This 60 page survey from 1997 gives an excellent overview over the concepts used in 2-dimensional conformal field theory (CFT). it sticks to the concepts, which are explained well, rather than drowning their meaning in detailed calculations or proofs. This makes this survey much more readable than other CFT texts - the latter are much easier to read after having seen the main outline here.
The survey should be considered as a tourist guide for visiting the beauties of the country of 2-dimensional conformal field theories. Everything of interest is described just enough to understand why it is worth a visit, i.e., an invitation for deeper study. Thus it is full of concepts that take time to digest thoroughly when met for the first time - understanding the concepts probably requires a true visit, hence going to a textbook or more specialized literature. (Many references are given.) However, to plan your tours the guide is likely to be very useful even with a partial understanding!
The material is grouped into four lectures of 6 or 7 sections each and a postlude with omissions and outlook. Reviewing the review here means only giving a bird's eye view of the field.
Lecture 1 explains in seven sections what conformal quantum theory (QFT) is about - confessing in Section 1 that the concept is ill-defined, different schools calling different but related subjects by this name. (Indeed, one can find texts with almost completely disjoint notation and content, and it takes expertise to recognize that it is in essence the same subject.) What one gathers is that there are fields and observables (though the latter become mostly invisible in approaches based on vertex algebras, Hopf algebras, or Riemannian surfaces). Fields of a quantum field theory are local fields of massless particles (though this in nowhere clearly stated); thus their spectrum is the light cone, which is invariant under the infinitesimal symmetries of the conformal Lie algebra (which figures prominently). This is the reason for calling the subject conformal field theory.
Sections 2 and 3 spell out what observables are meant to be. On p.8 they are identified as distinguished basis elements of a Lie algebra $\cal W$ of infinitesimal symmetries. The basis consists of Hamiltonians generating a Cartan subalgebra and ladder operators defining a triangular decomposition of $\cal W$. This gives an intuitive idea of how it relates to textbook quantum field theory (QFT), though there the ladder operators are distributions only. The fact that one can avoid distributions in 2 dimensions makes this case far more tractable than the 4-dimensional case in which QFTs such as realistic QED or QCD live.
Dimension 2 is special since there the light cone just consists of two perpendicular lines, so the theory more or less factors into a product of two one-dimensional chiral CFT, a left-handed and a right-handed one. To promote the infinitesimal conformal symmetries to global symmetries one must compactify the 1D spaces by a point at infinity, turning them into a circle, usually taken as the unit circle in the complex plane; fields are then periodic functions on the circle. (This removes all infrared problems, as it eliminates the possibility of scattering.)
The conjugate momentum variable is given by the Fourier transform and therefore takes integer values $n$. Thus the ladder operators in a chiral CFT are indexed by $n$; those with negative (positive) $n$ are creation (annihilation) operators. However, unlike in standard QFT, where the commutator of a creation operator and the corresponding annihilation operator is a simple complex number (the central charge), in a CFT, these commutators are linear combinations of Hamiltonians and the central charge. Moreover, operators corresponding to different momenta (i.e., different $|n|$) do not need to commute.
States annihilated by all annihilation operators are called highest weight vectors, a name borrowed from the representation theory of finite-dimensional semisimple Lie algebras. (However, the Lie algebra of a CFT is infinite dimensional.) The Hamiltonians act on these by multiplication with a complex number called the weight; the weight is therefore a vector in the dual of the Cartan subalgebra. Acting with all Lie algebra elements of a highest weight vector defines the elements of the corresponding highest weight module (in algebraic quantum field theory called a superselection sector). In a unitary chiral CFT, these modules define irreducible unitary representations of the Lie algebra. There is a distinguished highest state vector called the vacuum vector, and the corresponding highest weight module defines the vacuum sector of the theory, which in algebraic QFT is the Hilbert space of the corresponding Wightman theory.
