Showing papers by "Pierre Mathieu published in 1997"

Affine Lie Algebras

Abstract: This chapter is a basic introduction to affine Lie algebras, preparing the stage for their application to conformal field theory. In Sect. 14.1.1, after having introduced the affine Lie algebras per se, we show how the fundamental concepts of roots, weights, Cartan matrices, and Weyl groups are extended to the affine case. Section 14.2 introduces the outer automorphism group of affine Lie algebras, which is generated by the new symmetry transformations of the extended Dynkin diagram. The following section describes highest-weight representations, focusing on those whose highest weight is dominant. Characters for these representations are introduced in Sect. 14.4. Their modular properties are presented in the following sections, where various properties of their modular S matrices are also reported. The affine extension of finite Lie algebra embeddings is presented in Sect. 14.7. Four appendices complete the chapter. The first one contains the proof of a technical identity related to outer automorphism groups. The second appendix displays an explicit basis (in terms of semi-infinite paths) for the states in integrable representations of affine su(N). In the third one, the modular transformation properties of the affine characters are derived. The final appendix lists all the symbols pertaining to affine Lie algebras.

Logarithmic Yangians in WZW Models

Abstract: A new action of the Yangians in the WZW models is displayed. Its structure is generic and level-independent. This Yangian is the natural extension at the conformal point of the one unraveled in massive theories with current algebras. Hopefully, this new symmetry of WZW models will lead to a deeper understanding of the integrable structure of conformal field theories and their deformations.

Quantum Field Theory

Abstract: This chapter provides a quick—and therefore incomplete—introduction to quantum field theory. Those among our readers who know little about it will find here the basic material allowing them to appreciate and understand the remaining chapters of this book. Section 2.1 explains the canonical quantization of free fields, bosons and fermions, starting from a discrete formulation. It is appropriate for readers without any previous knowledge of quantum field theory; some experience with quantum mechanics remains an essential condition, however. Section 2.2 reviews the path-integral formalism of quantum mechanics for a single degree of freedom, and then for quantum fields, especially fermions. Section 2.3 introduces the central notion of a correlation function, both in the canonical and path-integral formalisms. The Wick rotation to imaginary time is performed, with the example of the free massive boson illustrating the exponential decay of correlations with distance. Section 2.4 explains the meaning of a symmetry transformation and the consequences of symmetries in classical and quantum field theories. This section deserves special attention—even from experienced readers—because the notion of a symmetry transformation and how it is implemented is fundamental to this work. Section 2.5 is devoted to the energy-momentum tensor, the conserved current associated with translation invariance, which plays a central role as the generator of conformal transformations when suitably modified.

Global Conformal Invariance

Abstract: This relatively short chapter provides a general introduction to conformal symmetry in arbitrary dimension. Conformal transformations are introduced in Sect. 4.1, with their generators and commutation relations. The conformal group in dimension d is identified with the noncompact group SO(d + 1,1). In Sect. 4.2 we study the action of a conformal transformation on fields, at the classical level. The notion of a quasi-primary field is defined. We relate scale invariance, conformal invariance, and the tracelessness of the energy momentum tensor. In Sect. 4.3 we look at the consequences of conformal invariance at the quantum level on the structure of correlation functions. The form of the two- and three-point functions is given, and the Ward identities implied by conformal invariance are derived. Aspects of conformal invariance that are specific to two dimensions, including local (not globally defined) conformal transformations, are studied in the next chapter. However, the proof that the trace Tμμvanishes for a two-dimensional theory with translation, rotation, and dilation invariance is given at the end of the present chapter.

Simple Lie Algebras

Abstract: This chapter presents a survey of the theory of Lie algebras. This might appear somewhat remote from our main subject of interest: affine Lie algebras and their applications to conformal field theory. However, it turns out that in many respects the theory of affine Lie algebras is a natural extension of the theory of simple Lie algebras, and as such cannot be studied efficiently in isolation. This is an immediate motivation for devoting a complete chapter to Lie algebras. But as subsequent developments will show, conformal field theories with nonaffine additional symmetries, such as W algebras, parafermions, and son on, as well as related exactly solvable statistical models, also have a deep Lie-algebraic underlying structure, which can only be appreciated with a minimal background on simple Lie algebras.

Conformal Invariance in Two Dimensions

Abstract: Conformal invariance takes a new meaning in two dimensions. As already apparent in Section 4.1, the case d = 2 requires special attention. Indeed, there exists in two dimensions an infinite variety of coordinate transformations that, although not everywhere well-defined, are locally conformal: they are holomorphic mappings from the complex plane (or part of it) onto itself. Among this infinite set of mappings one must distinguish the 6-parameter global conformal group, made of one-to-one mappings of the complex plane into itself. The analysis of the previous chapter still holds when considering these transformations only. However, a local field theory should be sensitive to local symmetries, even if the related transformations are not globally defined. It is local conformal invariance that enables exact solutions of two-dimensional conformal field theories.

Minimal Models I

Abstract: Chapters 5 and 6 dealt with general properties of two-dimensional conformal field theories. The present chapter is devoted to particularly simple conformal theories called minimal models. These theories are characterized by a Hilbert space made of a finite number of representations of the Virasoro algebra (Verma modules); in other words, the number of conformal families is finite. Such theories describe discrete statistical models (e.g., Ising, Potts, and so on) at their critical points. Their simplicity in principle allows for a complete solution (i.e., an explicit calculation of all the correlation functions). The discovery of minimal models and their identification with known statistical models at criticality constitutes the greatest application of conformal invariance so far. Since a detailed study of minimal models may rapidly become highly technical, we have split the discussion among two chapters (this one and the next). The present chapter first explains some general features of Verma modules (Sect. 7.1), and in particular the occurrence of states of zero norm, which must be quotiented out. In Sect. 7.2 the question of unitarity is discussed and the Kac determinant is introduced. In Sect. 7.3 a survey of the theory of minimal models is presented. In Sect. 7.4 various examples of the correspondence between minimal theories and statistical systems are described. The next chapter will be devoted to more technical issues and will provide proofs for some assertions of the present chapter.

Fusion Rules in WZW Models

Abstract: This chapter is mainly concerned with the calculation of fusion rules in WZW models. As in any rational conformal field theory, fusion rules can be calculated from the Verlinde formula, in terms of the modular transformation matrices S. For WZW models, all these matrices are known explicitly. However, working out a few examples is convincing enough that the Verlinde formula is not very convenient for explicit calculations. Indeed, the dimension of the matrix S grows very quickly with the level and the rank of the algebra. Nevertheless, the formula itself allows us to derive useful identities (which is done in Sect. 16.1), in addition to being the starting point for a more efficient approach: the affine extension of the character method used for the calculation of tensor products in finite Lie algebra. This method is described in Sect. 16.2. It leads to a very nice relation between tensor-product coefficients and fusion coefficients. The concept of quantum dimension, naturally related to the character method, is introduced in Sect. 16.3.

Probing integrable perturbations of conformal theories using singular vectors II: N=1 superconformal theories

Abstract: In this work we pursue the singular-vector analysis of the integrable perturbations of conformal theories that was initiated in hep-th/9603088. Here we consider the detailed study of the N=1 superconformal theory and show that all integrable perturbations can be identified from a simple singular-vector argument. We identify these perturbations as theories based on affine Lie superalgebras and show that the results we obtain relating two perturbations can be understood by the extension of affine Toda duality to these theories with fermions. We also discuss how this duality is broken in specific cases.