Cuerdas y D-branas en espacio- tiempos curvos

UNIVERSIDAD DE BUENOS AIRES


Facultad de Ciencias Exactas y Naturales

Departamento de Física

[8mm]

Cuerdas y -branas en

espacio-tiempos curvos

[9mm]

Tesis presentada para optar por el título de Doctor de la

Universidad de Buenos Aires en el área Ciencias Físicas

[1cm]

Walter Helmut Baron

[1cm]

Director de tesis: Dra. Carmen Alicia Nuñez
Consejero de estudios: Dr. Gustavo Sergio Lozano
Lugar de Trabajo: IAFE (CONICET/UBA)
Buenos Aires, 2012


Cuerdas y -branas en espacio-
tiempos curvos

En esta tesis estudiamos el modelo Wess-Zumino-Novikov-Witten. Calculamos la Expansión en Producto de Operadores de campos primarios y de sus imágenes bajo el automorfismo de flujo espectral en todos los sectores del modelo considerado como una rotación de Wick del modelo coset . Argumentamos que las simetrías afines del álgebra requieren un truncado que determina la clausura de las reglas de fusión del espacio de Hilbert. Estos resultados son luego utilizados para discutir la factorización de las funciones de cuatro puntos con la ayuda del formalismo conocido como bootstrap.

También realizamos un estudio de las propiedades modulares del modelo. Los caracteres sobre el toro Euclídeo divergen de una manera poco controlable. La regularización propuesta en la literatura es poco satisfactoria pues elimina información del espectro y se pierde así la relación uno a uno entre caracteres y representaciones del álgebra de simetría que forman el espectro. Proponemos estudiar entonces los caracteres definidos sobre el toro Lorentziano los cuales están perfectamente definidos sobre el espacio de funcionales lineales, recuperando así la biyección entre caracteres y representaciones. Luego obtenemos las transformaciones modulares generalizadas y las utilizamos para estudiar la conexión con los correladores que determinan los acoplamientos a las branas simétricas en tal espacio de fondo, obteniendo que en los casos particulares de branas puntuales o branas se recuperan resultados típicos de Teorías de Campos Conformes Racionales como soluciones tipo Cardy o fórmulas tipo Verlinde.

Palabras claves: teoría de cuerdas, teorías conformes no racionales, -branas, , reglas de fusión, transformaciones modulares.

Strings and -branes in curved
space-time

In this thesis we study the Wess-Zumino-Novikov-Witten model. We compute the Operator Product Expansion of primary fields as well as their images under the spectral flow automorphism in all sectors of the model by considering it as a Wick rotation of the coset model. We argue that the symmetries of the affine algebra require a truncation which establishes the closure of the fusion rules on the Hilbert space of the theory. These results are then used to discuss the factorization of four point functions by applying the bootstrap approach.

We also study the modular properties of the model. Although the Euclidean partition function is modular invariant, the characters on the Euclidean torus diverge and the regularization proposed in the literature removes information on the spectrum, so that the usual one to one map between characters and representations of rational models is lost. Reconsidering the characters defined on the Lorentzian torus and focusing on their structure as distributions, we obtain expressions that recover those properties. We then study their generalized modular properties and use them to discuss the relation between modular data and one point functions associated to symmetric D-branes, generalizing some results from Rational Conformal Field Theories in the particular cases of point like and branes, such as Cardy type solutions or Verlinde like formulas.

Keywords: string theory, non rational conformal field theories, -branes, , fusion rules, modular transformations.

Acknowledgments

Es el momento de agradecer apropiadamente a todas las personas e instituciones que de una u otra forma contribuyeron a la realización de esta tesis.

Desde luego voy empezar agradeciendo a mi hija Juanita y a mi mujer Gisela por el apoyo y por haberme dado tantos gratos momentos que tantas fuerzas me dieron a lo largo de estos años. A mis padres y mis suegros también por su continuo apoyo. A mis hermanos y a mis amigos por la compañia. Quiero agradecer muy especialmente a mi directora, Carmen por haberme guiado y escuchado tantas veces y con tanta paciencia durante este tiempo. A diversos colegas que tanto me han enseñado en valiosas e innumerables discusiones durante la elaboración de mis investigaciones: Adrian Lugo, Alejandro Rosabal, Carlos Cardona, Diego Marqués, Eduardo Andrés, Gerardo Aldazabal, Jan Troost, Jörg Teschner, Jorge Russo, Juan Martín Maldacena, Mariana Graña, Pablo Minces, Robert Coquereaux, Sergio Iguri, Sylvain Ribault, Victor Penas, Volker Schomerus y Yuji Satoh.

