Vector Bundles of Coinvariants for Admissible Affine Vertex Operator Algebras

Abstract

Simple affine vertex operator algebras L_k(g) at non-integral admissible levels k are generally neither C₂-cofinite nor rational. Despite the absence of these standard finiteness conditions, we prove that the sheaf of coinvariants (and, dually, of conformal blocks) of ordinary modules in the category 𝒪_k forms a vector bundle over the moduli space ℳ̅_0,n of stable n-pointed genus-zero curves. The proof relies on establishing the finite-dimensionality of genus-zero coinvariants, a vanishing theorem that isolates ordinary admissible modules at the boundary, and a smoothing theorem enabled by partial strong identity elements in the mode transition algebra. Consequently, admissible affine vertex operator algebras supply a rich class of non-rational, non-C₂-cofinite examples in which vector-bundle behavior is preserved for a well-behaved subcategory of modules.

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BibTeX RIS

BibTeX

@misc{alekseev2026vector,
  title = {{Vector Bundles of Coinvariants for Admissible Affine Vertex Operator Algebras}},
  author = {Victor Alekseev and Jianqi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09154},
  eprint = {2609.09154},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09154},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Vector Bundles of Coinvariants for Admissible Affine Vertex Operator Algebras
AU  - Victor Alekseev
AU  - Jianqi Liu
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09154
UR  - https://arxiv.org/abs/2609.09154
AB  - Simple affine vertex operator algebras L_k(g) at non-integral admissible levels k are generally neither C_2-cofinite nor rational. Despite the absence of these standard finiteness conditions, we prove that the sheaf of coinvariants (and, dually, of conformal blocks) of ordinary modules in the category 𝒪_k forms a vector bundle over the moduli space ℳ̅_0,n of stable n-pointed genus-zero curves. The proof relies on establishing the finite-dimensionality of genus-zero coinvariants, a vanishing theorem that isolates ordinary admissible modules at the boundary, and a smoothing theorem enabled by partial strong identity elements in the mode transition algebra. Consequently, admissible affine vertex operator algebras supply a rich class of non-rational, non-C_2-cofinite examples in which vector-bundle behavior is preserved for a well-behaved subcategory of modules.
ER  -

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