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

## TL;DR

This paper constructs holomorphic prefactorization algebras from complex manifolds with holomorphic fibrations, linking them to vertex algebras and toroidal algebras, thus extending the understanding of algebraic structures in complex geometry.

## Contribution

It introduces a new construction of holomorphic prefactorization algebras associated to holomorphic fibrations and relates them to vertex algebras and toroidal algebras.

## Key findings

- Construction of holomorphic prefactorization algebras from fibrations.
- Extraction of vertex algebras as vacuum modules in specific cases.
- Generalization of affine vacuum modules to toroidal algebras.

## 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.