Abstract: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.
From: Elchanan Nafcha [view email]
[v1]
Fri, 12 Jun 2026 00:04:27 UTC (36 KB)
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