

(2.23)

where 0(1) stands for regular ternis in the limit z w. (2.23) defines the notion of conformai primary field which transforms like a tensor of weight (h, h) under conformai transformations, and encodes the conformai transformation properties of (f>{w,w). In the regular part of (2.23), the primary field generates an infinité set of new local fields called descendant or secondary fields. Note that a generic field of dimension h — in particular a descendant field or a dérivative of a primary field — would generate in (2.23) higher pôles than the double pôle singularity.

2.4. Ward Identifies and Operator Product Expansions.

Ward identities are relations satisfied by corrélation functions as a conséquence of symmetries of the theory. In the case considered here, (2.14) is the fundamental Ward identity of a theory with (global) conformai symmetry. In two dimensions, and taking into account (2.14), (2.22) and (2.23), it can be seen that (local) conformai invariance implies also the following conformai Ward identity for a generic corrélation function of primary fields:

N N N

{T(z)n =Y.{ y ,+(jT^)(ii

j=l

i=] '

ji=]

(2.24)

Eqn.(2.24) states that corrélation functions are meromorphic fimctions of z with sin-

gularities at the positions of the fields <f>j and residues determined by the conformai

properties of these fields. It is straightforward to see that, for instance, the correlators (2.18) are solutions to (2.24). On the othet hand, corrélation functions in a conformally invariant field theory will satisfy more constraints, as we shall see in the foUowing.

The corrélation functions are computed in terms of the singularities that occur when pairs of operators approach each other. These singularities are conveniently encoded in the operator product expansion (OPE) or short distance expansion of two operator fields A(z) and B{w):

A(z) B{w) ~ Ci{z — w)Oi(w) (2.25)

t

where Oi{w) are a complété set of local operators, and Ci{z — w) are the Wilson c- number coefficients which describe the singularities. Because of radial ordering, we tacitly assume | 2 |>| u> | — usually we shall drop the R symbol in operator product expansions. The OPE is an asymptotic expansion, and it should be understood as a re­

lation between (radially ordered) corrélation functions involving A and B zmd a certain combination of other operators. Convergence of the OPE follows from the fact that, acting on the vacuum, the expansion (2.25) is nothing but the décomposition of uni- tary représentation of the conformai group into irreducible components [51]. FoUowing dimensional arguments, assuming fixed dimension for our operators, and taking into accoimt that the radius of convergence of the sériés dépends on the distance between operators in the corrélation function, we hâve for the coefficients Ci{z — w) the general expression

C,{z-w) . (2.26)

In operator current algebras, and in particular in CFT, the singular part of OPE’s yields Lie bracket structures (or canonical commutators) for the modes or Laurent coefficients of the operator fields, while the regular part yields a prescription of forming normal ordered products (see Appendix A and B). Henceforth OPE’s can be interpreted as a compact way of computing standard commutation relations of mode algebras, with the extra advantage of being independent of any Hilbert space interprétation.

AU fields in a given CFT are expected to obey a closed operator algebra under OPE.

In particulax, for the primary fields <j>i{z,z) of dimensions (hi,hj) we hâve

<i>i{z-,z)4>j{w,w) = ^ Cfj(2 - {z - \<i>k\{z,z,w,w) (2.27)

k

where the indices run over ail primeiries of the theory and [</>jt](2,2, tr, û)) stand for an expansion over secondaries of 4>k{ZiZ)i whose coefficients are (in principle) fixed by conformai invariance [2]. An important requirement of the operator algebra (2.27) is associativity, which translates into consistency conditions for the structure const^mts It is worth to note at this point that conformai field théories are classified depending on the operator content (i.e. the dimensions (/i^, of the primary fields) and the structure constants C^j. We shall explain in which sense and how these quantities détermine completely a conformally invariant quantum field theory.

2.5. The Virasoro Algebra.

The conformai properties of the stress tensor can be extracted by considering two conformai transformations in succession. It follows that

T{z)T{w)

c/2 2T{w) du,T{w)

(2

— wy

— wy

— w) + 0 ( 1 ) (2.28) which States that the stress tensor T{z) is a quasi-primary field of weight (2,0). The (2 — w)~‘^ term is allowed by analyticity, the bosonic character of T{z) and scale in­

variance, and is proportional to a free parameter c (called the central charge) w'hich is not determined by conformai invEiriance — such a central charge is related to the trace amomaly, see [52]. Since {T{z)T{w)) = (c/2)/(2 - w)*, the central charge c will satisfy (at least) the condition c > 0 in a positive semi-definite Hilbert space.

