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

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 VA(A, H, S), but we also show that the functors from 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.

Vertex algebras and $2$-monoidal categories

“…For a more comprehensive comparison between (C , H, A)-vertex algebras and classical vertex algebras, see [10].…”

“…We also note that υ A is trivially a morphism of algebras in C . Furthermore, since A(I) is a commutative algebra for all objects I in F , the Acknowledgements I thank Carina Boyallian for pointing out the interesting reference [10], and the referees for the careful reading of the manuscript.…”

“…[2,24]), as far as we know the categorical constructions in [9] and the singular tensor product have not been studied in general. However, in the interesting preprint [10] the author compares the definition of vertex algebra in [9] for the specific choice recalled in Thm. 5.9, and the classical definition of vertex algebra.…”