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Vertex operator algebras associated to representations of affine and Virasoro algebras. (English) Zbl 0848.17032

The first construction of the integrable highest-weight representations of affine Lie algebras or loop algebras by Kac was greatly inspired by the generalization of the Weyl denominator formula for affine root systems discovered earlier by Macdonald. Though the Macdonald identity found its natural context in representation theory, its mysterious modular invariance was not understood until the work of Witten 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. Independently (though two years later), R. E. Borcherds, in an attempt to find a conceptual understanding of a certain algebra of vertex operators invariant under the Monster, introduced in [Proc. Natl. Acad. Sci. USA 83, 3068-3071 (1986; Zbl 0613.17012)] a new algebraic structure. Here the authors call vertex operator algebras a slightly modified version of Borchered’s new algebras. In this paper the authors construct the vertex operator algebras corresponding to the highest weight representations of an affine Lie algebra \(\widehat {\mathfrak g}\) with the highest weight \(k \omega_0\), a multiple of the highest weight \(\omega_0\) of the basic representation. They sketch a similar construction for the Virasoro algebra and consider various applications of their results. In Section 1 they present the main definitions of vertex operator algebras, their representations, and their intertwining operators. They also define the universal enveloping algebra of a vertex operator algebra \(V\) and study its relation with the associative algebra \(A(V)\) introduced in [Y. Zhu, Vertex operator algebras, elliptic functions and modular forms, Ph. D. Dissertation, Yale Univ. (1990)]. They assign an \(A(V)\)-bimodule \(A(M)\) for every representation \(M\) of a vertex operator algebra \(V\), which provides an important tool to study the representations and the space of the intertwining operators.
In Section 2, after recalling some facts about affine Lie algebras and their representations, the authors find an explicit combinatorial expression for the matrix coefficients of the product of generating functions for affine Lie algebras, known as correlation functions for the currents in the physics literature. Using this formula, they are able to construct vertex operator algebras and their representations corresponding to certain induced representations of the affine Lie algebras for any value \(k\) of the central element \({\mathbf k}\).
The study is restricted to the case of the positive integral \(k\) in Section 3. They show that the integrable representations of the affine Lie algebras for a fixed value \(k\) provide the full set of irreducible representations of their vertex operator algebras. As an application of this construction, they compute the dimension of the space of intertwining operators for these algebras, known as the fusion rules in the physics literature. The first proof of the fusion rules in the case of \(\widehat {sl(2)}\) was given by Tsuchiya and Kanie; the general case has been stated without proof.
In Section 4 the authors show how their construction of vertex operator algebras can be extended to representations of the Virasoro algebra.
In conclusion they discuss various generalizations and applications of their results as well as some open problems. One of the interesting corollaries of the construction of vertex operator algebras associated to representations of affine Lie algebras combined with the theorem of the modular invariance (Zhu, loc. cit.) is the general structural explanation – without explicit calculation of characters – of the mysterious phenomenon of the modular invariance originated in Macdonald’s work.

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B68 Virasoro and related algebras

Citations:

Zbl 0613.17012
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References:

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