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

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 do indeed give 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.