Restricted conformal blocks and the braiding on Zhu algebra modules

Abstract

Let V be a vertex operator algebra (VOA), not necessarily rational or C₂-cofinite, such that the category 𝒞₁(V) of C₁-cofinite grading-restricted generalized V-modules is semisimple and contains the generalized Verma modules associated to finite-dimensional modules over the Zhu algebra A(V). Such VOAs include affine VOAs at admissible and generic levels. We prove that the bottom degree of Huang-Lepowsky's P(w)-tensor product module M¹boxtimes_P(w)M² is isomorphic to the left A(V)-module M¹⊙_w,0M² arising from the restricted conformal blocks on (¶¹,∞,w,0). By shifting the marked points and using the parallel transport of the restricted conformal blocks, we construct an explicit braiding isomorphism on the category of finite-dimensional A(V)-modules, which is the restriction of the braiding isomorphism of Huang-Lepowsky-Zhang to the bottom degrees. For the affine VOA L_k(g) at a positive integral level, it is given by the Casimir tensor of g on certain quotients of tensor products of finite-dimensional g-modules, and at generic levels it recovers Drinfeld's commutativity isomorphism.

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BibTeX RIS

BibTeX

@misc{cui2026restricted,
  title = {{Restricted conformal blocks and the braiding on Zhu algebra modules}},
  author = {Zhiguang Cui and Jianqi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41605},
  eprint = {2610.41605},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41605},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Restricted conformal blocks and the braiding on Zhu algebra modules
AU  - Zhiguang Cui
AU  - Jianqi Liu
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41605
UR  - https://arxiv.org/abs/2610.41605
AB  - Let V be a vertex operator algebra (VOA), not necessarily rational or C_2-cofinite, such that the category 𝒞_1(V) of C_1-cofinite grading-restricted generalized V-modules is semisimple and contains the generalized Verma modules associated to finite-dimensional modules over the Zhu algebra A(V). Such VOAs include affine VOAs at admissible and generic levels. We prove that the bottom degree of Huang-Lepowsky's P(w)-tensor product module M^1boxtimes_P(w)M^2 is isomorphic to the left A(V)-module M^1⊙_w,0M^2 arising from the restricted conformal blocks on (¶^1,∞,w,0). By shifting the marked points and using the parallel transport of the restricted conformal blocks, we construct an explicit braiding isomorphism on the category of finite-dimensional A(V)-modules, which is the restriction of the braiding isomorphism of Huang-Lepowsky-Zhang to the bottom degrees. For the affine VOA L_k(g) at a positive integral level, it is given by the Casimir tensor of g on certain quotients of tensor products of finite-dimensional g-modules, and at generic levels it recovers Drinfeld's commutativity isomorphism.
ER  -

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