# Chiral algebras, factorization algebras, and Borcherds' "singular commutative rings" approach to vertex algebras

Emily Cliff

## Affiliation:

University of Sydney

## Date:

Tue, 12/11/2019 - 12:00pm to 1:00pm

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

A vertex algebra describes the symmetries of a two-dimensional conformal field theory (CFT), while a factorization algebra (introduced by Beilinson and Drinfeld) over a complex curve consists of local data in such a field theory. Roughly, the factorization structure encodes collisions between local operators. The two perspectives are equivalent, in the sense that, given a fixed open affine curve $X$, the category of vertex algebras over $X$ is equivalent to the category of factorization algebras over $X$. In the late 1990s, Borcherds gave an alternate definition of some vertex algebras as "singular commutative rings" in a category of functors depending on some input data $(A,H,S)$. He proved that for a certain choice of $A$, $H$, and $S$, the singular commutative rings he defines are indeed examples of vertex algebras. In this talk I will explain how we can vary this input data to produce categories of chiral algebras and factorization algebras (in the sense of Beilinson--Drinfeld) over certain complex curves $X$. We'll also discuss the failure of these constructions to give equivalences of categories. I will not assume background in vertex algebras, factorization algebras, or chiral algebras.