Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras

Abstract

Let X be a complex manifold, π:E→X a locally trivial holomorphic fibration with fiber F, and (g,〈•,•〉) a Lie algebra with an invariant symmetric form. We associate to this data a holomorphic prefactorization algebra Fg,π on X in the formalism of Costello-Gwilliam. When X=C, g is simple, and F is a smooth affine variety, we extract from Fg,π a vertex algebra which is a vacuum module for the universal central extension of the Lie algebra g⊗H0(F,O)[z,z−1]. As a special case, when F is an algebraic torus (C⁎)n, we obtain a vertex algebra naturally associated to an (n+1)–toroidal algebra, generalizing the affine vacuum module.