Quiero agradecer al Instituto de Astronomía y Física del Espacio (IAFE) por haberme dado no sólo un lugar físico y multiples facilidades para llevar a cabo la tarea de investigación sino también por saber generar una apropiada atmósfera de trabajo. Al Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) por haber permitido iniciarme en la inverstigación gracias a su financiamiento. A las diversas instituciones que me recibieron a lo largo de estos años y me permitieron discutir con diversos colegas expertos durante la elaboración de mis trabajos: el Instituto Balseiro (IB), Instituto de Física de La Plata (IFLP), Institut de Physique Théorique (IPTh, CEA Saclay), Laboratoire Charles Coulomb (LCC, Universidad de Montpellier II), Deutsches Elektronen-Synchrotron (DESY, Universidad de Hamburgo) , Institució Catalana de Recerca i Estudis Avançats (ICREA, Universidad de Barcelona), Scuola Internazionale Superiore di Studi Avanzati di Trieste (SISSA) y el Abdus Salam International Centre for Theoretical Physics (ICTP).




A Juanita y a Gisela por su incansable apoyo.

Contents

Chapter 1 Introduction

String theory is one of the most ambitious projects in the field of high energy physics. Even though it was originally conceived to explain quark confinement, its development forced the opening of different roads and the extension of its original aims. Shortly after its beginning, people realized every consistent string theory had a massless spin two particle in the spectrum, a graviton. This opened a new possibility to reach a consistent theory of quantum gravity, with extended objects to quantize instead of point-like particles, and so to reach the dream of every theoretical physicist, a unified theory for all known interactions or a theory of everything.

This theory, yet under construction, is the best candidate to give a unified description of all interactions, but also has the beauty of having offered important results going beyond the theory itself. It gave rise to an amazing development of Conformal Field Theories (CFTs) providing several tools to deal with two dimensional statistical models. The consistency of the theory led to the idea of supersymmetry. The gauge/gravity duality, whose validity is not restricted to that of string theory, has offered the first realization of the holographic principle, leading not only to a theoretical frame relating gravitational theories with gauge theories in flat space, but a powerful tool allowing to study non perturbative regions of superconformal, but also of non conformal field theories or models with less supersymmetry or in different space-time dimensions [1] than those proposed in the first example of this duality [2, 3], including holographic renormalization group flows [4, 5, 6], rotating strings [7, 8, 9], applications to the study of baryonic symmetry breaking [10, 11], applications to cosmology [12, 13], applications to holographic QCD [14, 15, 16, 17], to electroweak symmetry breaking [18], and non relativistic quantum-mechanical systems [19, 20, 21, 22], among others.

Many years have passed since its birth and it appears frustrating not having found yet a solid confirmation of the theory. There is not known way to reduce the theory to one which exactly reproduces all aspects of the Standard Model, or which solves the cosmological constant problem, among other drawbacks. But these puzzles must not be seen as a failure of the theory, but as a consequence of the complexity of the problems to be solved. The knowledge of the theory has increased considerably. There is a good understanding of the perturbative regime, and the discovery of branes, extended dynamical objects on which the ends of open strings live, has been crucial in the development of dualities that allow to explore non perturbative regions of the theory.

One of the major challenges of the theory is to find a mechanism univocally leading to an effective reduced theory with a solution phenomenologically contrastable with the real world. Instead, one finds a large number of vacua, roughly , a problem frequently named the of string theory [23].

There is no doubt that D-branes play a fundamental role in the resolution of these problems and so a fundamental step consists in the study of branes in non trivial curved backgrounds.