There is a distinguished Hamiltonian $L_0$, the generator of dilations in the complex plane; the conformal weight of a highest weight vector or module is defined as the weight of $L_0$. $L_0$ is part of an indexed sequence $L_n$ of generators of the conformal symmetries, which span a Lie algebra, the so-called Virasoro algebra. The (fully understood) representation theory of the Virasoro algebra, discussed in Section 4, is an important ingredient for an understanding of 2-dimensional CFT.
Section 5 discusses chiral conformal fields. Transforming from the Fourier domain to the position domain, with complexified positions $z$ (whose physical values are on the unit circle) introduces singularities at $z=0$, which are handled by expanding the fields into Laurent series; the coefficients can be recovered from the fields by taking appropriate residues. If the Laurent series are taken as formal series only (i.e., if questions of convergence are ignored) one ends up with so-called vertex operators. Their properties can be formulated axiomatically, leading to so-called vertex operator algebras (VOAs). These algebras encode in particular the so-called state-field correspondence, which says that physical states (vectors in the Hilbert space) and local fields (Fourier transforms of the algebra $\cal W$) are in 1-1 correspondence. Fuchs discusses VOAs only in passing, but mentions some references in the omission section at the end. The fields corresponding to highest weight vectors in this correspondence are the primary fields.
Section 6 discusses the operator product expansion, which tells how chiral fields multiply. The singularities imply that chiral fields are distribution-valued operators only, and their multiplication at coinciding arguments is not defined. However, the product of chiral fields at distinct arguments $z$ and $w$ is defined and singularities develop only for $z-w\to 0$. The chiral CFT is called rational if these singularities are poles only; the expansion of the product in terms of a Laurent series is called the operator product expansion. The rationality assumption allows a far reaching analysis and covers the most important examples.
Section 7 discusses how a 2-dimensional CFT is built from its two chiral CFTs and how its $N$-point correlation functions (the Wightman functions of axiomatic QFT) are constructed. For up to 3 points, the correlation functions are determined through differential equations, the Ward identities corresponding to the conformal symmetries. For $4$-point correlations, one can use the conformal symmetry to place three points at $0,1,\infty$ and gets a decomposition of the dependence on the remaining point in terms of the chiral parts, leading to so-called conformal blocks. If this is expressed in terms of vertex operators, one gets a corresponding decomposition into chiral vertex operators. These are multivalued fields, i.e., fields on an infinite-sheeted covering of the complex plane. Their commutation properties are therefore governed by braiding symmetries rather than the permutations symmetries familiar from 4 dimensions. (As a result, there are interesting relations to quantum groups, Hopf algebras, and Yang-Baxter equations, but these are delegated to the omission section.)
This completes my bird's eye view of the first lecture. The other lectures are as rich in content but I'll describe them much more briefly.
Lecture 2 discusses in Sections 8-11 the representation theory of CFTs (fusion rules and characters, roughly analogous to the representation theory of semisimple Lie algebras), duality (crossing symmetry) expressing the locality of the fields, and modularity. The latter is an additional $sl(2,Z)$ symmetry that enables on to extend CFTs to general punctured Riemannian surfaces, the study of which is relevant for perturbative string theory. Again, the details are delegated to the omission section. (For CFT from the string theory point of view, see the introduction by Schweigert, Fuchs and Walcher.
It is remarkable that the most elementary examples of CFT (free bosons and - in passing - orbifold theories) are only met in Section 12; the reason is that their discussion involves many of the concepts introduced before. Further examples - the WZW theories, also called WZNW theories, in Section 21, coset theories in Section 25) appear much later in Lecture 4, as their construction and the discussion of their properties needs a thorough mathematical background about Kac-Moody algebras (an infinite-dimensional generalization of simple Lie
algebras).
The latter is provided in Lecture 3, which can be read independently of the other lectures. Although in principle self-contained, it may not be easy reading unless the reader is at least superficially acquainted with the structure of semisimple Lie algebras.
Now I invite you to read the tourist guide, make one or two of the tours, return to the guide, and iterate until you are happy with your understanding. You are welcome to ask questions that you can't resolve on your own in the Q&A section of this site. I wish you happy reading!
Q&A (4870)
