The first chiral homology group in higher genus
Abstract
We extend the theory of the first chiral homology group of vertex algebras, developed by van Ekeren and Heluani for elliptic curves, to compact Riemann surfaces of arbitrary genus. Our approach realizes a genus g surface by iterated self-sewing of g handles onto the Riemann sphere, each governed by a sewing parameter ρi in a punctured disc. We construct an explicit complex computing the chiral homology groups Hch0 and Hch1 of a vertex algebra V on a genus g surface with n marked points, and equip it with a projectively flat connection, with an explicit central-charge anomaly, over the g-dimensional space of sewing parameters. We prove a genus g Fourier-space Borcherds identity for the associated modified vertex operators. We show that two finiteness assumptions 1) finite dimensionality of the first Poisson homology HP1(RV) of the Zhu C2-algebra, and 2) finite generation of a certain Koszul homology of the associated graded algebra, imply finite dimensionality of Hch1(X,V) for every genus g and every vertex algebra V. Using the degeneration ρi → 0 together with the factorization theorem of Damiolini-Gibney-Tarasca, we relate the totally degenerate limit of H1ch to the Hochschild homology of an iterated construction on the Zhu algebra, and deduce vanishing of the first chiral homology group in every genus for rational vertex algebras.
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BibTeX
@misc{zuevsky2026first,
title = {{The first chiral homology group in higher genus}},
author = {A. Zuevsky},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.24719},
eprint = {2607.24719},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24719},
note = {CC0},
}
RIS
TY - UNPB
TI - The first chiral homology group in higher genus
AU - A. Zuevsky
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2607.24719
UR - https://arxiv.org/abs/2607.24719
AB - We extend the theory of the first chiral homology group of vertex algebras, developed by van Ekeren and Heluani for elliptic curves, to compact Riemann surfaces of arbitrary genus. Our approach realizes a genus g surface by iterated self-sewing of g handles onto the Riemann sphere, each governed by a sewing parameter ρ_i in a punctured disc. We construct an explicit complex computing the chiral homology groups H^ch_0 and H^ch_1 of a vertex algebra V on a genus g surface with n marked points, and equip it with a projectively flat connection, with an explicit central-charge anomaly, over the g-dimensional space of sewing parameters. We prove a genus g Fourier-space Borcherds identity for the associated modified vertex operators. We show that two finiteness assumptions 1) finite dimensionality of the first Poisson homology HP_1(R_V) of the Zhu C_2-algebra, and 2) finite generation of a certain Koszul homology of the associated graded algebra, imply finite dimensionality of H^ch_1(X,V) for every genus g and every vertex algebra V. Using the degeneration ρ_i → 0 together with the factorization theorem of Damiolini-Gibney-Tarasca, we relate the totally degenerate limit of H_1^ch to the Hochschild homology of an iterated construction on the Zhu algebra, and deduce vanishing of the first chiral homology group in every genus for rational vertex algebras.
ER -