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

Given a vertex algebra $V$, Zhu constructed an associative algebra $A(V)$, whose representation theory provides an approximation to the category of $V$-modules. We describe a geometric construction of a certain derived associative algebra ${\mathfrak{Z}}_{\mathcal{A}}^0$ associated to any universal factorization algebra $\mathcal{A}$, whose zeroth homology recovers Zhu’s associative algebra in the case where $\mathcal{A}$ is obtained from a vertex algebra. The construction is given by integrating $\mathcal{A}$ over stable configurations parametrized by two-pointed semistable genus zero curves. By a variant of this construction, we obtained a chiral $\mathcal{A}$-bimodule ${\mathfrak{Z}}_{\mathcal{A}}$ describing the value of a factorization algebra at a nodal point. We show that its zeroth homology recovers the bimodule underlying the mode-transition algebra $\mathfrak{A}(V)$ defined by Damiolini, Gibney, and Krashen, which they use to describes a vertex algebra at a node. Finally, the value over a formal smoothing of a node provides a deformation ${\tilde{\mathfrak{Z}}}_{\mathcal{A}}$ of ${\mathfrak{Z}}_{\mathcal{A}}$, whose zeroth homology describes a deformation of the mode-transition algebra in the case of a vertex algebra.