The antianalytical component of the stress tensor T{z) has a similar OPE with central charge c, and the OPE of T{z) with T{z) has no singular term. It can be seen that c = c in a theory with Lorentz-invariance and conserved two-point function {Tft^{a)Ta^{—a)) (see for instance [46]). Note that T{z) must satisfy the foUowdng condition of regularity at infinity

T{z) ~ ¹ as

—> cx) (2.29)

and the corresponding one for the antiholomorphic component T{z).

The Laurent expansion coefficients (Ln^L^) oî the stress-energy tensor

+

+ 00

T(r)= Y.

n= —oo

T(Z)= Y

n=-«. (2.30)

= f(l)

are the operators generating the infinitésimal transformation (2.5). Indeed, (2.23) be- comes

[X„, (!>{w)] = h{n + l)w^<P{w) + w^-^^d^4>{w). (2.31) By use of Cauchy’s theorem, (2.28) is converted into commutation relations for the modes {^n} — fhis procedure makes contact between local operator products and commutators of operator modes, see Eqns.(A.22-26):

[^ni ^m] ~ ^)-^n+m "k ^2^^ ^)^n+m,0

[Z„,Z,n] = (n - m)L„+m + ~ «)<5n + m,0 {2.Z2) [Xn, Xm] = 0.

The first term on the right hand side of (2.32) is what we would expect from the composition of (2.5) (for c = 0, (2.32) reduces to (2.7)). The c extra term is the conséquence of the projective représentation in the Hilbert space of the quantum field theory ((2.32) is a ‘central extension’ of the Lie algebra of the diffeomorphism group Dif f{S^) on the circle [53]). We thus hâve the direct product Vir ® Vir of two copies of an infinité dimensional Lie algebra Vir called the Virasoro algebra [54], with a — for the time being free — parameter c which parametrizes the représentations. Note that the global conformai group 5X(2,C) generated by {Xo,X±i} is not affected by the central charge in (2.32), and it remains an exact symmetry group.

Taking n = 0 and real e in (2.5) allo^ws us to interpret X

+ Xo as the radial ‘hamil- tonian’. Taking now n = 0 in (2.31), and considering the isomorphic antiholomorphic sector, allows us to identify {h + h) with the scaling ((X

+ Xo)-eigenvalue) dimension of Eind (h — h) with its spin ((X

— Xo)-eigenvalue). For local fields, the vnlue of the spin can take only integer** (Bose fields) or half-integer (Fermi fields) values.

t; Note that requiring the corrélation functions (2.18) to be single valued when z = z* restricts the

spin h — h to he intégral.

Since the stress components are hermitian, it can be seen that

Li = L.„ , = (2.33)

which imply the hermiticity relations

î”W = pr(i) , f'(z) = ±f(l). (2.34)

It is useful to extend (2.34) to the concept of adjoint of a generic field A{z,z):

Now, we can associate States with primary fields through the relation

lAn) = lim A{z,z) |0) (2.36)

and define the states |^out)

(AoutI = lAn)^ = lim (0|^^(2,f) = lim {0\ A{w,w)w^^w^'^ (2.37)

2,2 — 0 ^OO

where we hâve make use of (2.35) and we hâve performed the map z = 1/w. The extra z^z factors in (2.35) give the adjoint the proper tensorial properties, and moreover they accoimt for the fact that if {A{z,z)A{w,w)) = {z— w)~^^(z — w)~^^, the corresponding State |A) has norm one, (A|A) = 1.

In (2.36)-(2.37) |0) is the SL{2, C) invariant vacuum. The invariance of |0) under {Lo,T±i} is équivalent to regularity of T{z) |0) at z = 0. Indeed, in |0) only terms with m < —2 are aJlowed at z = 0, and therefore

im|0) = 0 , m>-l. (2.38)

It follows that Lo |0) = L±\ |0) = 0, i.e. the vacuum |0) is SL{2,C) invariant. Note

that only the states |0) with n < —2 survive, giving rise to new states.