A very powerful approach to explore string theories on non trivial curved backgrounds is to consider vacua with a lot of symmetry. This was successfully reached by considering the space-time to be given by the group manifold of a continuous compact group, . In such a case the worldsheet theory is a Rational Conformal Field Theory (RCFT), it contains a finite number of primary states. To date these theories can be solved exactly by using only algebraic tools [24, 25, 26]. The situation is very different in the non compact case, as a consequence of the presence of a continuous spectrum of states (see for instance [27] for a review). This thesis is devoted to the study of the worldsheet of string theory on a three dimensional anti de Sitter () space time. This theory is very interesting because it is one of the simplest models to test string theory in curved non compact backgrounds and with a non trivial timelike direction, it can be used to learn more on the not well known non RCFT, but also because of its relevance in the AdS/CFT correspondence.

Most of the studies regarding the AdS/CFT duality were explored only within the supergravity approximation. An example where the correspondence was successfully explored beyond the supergravity approximation is string theory in PP waves backgrounds [28, 29] with RR fields, obtained by taking the Penrose limit of AdS. Another accessible background is with Neveu Schwarz (NS) antisymmetric field, appearing within backgrounds like AdS, with or (obtained as the near horizon limit of the brane setup in the background ) [30, 31, 32, 33, 34, 35].

The worldsheet of the bosonic string propagating on is described by the Wess Zumino Novikov Witten (WZNW) model associated to the universal cover of the group, but for short we will refer to this as the WZNW model in order to avoid confusion with the WZNW model on the single cover of . So it is expected to be exactly solvable. The studies on this model started within the seminal work of O‘Raifeartaigh, Balog, Forgacs and Wipf [36] on the WZNW model more than twenty years ago, but it was necessary to wait for more than ten years until the work of Maldacena and Ooguri [37] correctly defined the spectrum, which is considerably more involved that those of RCFT. It consists of long strings with continuous energy spectrum arising from the principal continuous representation of and its spectral flow images, and short strings with discrete physical spectrum resulting from the highest-weight discrete representations and their spectral flow images.

By extending the result of Evans, Gaberdiel and Perry [38] to nontrivial spectral flow sectors, a no ghost theorem for this spectrum was proved in [37] and verified in [39] through the computation of the one-loop partition function on a Euclidean background at finite temperature. Amplitudes of string theory in on the sphere were computed in [40], analytically continuing the expressions obtained for the Euclidean H (gauged) WZNW model in [41, 42]. Some subtleties of the analytic continuation relating the H and models were clarified in [40] and this allowed to construct, in particular, the four-point functions of unflowed short strings. Integrating over the moduli space of the worldsheet, it was shown that the string amplitude can be expressed as a sum of products of three-point functions with intermediate physical states, the structure of the factorization agrees with the Hilbert space of the theory.

A step up towards a proof of consistency and unitarity of the theory involves the construction of four-point functions including states in different representations and the verification that only unitary states corresponding to long and short strings in agreement with the spectral flow selection rules are produced in the intermediate channels. To achieve this goal, the analytic and algebraic structure of the AdS WZNW model should be explored further.

Most of the important progress achieved is based on the better understood Euclidean model. The absence of singular vectors and the lack of chiral factorization in the relevant current algebra representations obstruct the use of the powerful techniques from rational conformal field theories. Nevertheless, a generalized conformal bootstrap approach was successfully applied in [41, 42] to the model on the punctured sphere, allowing to discuss the factorization of four-point functions. In principle, this method offers the possibility to unambiguously determine any point function in terms of two- and three-point functions once the operator product expansions of two operators and the structure constants are known.

We carried out some initial steps along the development of this thesis [43], by examining the role of the spectral flow symmetry on the analytic continuation of the operator product expansion from to the relevant representations of and on the factorization properties of four-point functions. These results give fusion rules establishing the closure of the Hilbert space and the unitarity of the full interacting string theory.

In RCFT, a practical derivation of the fusion rules ( of the representations contained in the Operator algebra) can be performed through the Verlinde theorem [44], often formulated as the statement that the matrix of modular transformations diagonalizes the fusion rules. Moreover, besides leading to a Verlinde formula, the matrix allows a classification of modular invariants and a systematic study of boundary states for symmetric branes. It is interesting to explore whether analogous of these properties can be found in the WZNW model. However, the relations among fusion algebra, boundary states and modular transformations are difficult to identify and have not been very convenient in non compact models [45]. In general, the characters have an intricate behavior under the modular group [46]-[48] and, as is often the case in theories with discrete and continuous representations, these mix under transformations. In the forthcoming chapters we will discuss these subjects based on our previous results [49].

