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

TL;DR

This paper explores Borcherds's approach to vertex algebras through singular commutative rings, introduces new examples, and compares their relationships with chiral and factorization algebras, revealing non-equivalence in realizations.

Contribution

It introduces new categories for Borcherds's constructions and demonstrates their capacity to realize all vertex and chiral algebras, while highlighting non-equivalence of realizations.

Findings

All vertex and chiral algebras can be realized in the new categories.

The functors from these categories to traditional algebras are not equivalences.

A single algebra may have multiple non-equivalent realizations.

Abstract

We recall Borcherds's approach to vertex algebras via "singular commutative rings", and introduce new examples of his constructions which we compare to vertex algebras, chiral algebras, and factorization algebras. We show that all vertex algebras (resp. chiral algebras or equivalently factorization algebras) can be realized in these new categories $\text{VA}(A,H,S)$, but we also show that the functors from $\text{VA}(A,H,S)$ to vertex algebras or chiral algebras are not equivalences: a single vertex or chiral algebra may have non-equivalent realizations as an $(A, H,S)$-vertex algebra.