Representation theory of the vertex algebra \(W_{1+\infty}\). (English) Zbl 0862.17023

Representations of the (unique) central extension \(\widehat {\mathcal D}\) of the Lie algebra of differential operators (with a finite number of Fourier modes) on the circle are studied. The authors together with other collaborators began these studies in the articles [V. Kac and A. Radul, Commun. Math. Phys. 157, 429-457 (1993; Zbl 0826.17027)] and [E. Frenkel, V. Kac, A. Radul and W. Wang, Commun. Math. Phys. 170, 337-357 (1995; Zbl 0838.17028)]. The representations are studied with the help of irreducible highest weight representations of the central extension \(\widehat {gl} (\infty)\) of the Lie algebra of infinite matrices with only finitely many diagonals and with the help of vertex algebras. For every central charge \(c\in \mathbb{C}\) there is an induced \(\widehat {\mathcal D}\)-module \(M_c\). This module admits a canonical vertex algebra structure. The unique irreducible quotient is denoted by \(W_{1+ \infty}\). The highest weight representations of the vertex algebra \(M_c\) are in canonical \(1-1\) correspondence to the highest weight representations of \(\widehat {\mathcal D}\) with central charge \(c\). The goal is to describe the representations of the simple vertex algebra \(W_{1+ \infty}\). By suitable normalisation of the defining cocycle for \(\widehat {\mathcal D}\) the representation \(M_c\) is already irreducible for \(c \notin \mathbb{Z}\). The representations with \(c= N \in \mathbb{N}_0\) were studied in the above mentioned articles. The case \(c = -N\), \(N\in \mathbb{N}\) is considered in the article under review.
The main result is a decomposition of the vertex algebra of \(N\) charged free bosons with respect to the commuting pair of Lie algebras \(gl(N)\) and \(\widehat {gl} (\infty)\). In this way a large class of irreducible modules are produced. The authors conjecture that all irreducible modules can be obtained by applying certain constructions to them. Explicit character formulas are given. A basic tool is a modified theory of dual Howe pairs.


17B69 Vertex operators; vertex operator algebras and related structures
17B68 Virasoro and related algebras
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B70 Graded Lie (super)algebras
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