The Virasoro vertex algebra and factorization algebras on Riemann surfaces

TL;DR

This paper constructs the Virasoro vertex algebra from local Lie algebras, extends it to Riemann surfaces, and explores its role in conformal symmetry and factorization homology.

Contribution

It provides a new construction of the Virasoro vertex algebra from local Lie algebras and extends factorization algebras to Riemann surfaces, linking to conformal symmetry.

Findings

Constructed Virasoro vertex algebra from local Lie algebra.

Extended factorization algebra to Riemann surfaces.

Computed factorization homology and correlation functions.

Abstract

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as are the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization.