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 \(\text {VA}(A,H,S)\).
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Notes
More generally, these are the axioms given by Borcherds for a (symmetric) bicharacter from an arbitrary commutative cocommutative bialgebra into \({\mathbb {C}}[(x_1 - x_2)^{\pm 1}]\); he uses this general set-up to produce other examples of \((A, H, S_B)\)-vertex algebras, but this example is sufficient for our purposes.
In the notation of Frenkel–Ben-Zvi, this map, defined on a formal disk around a point \((x,x) \in X^2\), is called \({\mathcal {Y}}_x^2\).
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Acknowledgements
I thank Dominic Joyce for introducing me to the paper [4], and for providing, via his work [9] on vertex algebra structures on the homology of moduli spaces, the motivation to study the relationship between Borcherds’s constructions and factorization algebras. Thanks also to Kobi Kremnizer, Thomas Nevins, and Reimundo Heluani for helpful discussions, and to Yi-Zhi Huang for useful email communications. I am grateful as well to Anna Romanov for comments on a draft of the paper. Finally, I thank the anonymous referee for their careful reading and extremely useful suggestions.
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Communicated by Y. Kawahigashi.
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Cliff, E. Chiral Algebras, Factorization Algebras, and Borcherds’s “Singular Commutative Rings” Approach to Vertex Algebras. Commun. Math. Phys. 392, 399–426 (2022). https://doi.org/10.1007/s00220-021-04307-4
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DOI: https://doi.org/10.1007/s00220-021-04307-4