A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA ₁ ⁽¹⁾ .     

Generalized Drinfel'd-Sokolov hierarchies

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct theW _n ^(l) algebras, first discussed for the casen=3 andl=2 by Polyakov and Bershadsky.     

From geometric quantization to conformal field theory

Investigation of 2d conformal field theory in terms of geometric quantization is given. We quantize the so-called model space of the compact Lie group, Virasoro group and Kac-Moody group. In particular, we give a geometrical interpretation of the Virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of CFT (minimal models, 2d-gravity, WZNW theory). In the appendix we discuss the relation between classical (constant)r-matrices and this geometrical approach.     

Current algebra and Wess-Zumino model in two dimensions

We investigate quantum field theory in two dimensions invariant with respect to conformal (Virasoro) and non-abelian current (Kac-Moody) algebras. The Wess-Zumino model is related to the special case of the representations of these algebras, the conformal generators being quadratically expressed in terms of currents. The anomalous dimensions of the Wess-Zumino fields are found exactly, and the multipoint correlation functions are shown to satisfy linear differential equations. In particular, Witten's non-abelean bosonisation rules are proven.     

Unitary representations of the Virasoro and super-Virasoro algebras

It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.     

Constrained fermionic system and its equivalence with the free parafermionic theory and nonlinear sigma model

We construct the parafermionic (of orderq) representation of the Kac-Moody and Virasoro algebra and compare it with a constrained fermionic system. We find that the central charge of the Virasoro algebra of the constrained fermionic system depends on the regularization scheme. Using the path integral method, we demonstrate this dependence for theq=2 case and find that it can have the same central charge as the free parafermionic theory or the non-linear sigma model depending on the regularization scheme. We point out some ambiguity in the quantization of the constrained system in Hamiltonian formulation.     

Level one representations of the simple affine Kac-Moody algebras in their homogeneous gradations

Using the central charge of the Virasoro algebra as a clue, we recall the known constructions of theA, D, E algebras and discuss new Bosonic constructions of the non simply laced affine Kac-Moody algebras: the twistedA, D, E and theB, C, F, andG algebras. These involve interacting Fermions and a generalization of the Frenkel-Kac sign operators which do not form a 2-cocycle when the horizontal algebra has more than one short simple root.     

Sugawara construction for higher genus Riemann surfaces

By the classical genus zero Sugawara construction one obtains representations of the Virasoro algebra from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type). In this lecture, the classical construction is recalled first. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in a joint work with O. K. Sheinman. We were able to show that also in the higher genus, multi-point situation one obtains (from representations of the global algebras of affine type) representations of a centrally extended algebra of meromorphic vector fields on Riemann surfaces. The latter algebra is a generalization of the Virasoro algebra to higher genus.     

The Virasoro vertex algebra and factorization algebras on Riemann surfaces

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello–Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta–gamma system using the method of effective BV quantization.     

The A-D-E classification of minimal andA 1 (1) conformal invariant theories

We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant sesquilinear forms, with positive integral coefficients, in the characters of the Virasoro or of theA ₁ ⁽¹⁾ Kac-Moody algebras, which describe the corresponding partition functions on a torus. A remarkable correspondence emerges with simply laced Lie algebras.     

Systematic approach to conformal systems with extended Virasoro symmetries

The Toda field theories, which exist for every simple Lie group, are shown to give realizations of extended Virasoro algebras that involve generators of spins higher than or equal to two. They are uniquely determined from the canonical lagrangian formalism. The quantization of the Toda field theories gives a systematic treatment of generalized conformal bosonic models. The well-known pattern of conformal field theories with non-extended Virasoro algebra, appears to be repeated for any simple group, leading to a “periodic table”, parallel to the mathematical classification of simple Lie groups.     

Remarks on the quantization of conformal fields

The quantization of a general (b, c) system in two dimensions is formulated in terms of an infinite hierarchy of modules for the Virasoro algebra that interpolate between the space of classical conformal fields of weight j and the Dirac sea of semi-infinite forms. This provides a natural framework in which to study the relation between algebraic geometry and representations of the Virasoro algebra with central charge c_j=−2(6j²−6j+1). The importance of the construction is discussed in the context of string theory.     

On the structure of unitary conformal field theory II: Representation theoretic approach

Two-dimensional, unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras. Chiral algebras are equipped with a family of co-multiplications which serve to define tensor product representations. Chiral vertices arise as Clebsch-Gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra. The algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by Yang-Baxter (braid) matrices. Chiral fusion is analyzed. It is shown that the braid- and fusion matrices determine invariants of knots and links. Connections between the representation theories of chiral algebras and of quantum groups are sketched. Finally, it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras.     

The properties of conformal blocks,the AGT hypothesis,and knot polynomials

Various properties of correlators of the two-dimensional conformal field theory are discussed. Specifically, their relation to the partition function of the four-dimensional supersymmetric theory is analyzed. In addition to being of interest in its own right, this relation is of practical importance. For example, it is much easier to calculate the known expressions for the partition function of supersymmetric theory than to calculate directly the expressions for correlators in conformal theory. The examined representation of conformal theory correlators as a matrix model serves the same purpose. The integral form of these correlators allows one to generalize the obtained results for the Virasoro algebra to more complicated cases of the W algebra or the quantum Virasoro algebra. This provides an opportunity to examine more complex configurations in conformal field theory. The three-dimensional Chern–Simons theory is discussed in the second part of the present review. The current interest in this theory stems largely from its relation to the mathematical knot theory (a rather well-developed area of mathematics known since the 17th century). The primary objective of this theory is to develop an algorithm that allows one to distinguish different knots (closed loops in three-dimensional space). The basic way to do this is by constructing the so-called knot invariants.     

Current algebra and conformal discrete series

We describe a large class of two-dimensional conformal field theories based on a current algebra construction of Virasoro representations due to Goddard, Kent, and Olive. The basic tool is a generalization of the Feigin-Fuchs representation. All the theories are organized by chiral algebras, the simplest examples being the Virasoro and super-Virasoro algebras.     

Angular quantization and form factors in massive integrable models

We discuss an application of the method of angular quantization to the reconstruction of form factors of local fields in massive integrable models. The general formalism is illustrated with examples of the Klein-Gordon, sinh-Gordon and Bullough-Dodd models. For the latter two models the angular quantization approach makes it possible to obtain free field representations for form factors of exponential operators. We discuss an intriguing relation between the free field representations and deformations of the Virasoro algebra. The deformation associated with the Bullough-Dodd models appears to be different from the known deformed Virasoro algebra.     

Determinant bundles and Virasoro algebras

We consider the interplay of infinite-dimensional Lie algebras of Virasoro type and moduli spaces of curves, suggested by string theory. We will see that the infinitesimal geometry of determinant bundles is governed by Virasoro symmetries. The Mumford forms are just invariants of these symmetries. The representations of Virasoro algebra define (twisted)D-modules on moduli spaces; theseD-modules are equations on correlators in conformal field theory.To the memory of Vadik Knizhnik (20. 2. 1962–25. 12. 1987)     

Toward classification of conformal theories

By studying the representations of the mapping class groups which arise in 2D conformal theories we derive some restrictions on the value of the conformal dimension h_i of operators and the central charge c of the Virasoro algebra. As a simple application we show that when there are a finite number of operators in the conformal algebra, the h_i and c are all rational.