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

Abstract

Given a universal factorization algebra 𝒜, we constructed in our previous paper a derived associative algebra Z_𝒜⁰, together with chiral modules mathfrakZ_𝒜⁺ and Z_𝒜^- associated to a puncture, and a chiral bimodule Z_𝒜 associated to a node. Furthermore, these constructions assemble to a factorization 𝒜-module over any family of nodal punctured curves. In this paper, we show that in the case where 𝒜 is constructed from a quasi-conformal vertex algebra V, the zeroth homology algebra H⁰Z_𝒜 is naturally isomorphic to Zhu's associative algebra A(V), and we identify H⁰Z_𝒜 with the bimodule underlying the mode-transition algebra of Damiolini-Gibney-Krashen. We also give an explicit description of the smoothing module H⁰Z̃_𝒞 which describes the deformation of H⁰Z_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.

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BibTeX RIS

BibTeX

@misc{nafcha2026nodal,
  title = {{Nodal degeneration of chiral algebras II: Local structure and chiral Zhu algebras}},
  author = {Elchanan Nafcha},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.13986},
  eprint = {2606.13986},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.13986},
}

RIS

TY  - UNPB
TI  - Nodal degeneration of chiral algebras II: Local structure and chiral Zhu algebras
AU  - Elchanan Nafcha
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2606.13986
UR  - https://arxiv.org/abs/2606.13986
AB  - Given a universal factorization algebra 𝒜, we constructed in our previous paper a derived associative algebra Z_𝒜^0, together with chiral modules mathfrakZ_𝒜^+ and Z_𝒜^- associated to a puncture, and a chiral bimodule Z_𝒜 associated to a node. Furthermore, these constructions assemble to a factorization 𝒜-module over any family of nodal punctured curves. In this paper, we show that in the case where 𝒜 is constructed from a quasi-conformal vertex algebra V, the zeroth homology algebra H^0Z_𝒜 is naturally isomorphic to Zhu's associative algebra A(V), and we identify H^0Z_𝒜 with the bimodule underlying the mode-transition algebra of Damiolini-Gibney-Krashen. We also give an explicit description of the smoothing module H^0Z̃_𝒞 which describes the deformation of H^0Z_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.
ER  -

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