Vertex algebras and algebraic curves (Q2735995)

Vertex operators were mathematically introduced by \textit{J. Lepowsky} and \textit{R. L. Wilson} [Commun. Math. Phys. 62, 43-53 (1978; Zbl 0388.17006)] in order to find explicit irreducible integral modules of affine Kac-Moody algebras. Vertex algebra was introduced by \textit{R. I. Borcherds} [Proc. Natl. Acad. Sci. USA 83, 3068-3071 (1986; Zbl 0613.17012)], and the slightly revised notion ``vertex operator algebra'' was given by \textit{I. Frenkel, J. Lepowsky} and \textit{A. Meurman} [Vertex Operator Algebras and the Monster, Pure Appl. Math., Academic Press (1988; Zbl 0674.17001)], in connection with the moonshine representation of the Monster group. The equivalent concept ``Chiral algebra'' appeared in physics as the fundamental algebraic structure in conformal field theory. Beilinson and Drinfeld gave an algebraic geometric interpretation of vertex algebra.NEWLINENEWLINENEWLINEThe aim of this book is to bridge the gap between the algebraic approach of vertex (operator) algebras and the algebraic geometric one. The key point is to make vertex operators coordinate-independent, thus effectively getting rid of the formal variable. This was achieved by attaching to each vertex algebra a vector bundle with a flat connection on the (formal) disc, equipped with an intrinsic operation. The formal variable is restored when a coordinate on the disc is chosen. The fact that the operation is independent of this choice follows from the axioms of vertex algebra. The authors also attached to each vertex algebra and any pointed algebraic curve the space of coinvariants and conformal blocks. When the curve \(X\) and other data on \(X\) (such as \(G\)-bundles) vary, these spaces combine into a sheaf on the relevant moduli space. A relation with Drinfeld-Sokolov reductions is also given.