This thesis is organized as follows, in Chapter 2 we review the geometry, symmetries, and give some basis functions of and related spaces which are the basic objects in the minisuperspace limit. Chapter 3 is devoted to introduce the WZNW models. We begin with a short introduction on general WZNW models and then present the WZNW model, and related ones like and the models. In Chapter 4 we discuss interactions in the model. We consider two and three point functions of the Euclidean model, and assuming they are related to those of the model [40], we compute the Operator Product Expansion (OPE) for primary fields as well as for their images under spectral flows in all the sectors of the theory. After discussing the extension to descendant fields, we show that the spectral flow symmetry requires a truncation of the fusion rules determining the closure of the operator algebra on the Hilbert space of the theory. In Chapter 5 we consider the factorization of four-point functions and study some of its properties. Chapter 6 is devoted to the characters of the relevant representations of the WZNW model. Since the standard Euclidean characters diverge and lack good modular properties, extended characters were originally introduced in [50] (see also [76])111Similar problems in non-compact coset models have also been considered in [52]-[54]. A different approach was followed in [37] where the standard characters were computed on the Lorentzian torus and it was shown that the modular invariant partition function of the model obtained in [55] is recovered after performing analytic continuation and discarding contact terms. However, this trivial regularization removes information on the spectrum and the usual one to one map between characters and representations of rational models is lost. With the aim of overcoming these problems, we review (and redefine) the characters on the Lorentzian torus, focusing on their structure as distributions and compute the full set of generalized modular transformations in Chapter 7.

In order to explore the properties of the modular S matrix, in Chapter 8 we consider the maximally symmetric D-branes of the model. We explicitly construct the Ishibashi states and show that the coefficients of the boundary states turn out to be determined from the generalized matrix, suggesting that a Verlinde-like formula could give some information on the spectrum of open strings attached to certain D-branes. Furthermore, we show that a generalized Verlinde formula reproduces the fusion rules of the finite dimensional degenerate representations of ) appearing in the boundary spectrum of the point-like D-branes.

In chapter 9 we give a summary of the thesis and discuss the actual status and future challenges and perspectives regarding open problems. We also list the original contributions to the subject presented along the thesis.

Some basic facts about CFTs are reviewed in appendix A, some technical details of the calculations are included in appendices B, D and E and a discussion of the moduli space of the Lorentzian torus is found in appendix C.

Chapter 2 Geometric aspects of maximally symmetric spaces

Before introducing the WZNW model, it is instructive to spend some pages reviewing some aspects of the geometry of maximally symmetric spaces and the minisuperspace limit in and related models.

2.1 Geometry and symmetries

Along the bulk of the thesis the reader will find discussions concerning different types of geometries such as hyperbolic, Anti de Sitter or de Sitter spaces, so a good point to begin with is by defining all these geometries.

2.1.1 Maximally symmetric spaces

Maximally symmetric spaces are defined as those metric spaces with maximal number of isometries in a given spacetime dimension111In the concrete case of D spacetime dimensions these spaces admit linear independent Killing vectors.. Due to this important property such geometries were extensively studied in the literature. Here we will give a short . For a deeper study of the subject we redirect the reader to [56].

Every D dimensional maximally symmetric space has constant curvature and can be realized as a pseudosphere embedded in a D+1 dimensional flat space.

To be more precise, let represent the Cartesian coordinates of a particular point in such a flat space, the symmetric spaces can be realized as the hypersurfaces constrained by

(2.1.1)

where the indices are lowered with the background metric

(2.1.2)

and the ’s are signs. is frecuently called the radius of the space because of the similarity with the radius of a sphere, but it must not be confused with the Ricci curvature scalar, , which is given by

(2.1.3)

We are specially interested in Euclidean or Lorentzian (one timelike direction) D dimensional spaces, so we fix .

The cases of interest for us are anti de Sitter, Hyperbolic and de Sitter spaces.

Anti de Sitter space: AdS

This geometry corresponds to the case where . The space has Lorentzian signature and isometry group. The time like direction is compact.