The same regularity condition can be applied to <f>{z) |0), where 4>{z) is a holomorphic primary field of conformad weight (/i,0). Putting

4>(z)= , 4>n= idzz^^^-'^(f>{z), (2.39) nez-h

regularity of <t>{z) |0) at

= 0 requires (j>n [O) = 0 for n > —h. Hence the State \h) =

^(0) |0) is in fact created by the mode <f>-h, 4>-h |0) = |/i) (note that (2.39) implies [Io,<^m] = i-e. (j>m bas scaling weight -m).

To einaJyze the représentations of the Virasoro aJgebra Vir (2.32) — equivalently, of the Vir operator algebra (2.28)— we can choose the hermitian operators L q and L q to be diagonal. Since L q + L q generates translations in r (or dilatations on the 2-plane), in any sensible quantum field theory L q and L q must be (separately) bounded from below.

Moreover, since the operators > 0} lower the eigenvalue of L q by n units, by repeatedly acting with such operators on any state, we must reach a state annihilated by ail the L„{n > 0). Any such state is called a heighest weight state (h.w.s.) of the Virasoro algebra (2.32), and it generates an infinite-dimensional représentation of Vir labeled by its I-o-eigenvalue:

Lo\h) = h\h) , T„|/i)=0 if n>0. (2.40) It can be shown (see for instance [47]) that every state in the space is obtained by acting with algebraic polynomials of {Ln, n < 0} on some h.w.s., and they are called descendants. The représentation space (or Verma modulus) will be spanned by vectors of the form

T_„,...I_„Jh) = |h;ni,...,nr) , n, > 0. (2.41) Hence, the Hilbert space of the theory is built up of ail possible irreducible highest weight représentations Vir(.(h) at fixed value of c, each représentation being labeled a number h{c). Taking now into accoimt the antiholomorphic sector, the complété Hilbert space 7i of the theory corresponding to the full two-dimensional conformai group Vir ® Vir is:

®h:h-^h:hVirc{h)®Vir^{h) (2.42)

where the non-negative integers h stand for the multiplicity of the représentations.

For a generic h.w.s. \h), it foUows from (2.32) that:

Since L„ = and we assume that the metric of the Hilbert space is positive, we hâve (2nh + ^( n* — n)) > 0. Taking n >> 1 implies c > 0, while n = 1 implies h > 0. Since Il L_] \h) |p= {h\LiL^i |/i)= 2{h\Lo \h)= 2h, it follows that h = 0 only if the state is annihilated by L_i — i.e. if the state is the projective or SL{2,C) invariant vacuum.

The vedue of c is actually further constrained by unitarity, as we will see below. For instance, positivity of (û|a), with

|û) = [ST—9 + GL—yL—2 "t* \2L—^L—^ — %L—

L—^L

12i_3T_4i_2 — |0) {h\ \h) = {h\\h) = [2nh + - n)){h \ h). (2.43)

constrains c > 1/2 or c = 0 in any CFT. It can be also shown [55] that if c = 0 the only unitary représentation of the Virasoro algebra is the trivial one: h = G and |0) = 0.

Note that from (2.43) it follows that a field (j>{z,z) of conformai weights {h, h = 0) is purely holomorphic: || [X_i,<^]|0) |p= 0 implies that = di4>{z^z) has to be identified with zéro.

If (f){z) is a primary field of dimension (/ î ,0), the state <^(0) |0) = |/i) introduced in (2.36) is annihilated by the positive Virasoro generators. Henceforth it vérifiés (2.40), and it generates a h.w.s. of weight h. Thus, primary fields are those which create h.w'.s.

from the vacuum. The same correspondence is established between the descendant States (2.41) and the secondary fields of 4>{z)\

’-"^)(2) = jdzi ...j c, c.

dz. T{z,)...T{Zr)<t>{z)

(zi — u;)"i“V..(2r ~ u;)"’'~^ (2.44)

of dimension h + It can be shown that any dérivative of the primary field (f>{z)

can be expressed this way, and that any secondary field can be described zis a differential

operator acting on the primary field. In particular = h4> and = d4>. Note also

that ^ dzT{z)/{z — w) = T(w), i.e. the stress-energy tensor is a descendant of

the identity operator.

