On generators and relations for On∘(V) in higher-level Zhu algebras, Part II
Abstract
We study circle relations in the higher-level Zhu algebra of a vertex operator algebra V. For regular vacuum words in homogeneous letters, we prove the termwise correction-residue conjecture of Part I: every correction in the one-sided iterate expansion belongs separately to the finite-length plus subspace. The first correction formed with the left truncated field vanishes identically. Multiplying the diagonal commutator kernel by a power of 1+x turns it into a Laurent polynomial. This proves the vanishing and writes every other correction as a finite combination of product residues with fewer YL factors, coming from the word obtained by replacing one negative mode by a nonnegative one. These results hold at all levels n≥0, without an auxiliary-degree hypothesis. For n≥1 and a homogeneous ordered pair (u,v), two explicit triangular families span the quadratic residue operators modulo the length-one and singular-iterate relations, giving dim𝒬n((u,v))≤ n(n+1). The universal product-coefficient arrays have dimension exactly n(n+1) and three natural bases; restriction to the regular mode square is an isomorphism. This proves operator spanning and gives invertible matrices for two complementary diagonal eliminations. For equal inputs, mode interchange reduces the families to their odd-indexed members, giving dim𝒬n((u,u))≤ 2⌊ (n+1)2/4⌋. The pairwise estimates give an explicit global quadratic bound for any finite homogeneous strong generating set. Heisenberg, Virasoro, and affine sl2 examples illustrate equality and strict inequality in the operator bounds.
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BibTeX
@misc{werf2026generators,
title = {{On generators and relations for $O_n^\circ(V)$ in higher-level Zhu algebras, Part II}},
author = {Nathan P Vander Werf},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35527},
eprint = {2610.35527},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35527},
}
RIS
TY - UNPB
TI - On generators and relations for O_n^∘(V) in higher-level Zhu algebras, Part II
AU - Nathan P Vander Werf
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35527
UR - https://arxiv.org/abs/2610.35527
AB - We study circle relations in the higher-level Zhu algebra of a vertex operator algebra V. For regular vacuum words in homogeneous letters, we prove the termwise correction-residue conjecture of Part I: every correction in the one-sided iterate expansion belongs separately to the finite-length plus subspace. The first correction formed with the left truncated field vanishes identically. Multiplying the diagonal commutator kernel by a power of 1+x turns it into a Laurent polynomial. This proves the vanishing and writes every other correction as a finite combination of product residues with fewer Y_L factors, coming from the word obtained by replacing one negative mode by a nonnegative one. These results hold at all levels n≥0, without an auxiliary-degree hypothesis. For n≥1 and a homogeneous ordered pair (u,v), two explicit triangular families span the quadratic residue operators modulo the length-one and singular-iterate relations, giving dim𝒬_n((u,v))≤ n(n+1). The universal product-coefficient arrays have dimension exactly n(n+1) and three natural bases; restriction to the regular mode square is an isomorphism. This proves operator spanning and gives invertible matrices for two complementary diagonal eliminations. For equal inputs, mode interchange reduces the families to their odd-indexed members, giving dim𝒬_n((u,u))≤ 2⌊ (n+1)^2/4⌋. The pairwise estimates give an explicit global quadratic bound for any finite homogeneous strong generating set. Heisenberg, Virasoro, and affine sl_2 examples illustrate equality and strict inequality in the operator bounds.
ER -