Nodal degeneration of chiral algebras II: Local structure and chiral Zhu algebras

Abstract

Given a universal factorization algebra $\mathcal{A}$, we constructed in our previous paper a derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, together with chiral modules $\\mathfrak{Z}_{\mathcal{A}}^+$ and $\mathfrak{Z}_{\mathcal{A}}^-$ associated to a puncture, and a chiral bimodule $\mathfrak{Z}_{\mathcal{A}}$ associated to a node. Furthermore, these constructions assemble to a factorization $\mathcal{A}$-module over any family of nodal punctured curves. In this paper, we show that in the case where $\mathcal{A}$ is constructed from a quasi-conformal vertex algebra $V$, the zeroth homology algebra $H^0\mathfrak{Z}_{\mathcal{A}}$ is naturally isomorphic to Zhu's associative algebra $A(V)$, and we identify $H^0\mathfrak{Z}_{\mathcal{A}}$ with the bimodule underlying the mode-transition algebra of Damiolini-Gibney-Krashen. We also give an explicit description of the smoothing module $H^0\tilde{\mathfrak{Z}}_{\mathcal{C}}$ which describes the deformation of $H^0\mathfrak{Z}_{\mathcal{A}}$ which we attach to a smoothing family of a nodal curve. We therefore get a geometric interpretation of the Zhu algebra and the mode-transition algebra, as the integration of a factorization algebra over a certain compactification of configuration spaces of punctured nodal curves.