The set of secondary fields constitute the conformai family [4>\ generated by the corresponding primary field 4>{z) (usually called the ‘ancestor’). Each conformai family

— or eqnivalently, each Verma modulus — is organized in Lo-eigenvalues, and the représentation space Virc{h) sphts into subspaces Virc{h)i<i of States with Xo-eigenvalue {h + N), N being a positive integer:

Lo level states

h 0 \h)

h + 1 1 L-j (h)

h + 2 2 X_2 \h),Ll,\h)

h + 3 3 L-3 |h), X_2X_

Each h.w.s. — each conformai primary field — détermines in this way a représen­

tation of the Virasoro algebra. The action of the Virasoro operators in this space, as well as the scalar products of states in such a space, is completely determined by the Virasoro algebra and the relation L„ |/i) = èn^^h \h) for n > 0. For the Verma modulus to constitute a unitary représentation, the matrices of inner products [56] of states (at a particulax value of h) hâve to verify the positivity condition^ It tums out that if c > 1 and h > 0, unitarity imposes no further restrictions, but that only a discrète set of values (c, h) is allowed for 0 < c < 1 [10]:

6 , , P + 3)p-(l + 2)?p-l

(k + 2)(k + 3) ’ "-'“ m W- 4(i + 2)(it + 3) (2.45) where k = 0,1,2,..., p = 1,2,..., k + l and g = 1, (Note that (2.45) is unchanged if we perform the substitution p k — p + 2 and k — q + 3.) The sériés (2.45) turns out to classify ail unitary — and minimal, see below — models of Vir [58]. Goddard, Kent and Olive hâve shown that ail éléments of this sériés are in fact unitary by con- structing manifestly unitary realizations for ail of them [11]. Some of the éléments of this sériés appear to be realized in nature, with the dimensions (2.45) appeaxing as crit- ical exponents: A: = 1,2,3 and 4 correspond respectively to the Ising, the critical Ising,

I] There are, however, realistic statistical Systems which are not unitary, such as the Lee-Yang edge

singularity and some percolating phenomena [57].

the three-state Potts and the tricritical three-state Potts models [27]. CriticaJ points of restricted solid-on-solid (RSOS) models hâve been also shown to provide realizations of the discrète sériés (2.45) [59].

One of the key properties of two-dimensional CFT is that conformai Ward identi­

fies give differential équations that détermine also the corrélation functions of arbitrary descendant fields in terms of those of primaries. Indeed, since second^lries can be ex- pressed in term of primaries, it can be shown that any corrélation function involving a secondary field like (see (2.44)) can be reduced mechanically to another one involving the corresponding primai^’ field

.... (2.46)

=(-ir n É {^10^ +... ).

i=l

''

The same procedure implies that the OPE coefficients of descendant fields are also determined from those of the primary fields (2.27) and the quantifies (/i,, h,).

This show's that the complété information necessary to specify a conformally in­

variant two dimensional quantum field theory is the Virasoro h.w.r. content (/i,, h^), together with the operator product coefficients C,* determining the OPE’s (2.27) be- tween the primary fields creating the respective h.w.r.’s. The coefficients can be computed once we know the four-point functions between the primary fields of the theory, and in this sense these ‘Clebsch-Gordan’ coefficients encode the dynamical in­

formation of the CFT. Let us stress that there are strong constraints which hâve to be satisfied by the coefficients which corne from the requirement of associativity of the OPE algebra (in [2] it hais been shown that such a requirement is équivalent to ask duality (crossing symmetry) of the four-point function).

This is the power of conformai invariance in two dimensions: the four-point func­

tions and ail higher Green’s functions of (primary and secondary) fields are completely determined in terms of dérivatives of the two- and three-point functions of the pri­

maries by picking successively two fields and writing them in terms of OPE’s of the respective primary fields. Since the two- zind three-point functions are fixed by confor­

mai invariance to hâve the form (2.18), these models are exactly solvable. BPZ hâve

suggested that it may be possible to classify and solve ail two-dimensional conformai field théories this way (the ‘conformai bootstrap’ program of Polyakov [9] applied to the two-dimensional case). The difficulties in this program zire first the technical one of evaluating the ‘conformai blocks’ — i.e. the contribution of an entire conformzd family to a given correlator — and second the possibility of a large number of coupled pri- maries. Additional constraints are, of course, imitarity and modulzir invariance. In the following we will see other steps in performing such a program.