The particular case will be of special interest for us.

Notice that the topology of coincides with that of the group manifold as can be checked from the following parametrization of this group

(2.1.4)

This relation is intimately linked to the fact that the isometry group is locally isomorphic to .

For obvious reasons we will decompactify the time-like direction when discussing physical applications. And from now on we will denote this space as , which is nothing but the group manifold of the universal covering of , .

Another useful coordinate system, frequently used in the literature, is the so called global or cylindrical coordinate system , related to the previous one via

(2.1.5)
(2.1.6)
Hyperbolic space: H

This geometry is realized when . It has Euclidean signature, isometry and decomposes in two disconnected branches, the upper sheet () denoted by H and the lower one () denoted by H.

For the special case of , this space has the topology of the group coset manifold of the subset of hermitian matrices of unit determinant in , . The subspace can be interpreted as a Euclidean Wick rotation of , in fact it can be parametrized with the analytic continuation of the cylindrical coordinate system of so breaking the periodicity of the time-like direction.

Another useful coordinate system is the one defined by the complex variables , where

(2.1.7)
(2.1.8)
(2.1.9)

In terms of these coordinate, the elements of the coset are parametrized by

(2.1.10)

and the coset structure is manifest when is written as , with

(2.1.11)

Clearly is invariant under , with and this explicit realization of the coset structure is the reason why these coordinates are implemented when constructing the model by gauging the subgroup of the WZNW model.

de Sitter space: dS

In this case only . The space has Lorentzian signature, isometry group and its topology coincides with the group manifold of unimodular antihermitian matrices

(2.1.12)

Other geometries (not discussed in the thesis) are the (Euclidean) D-dimensional sphere with SO(D+1) isometry group () and the “two time” pseudosphere with isometry corresponding to .

2.2 Basis functions

In this section we describe some of the basis functions for and , where by abuse of notation we refer to as the group manifold of the single cover of . The fact that both and admit matricial representations makes the study of functions over these spaces much easier than in the case where such representations are not present.

Let us begin with , and the parametrization (2.1.6), then the elements of the group, are written as

(2.2.1)

As we commented in the previous section, is obtained by decompactifying . If we write , where , the elements of admit a parametrization in terms of

(2.2.2)

The product is built from the one in the single cover as

(2.2.3)

Notice that both and the universal cover carry a natural left and right multiplication by themselves. For instance, for we have and this will be the symmetry of the model under consideration (see section 3.1).

The geometric symmetry group of is , where is the center of the group. In the case of the symmetry group is , where the center is isomorphic to . Indeed, it can be easily checked, from (2.2.3), that it is given by the subgroup , which is the subgroup freely generated by the element .

A useful parametrization for , different from the ones considered in the previous section is given by

(2.2.4)

Notice that the product of two matrices in , and with is outside of the hyperbolic space. This is a not surprising because is not a group, but the group coset . In this case the left and right action take the element out of the space and the geometric symmetry group acts as , with . So it is given by .

2.2.1 Continuous basis

As we commented at the beginning of this section, the fact that and admit matrix representations simplifies matters.

A useful basis for is the well known -,

(2.2.5)

whit . These functions have simple behavior under symmetry transformations. If is parametrized by then

(2.2.6)

where

(2.2.7)

It was shown [57] that a complete basis of functions in the Hyperbolic space is generated by , with .

Functions with and are related by the reflection symmetry

(2.2.8)

where the reflection is such that

(2.2.9)

A continuous basis for is given by the so called -

(2.2.10)

The parity , and the symmetry group acts as

(2.2.12)

where

(2.2.13)

A complete basis of functions is known to be mod

The situation is subtler in where the correct behavior under symmetry transformations was found in [58]

(2.2.15)

with , , being the projection of . is a function with the following properties

(2.2.16)
(2.2.17)

For instance can be taken as the number of times crosses infinity when moves from to . A function satisfying (2.2.15) was found in [59] and is given by

(2.2.18)

where is the function defined such that gives the number of times crosses as moves from to .

2.2.2 Discrete basis

The continuous and basis have simple properties under symmetry transformations. But as in (2.2.1) and in (2.2.4) are not related by Wick rotation, neither the continuous functions defined above are related by Wick rotation.

