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

Abstract

Let $X$ be a complex manifold, $\pi :E\to X$ a locally trivial holomorphic fibration with fiber $F$, and $\mathfrak{g}$ a Lie algebra with an invariant symmetric form. We associate to this data a holomorphic prefactorization algebra ${\mathcal{F}}_{\mathfrak{g},\pi }$ on $X$ in the formalism of Costello-Gwilliam. When $X=\mathbb{C}$, $\mathfrak{g}$ is simple, and $F$ is a smooth affine variety, we extract from ${\mathcal{F}}_{\mathfrak{g},\pi }$ a vertex algebra which is a vacuum module for the universal central extension of the Lie algebra $\mathfrak{g}\otimes H^0(F,\mathcal{O})[z,z^{-1}]$. As a special case, when $F$ is an algebraic torus $({\mathbb{C}}^{\ast }{)}^n$, we obtain a vertex algebra naturally associated to an $(n+1)$--toroidal algebra, generalizing the affine vacuum module.