2.6. Null Vectors and Minimal Models.

The Friedcin, Qiu and Shenker discrète sériés (2.45) tums out to be a subset of the set of minimal models obtained by Belavin, Polyakov and Zamolodchikov [2]. These are exactly solvable conformai field théories which contain only a finite number of conformai primary fields and whose OPE algebra closes. This is possible due to the appearance of degenerate représentations of the Vireisoro algebra. Indeed, suppose that in some Verma modulus we find at level N a descendant State |x) of \h) which is also a h.w.s.:

Lo Ix) = (h + N) Ix) and |x) = 0, n > 0. Since such a state — as weU ELS ail of its descendants — tums out to be a null state (x I x) == is orthogonal to ail other States {j^)} in the Verma module: (>s | x) = 0- We can now set consistently ail of these States equal to zéro, and the représentation becomes a true irreducible représentation. In the new représentation, the polynomials in which previously created the null states now annihilate the h.w.s. of the Verma module. If x(^) is the (null) conformai primary field which créâtes the null state |x) from the SL{2,C) vacuum, then x('2^) = T^z{Ln)4>{z) for some primary field <^(2) and some differential operator T>i{L„). Now, any corrélation function involving xi^) vauishes. Therefore

0= (x(2)ç^l(^l)...<^„(2„))=P.(i„)(<;i(2)<^l(^l)-<?^n(2n)) (2.47)

and we obtain diferential équations for the corrélation fonctions of 4>{z) with any set of fields.

Kac [12,12] hâve listed aJl the spécial values of the dimensions A for which the

représentation turns out to be degenerate. For a given value of c, they are labeled by

(2.48) two positive integers n and m, and aire given by

A(n, m) = A(0,0) + ^{na+ + moi_Ÿ where

. c-1 y/\ — c± >/25 — c

A(0,0) = and a± = r- . (2.49)

24 ^ ^

The null vectors hâve conformai weights A(n,m) + nm and (2.48)-(2.49) constrain 0 < c < 1. BPZ [2] showed that in the spécial case in which — p/q., i.e.

y/25 — c — y/\ — c

(2.50) y/25~~~c -|- \/l — c q

for some positive integers p and q, the corresponding models satisfy the following prop- erties:

— The operator product algebra closes involving only a finite mnnber of conformai familles, ail of them being degenerate.

— The spectrum of dimensions of the fields is determined.

— Ail corrélation functions of the theory can be computed via differential équations of the form (2.47).

— Since the four-point functions can be computed, the structure constants are determined, and exact solvability of the models is ensured.

Therefore, this powerful method can select a finite number of primary fields imder which the OPE-algebra closes, and only a finite mnnber of représentations are used to build up these exactly solvable models. These are the so called ‘minimal’ models of Vir, which only exist for c < 1 [22]. Since (2.50) can be written as

{k -|- 2)(/c -|- 3) where k =

—^---2 if g > P q-p

—--- 2 if P > g p-q

(2.51)

the Vir minimal sériés tums out to include, actually, the Vir unitary sériés (2.45).

After considering other complementary methods in characterizing such Vir minimal models, we shall show in the next chapter how to extend this concept of ‘minimality’

to models with c > 1.

2.7. The Free Boson.

The single free massless scalar field constitutes the simplest example of conformai System, and at the same time a very rich one. The relevance of this model in CFT cornes from the fact that ail exactly solvable conformai field models discovered to date can be represented in terms of a finite number of such free fields. The procedme of representing and characterizing a conformai model in terms of free scalar fields is usually called bosonization.

In Euclidean space the équation of motion of a massless scalar field ip{z,z) with action

5 = — / dzdz dipdif

27T J (2.52)

is ddip{z,z) = 0, which general solution is a linear combination of an analytic and an antianalytic function of z. Hence, (p{z,z) splits into left- and right-mover pièces, (f{z, z) = (f(z) + <f{z), where

<^(2) = q — iplnz + i —z "

n^O

(f{z) = q — ip\nz + i ^ —2~".

(2.53) n^O

The functions ip(z) and (p(z) axe not single valued operators on the complex plane, but the conserv'ed currents

j(2) = id<i>(2) , j(z) = id(p(z) (2.54)

are. By analyticity, the line intégrais of j(z) and j(z) around contours circling the origin (p and P, respectively) are independent of the contour; this is in radial quantization the statement of conservation law for the quantities p ± p (which correspond to the momentum and the winding number in a string model).

The quantum theory of the scalar field ip(z,z) is defined by assuming the commu­

tations rules

~ ^ ^n,

m t ffliPl — î

[â„, 0,n] ~ ^

>

[9,P] = i [9,p] = i

(2.55)