So it is useful to introduce another type of basis, the so called “-”. These are sets of functions parametrized by discrete parameters , which even though not having a simple behavior under symmetry transformations have a simple connection via Wick rotation.

The - of can be defined as a kind of Fourier transformation of the -

(2.2.19)

The combinations and are proportional to the momentum numbers along the compact -direction and the non-compact direction respectively, which implies the decomposition

(2.2.20)
(2.2.21)

with . Notice that ensures the monodromy in (2.2.19).

The explicit computation of (2.2.19) gives

(2.2.23)

The reflection property (2.2.8) translates in the - to

(2.2.24)

where

(2.2.25)

We take the - of the Lorentzian models as the Wick rotation of the basis above and by abuse of notation we use the same name, . The first difference of the Wick rotated functions is that must be real in order to ensure the (delta function) normalization.

If we introduce the parameter such that , the change of basis between and -basis is

(2.2.26)
(2.2.27)

where

(2.2.28)

All the functions introduced in this section have a relevant role in the field theory description as they describe what is known as the minisuperspace limit of the WZNW models associated with or their Euclidean rotation, the space. In such a limit these functions represent the zero mode contribution of the primary fields. A thorough study of the minisuperspace limit of the coset model and the WZNW model was presented in [57] and [59] respectively. The basis functions discussed in this section were proved to be a complete set of functions in each space. These functions not necesarily belong to the squared integrable set but form an orthogonal basis in the same sense that is a complete basis over the space of real functions. The completeness of the continuous and discrete basis of functions on was proved in [57], and as commented in [59] the proof of the completeness of the discrete basis of follows from the results of [60] where a plancherel formula for was proved. The completeness of the continuous basis is a consequence of the integral relation (2.2.27). And finally the completeness of the basis functions for follows from the observation that these are nothing but the functions of with periodicity on .

Chapter 3 WZNW Model

In this chapter we present the WZNW model and other coset models related to the former by Wick rotations. In next section we present a brief introduction on nongauged and gauged WZNW models, with the aim of introducing some basics tools, setting the notation and to present the formulae we will be using along the thesis. Then we turn to a description of the WZNW model, the or Coset model and the Coset model.

3.1 WZNW models

WZNW models are CFTs with a Lie group symmetry, where the spectrum is built over representations of the affine algebra. These theories have the peculiarity of being defined with an action, a feature that usually does not occur in CFT’s.

Sigma models defined with semisimple group manifolds as target space constitute a natural starting point to construct a theory with the properties mentioned above. Nevertheless, even though these theories are classically scale invariant, the function of the renormalization group is nonzero and so the effective theory becomes massive and a scale anomaly emerges at the quantum level.

This observation is sufficient to realize that this is not the theory we are looking for. Another indication follows from the fact that the conserved currents do not satisfy the factorization property of CFTs, they do not factorize in a holomorphic current and an antiholomorphic one.

The requested theory is obtained when the sigma model action is supplemented with a Wess-Zumino term. The WZNW action is [61, 62, 63]

(3.1.1)
(3.1.2)
(3.1.3)

where is the space whose boundary is the compactification of the space on which we defined the sigma model. The prime in the trace means a normalization in the trace such that in any representation the generators of the Lie algebra satisfy

(3.1.4)

The field lives in a unitary representation of the semisimple group G in order to ensure the sigma model action be real. is the extension to the three-dimensional space . The coupling constant , usually referred to as the level, must be quantized because has the topology of a sphere.

Even though the Wess-Zumino term is an integral over a three dimensional space, its variation being a divergence can be written as an integral over the two dimensional boundary and the solution to the Euler-Lagrange equation of the WZNW action is, after the change of variable , with independent holomorphic and antiholomorphic functions. The conserved currents are

(3.1.5)
(3.1.6)

where the notation was used. They are associated with the following invariance of the action

(3.1.7)

with two arbitrary functions living on G so that the global symmetry of the sigma model was lifted to a local one by adding the Wess-Zumino term. The transformation law of the currents is easily read out from (3.1.7) and (3.1.6) and the current algebra can be determined using the Ward identities. It is found to be

(3.1.8)

where means equal up to regular terms and are the components of in the basis. So defining the Laurent modes as

(3.1.9)

the current algebra leads to the desired affine Lie algebra

(3.1.10)

and similarly for the antiholomorphic sector. The OPE of holomorphic and antiholomorphic currents has no singular terms implying that the modes commute with each other.

