On generators and relations for O_n^∘(V) in higher-level Zhu algebras, Part I

Abstract

We study the circle-relation subspace O_n^∘(V) in the defining relations of the level-n Zhu algebra of a vertex operator algebra V. We assemble the residue relations generating O_n^∘(V) into generating functions and obtain an explicit recursion that replaces every sufficiently deep mode by retained modes with indices at least −2n−1, modulo O_n^∘(V). This recursion is encoded in a left truncated field Y_L. Since such mode replacements are valid only in the leftmost position of an operator product, we prove the replacement, derivative, commutator, and one-step iterate identities for Y_L needed for finite-length calculations. For a finite homogeneous strong generating set S, we define the finite-length circle subspaces O_n^∘(N,S,V) and their plus subspaces. We formulate a finite-length conjecture under an explicit auxiliary-degree hypothesis and a separate termwise conjecture for the correction residues of one-sided iterates. We prove reductions to the k=0 strip and to last vacuum mode −1, recover the length-one relation spaces, and give explicit Virasoro computations at word length two for n=1 and n=2. The Y_L identities are the input for the companion paper (Part II).

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BibTeX RIS

BibTeX

@misc{werf2026generators,
  title = {{On generators and relations for $O_n^\circ(V)$ in higher-level Zhu algebras, Part I}},
  author = {Nathan Vander Werf},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19501},
  eprint = {2610.19501},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19501},
}

RIS

TY  - UNPB
TI  - On generators and relations for O_n^∘(V) in higher-level Zhu algebras, Part I
AU  - Nathan Vander Werf
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19501
UR  - https://arxiv.org/abs/2610.19501
AB  - We study the circle-relation subspace O_n^∘(V) in the defining relations of the level-n Zhu algebra of a vertex operator algebra V. We assemble the residue relations generating O_n^∘(V) into generating functions and obtain an explicit recursion that replaces every sufficiently deep mode by retained modes with indices at least −2n−1, modulo O_n^∘(V). This recursion is encoded in a left truncated field Y_L. Since such mode replacements are valid only in the leftmost position of an operator product, we prove the replacement, derivative, commutator, and one-step iterate identities for Y_L needed for finite-length calculations. For a finite homogeneous strong generating set S, we define the finite-length circle subspaces O_n^∘(N,S,V) and their plus subspaces. We formulate a finite-length conjecture under an explicit auxiliary-degree hypothesis and a separate termwise conjecture for the correction residues of one-sided iterates. We prove reductions to the k=0 strip and to last vacuum mode −1, recover the length-one relation spaces, and give explicit Virasoro computations at word length two for n=1 and n=2. The Y_L identities are the input for the companion paper (Part II).
ER  -

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