Vertex operator algebras associated to representations of affine and Virasoro Algebras
TL;DR: In this paper, the integrable highest-weight representations of affine Lie algebras or loop algesbras were constructed by Kac i-K, inspired by the generalization of the Weyl denominator formula for affine roots systems discovered earlier by Macdonald [M].
Abstract: The first construction of the integrable highest-weight representations of affine Lie algebras or loop algebras by Kac i-K] was greatly inspired by the generalization of the Weyl denominator formula for affine roots systems discovered earlier by Macdonald [M]. Though the Macdonald identity found its natural context in representation theory, its mysterious modular invariance was not understood until the work of Witten [W-I on the geometric realization of representations of the loop groups corresponding to loop algebras. The work of Witten clearly indicated that the representations of loop groups possess a very rich structure of conformal field theory which appeared in physics literature in the work of Belavin, Polyakov, and Zamolodchikov [BPZ-I. Independently (though two years later), Borcherds, in an attempt to find a conceptual understanding of a certain algebra of vertex operators invariant under the Monster [FLM1], introduced in [B-I a new algebraic structure. We call vertex operator algebras a slightly modified version of Borcherd’s new algebras [FLM2].
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TL;DR: In this article, it was shown that the characters of the integrable highest weight modules of affine Lie algebras and the minimal series of the Virasoro algebra give rise to conformal field theories.
Abstract: In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z) is manifest. In the study of conformal field theory, physicists arrived at the notion of chiral algebras (see e.g. [MS]). Independently, in the attempt to realize the Monster sporadic group as a symmetry group of certain algebraic structure, an infinite dimensional graded representation of the Monster sporadic group, the so called Moonshine module, was constructed in [FLM1]. This algebraic structure was later found in [Bo] and called the vertex algebra; the first axioms of vertex operator algebras were formulated in that paper. The proof that the Moonshine module is a vertex operator algebra and the Monster group acts as its automorphism group was given in [FLM2]. Notably the character of the Moonshine module is also a modular function, namely j(τ) − 744. It turns out that the vertex operator algebra can be regarded as a rigorous mathematical definition of the chiral algebras in the physical literature. And it is expected that a pair of isomorphic vertex operator algebras and their representations (corresponding to the holomorphic and antiholomorphic sectors) are the basic objects needed to build a conformal field theory of a certain type.
1,122 citations
TL;DR: In this paper, the authors derived new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk.
Abstract: We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra
representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the “face” formulation for any type of Lie algebra
$$\mathfrak{g}$$
and arbitrary finite-dimensional representations of
. We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq→1 these solutions degenerate again into
solutions with
$$q' = \exp \left( {\frac{{2\pi i}}{{k + g}}} \right)$$
. We also study the simples examples of solutions of our holonomic difference equations associated to
$$U_q (\widehat{\mathfrak{s}\mathfrak{l}(2)})$$
and find their expressions in terms of basic (orq−)-hypergeometric series. In the special case of spin −1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.
683 citations
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TL;DR: In this paper, the relation between the g-twisted V-modules and Ag(V)-modules is established, and it is proved that if V is g-rational, then Ag (V) is finite-dimensional semi-simple associative algebra and there are only finitely many irreducible g-two-stuck Vmodules.
Abstract: This paper gives an analogue of Ag(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and Ag(V)-modules is established. It is proved that if V is g-rational, then Ag(V) is finite-dimensional semi-simple associative algebra and there are only finitely many irreducible g-twisted V-modules.
486 citations
TL;DR: In this article, it was shown that the vertex operator superalgebra V has a vertex (super)algebra structure with M as a module, and for a vertex operator V, giving a V-module M is equivalent to giving a vertex operators homomorphism from V to some local system of vertex operators on a vector space.
480 citations
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TL;DR: In this paper, the authors present an investigation of the massless, two-dimentional, interacting field theories and their invariance under an infinite-dimensional group of conformal transformations.
4,595 citations
TL;DR: In this article, a non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented, which is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.
Abstract: A non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented. Any fermi theory in 1+1 dimensions is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.
2,231 citations
Journal Article•
TL;DR: In this paper, complex realizations of vertex operator algebraic expressions are presented, and the main theorem of complex realisation of vertices operator algebra is proved. But the complexity is not discussed.
Abstract: Lie Algebras. Formal Calculus: Introduction. Realizations of sl(2) by Twisted Vertex Operators. Realizations of sl(2) by Untwisted Vertex Operators. Central Extensions. The Simple Lie Algebras An, Dn, En. Vertex Operator Realizations of An, Dn, En. General Theory of Untwisted Vertex Operators. General Theory of Twisted Vertex Operators. The Moonshine Module. Triality. The Main Theorem. Completion of the Proof. Appendix: Complex Realization of Vertex Operator Algebras. Bibliography. Index of frequently used symbols. Index.
2,010 citations
TL;DR: An integral form is constructed for the universal enveloping algebra of any Kac-Moody algebras that can be used to define Kac's groups over finite fields, some new irreducible integrable representations, and a sort of affinization of anyKac-moody algebra.
Abstract: It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The “Moonshine” representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
1,517 citations
TL;DR: In this article, the authors define the classical limit of a conformal field theory as a limit in which the conformal weights of all primary fields vanish, and define chiral vertex operators and duality matrices and review the fundamental identities they satisfy.
Abstract: We define chiral vertex operators and duality matrices and review the fundamental identities they satisfy. In order to understand the meaning of these equations, and therefore of conformal field theory, we define the classical limit of a conformal field theory as a limit in which the conformal weights of all primary fields vanish. The classical limit of the equations for the duality matrices in rational field theory together with some results of category theory, suggest that (quantum) conformal field theory should be regarded as a generalization of group theory.
1,305 citations