The classical energy momentum tensor is obtained from varying the action with respect to the metric111 From now we will work only with the holomorphic sector, the antiholomorphic sector being analogous..

(3.1.11)

Fields are not free, so that this expression will be corrected at quantum level. If the product of currents in (3.1.11) is replaced by a normal ordered product, namely

(3.1.12)

and the coefficient is left as a free parameter to be fixed by the requirement that the OPE between two energy momentum tensors be as required by a CFT,

(3.1.13)

one finds the quantum corrected energy momentum tensor

(3.1.14)

where is the dual coxeter number and the central charge is found to be

(3.1.15)

It is bounded from below

(3.1.16)

so that .

This realization of the energy momentum tensor in terms of the currents is usually named in the literature as the Sugawara construction.

After expanding according to

(3.1.17)

the Virasoro algebra is realized, namely

(3.1.18)

The commutator between Virasoro and current modes is

(3.1.19)

Of course holomorphic and antiholomorphic modes commute with each other.

There is much more to say about WZNW models, like discussing the Knizhnik-Zamolodchikov (KZ) equation, free field representations, the modular data, fusion rules and many other matters. We will only discuss the properties we need and at the appropriate time. We end this short review with a few comments on primary fields.

In conformal field theory one defines primary fields, , as those transforming covariantly with respect to scale transformations and satisfying

(3.1.20)

where is the conformal dimension of . So that

(3.1.21)
(3.1.22)

On the other hand, in the case of WZNW models primary fields are those transforming covariantly under and so satisfying the following OPE with the current

(3.1.23)

where denote the holomorphic and antiholomorphic representations of the field, and is the realization of the generator in such representation. These conditions translate into

(3.1.24)
(3.1.25)

where represents the primary state .

As a consequence of the realization of the conformal symmetry via the Sugawara construction, WZNW primaries are also conformal primaries. They satisfy

(3.1.26)
(3.1.27)

where the conformal weight is

(3.1.28)

and is the quadratic Casimir.

But the reader has to bear in mind that the inverse is not always true. A Virasoro primary can be a WZNW descendant.

3.1.1 Gauged WZNW models

We now consider the construction of Coset or Gauged WZNW models. These are constructed from two WZNW models where the first group is a subgroup of the second one. Contrary to what happens in WZNW models (see (3.1.16)), these theories are less restrictive as there are no bounds for the central charge222The central charge of the Coset is the difference of the central charges of both WZNW models.. Moreover it is expected that this framework provides a full classification of all RCFT [24].

As we saw above a WZNW model with group G is invariant under , thus it has a global symmetry . Given two subgroups it is sometimes possible to obtain a theory with local symmetry.

The gauged action was obtained in [64]. This is given by

(3.1.29)

where

(3.1.30)
(3.1.31)

denotes the covariant derivative and are the gauge fields associated to respectively. It was found that (3.1.29) is invariant under local transformation ()

(3.1.32)

when the gauge fields transform as

(3.1.33)

and are anomaly free subgroups, their Lie algebra generators () satisfy [64]

(3.1.34)

The origin of this unusual restriction can be traced back to the fact that the is a three dimensional term which defines a two dimensional theory and the standard machinery implemented to gauge a field theory fails.333Of course the standard approach perfectly works for , replacing a derivative with a covariant derivative gives a gauge invariant expression.

3.2 WZNW and related models

3.2.1 The spectrum

The spectrum of the WZNW model is built with representations of the affine algebra, but which are the representations to consider is a subtle question.

The representations of the affine algebra are generated from those of the global algebra by freely acting with the current modes , obeying the following commutation relations

(3.2.1)
(3.2.2)
(3.2.3)

with level .

In the first attempts to define a consistent spectrum for the worldsheet theory of string theory on and [36, 38][65]-[69] only representations with bounded from below were considered. These decompose into direct products of the normalizable continuous, highest and lowest weight discrete representations.

The principal continuous representations contain the states