Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras
Files
First author draft
Date
2019-04-05
DOI
Authors
Szczesny, Maciej
Walters, Jackson
Williams, Brian
Version
First author draft
OA Version
Citation
Maciej Szczesny, Jackson Walters, Brian Williams. 2019. "Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras." https://arxiv.org/abs/1904.03176
Abstract
Let $X$ be a complex manifold, $\pi: E \rightarrow X$ 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 $(\mathbb{C}^{*})^n$, we obtain a vertex algebra naturally associated to an $(n+1)$--toroidal algebra, generalizing the affine vacuum module.