Completing the Arakawa--Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras
Abstract
Arakawa and Moreau constructed explicit singular vectors in a family of negative-level universal affine vertex algebras of types D and E and conjectured that the ideals generated by these vectors are maximal. Previous work established the n=0 cases for D4, E6, E7, and E8, as well as the level −2 case for Dℓ with ℓ≥ 5. We prove all the remaining cases: the level −1 case for Dℓ with ℓ≥ 5, and the negative-level cases with n>0 for D4, E6, E7, and E8. Together with the previously known results, this completes Arakawa--Moreau Conjecture 1. The proof determines the images of the prescribed singular vectors under minimal Drinfeld--Sokolov reduction and establishes simplicity of the reduced quotients by combining a Ramond--Zhu algebra argument, a Casimir-gap argument, and Li's spectral flow. Exactness and a nonvanishing theorem for the reduction functor then lift simplicity to the corresponding affine quotients. We also formulate a general maximality principle based on minimal reduction, give an alternative reduction-theoretic proof of the known level −2 result for Dℓ, and obtain a rank-reduction proof of the maximal-ideal theorem for the collapsing family V2−2r(D2r). Consequently, every candidate quotient appearing in Arakawa--Moreau Conjecture 1 is the corresponding simple affine vertex algebra.
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BibTeX
@misc{jin2026completing,
title = {{Completing the Arakawa--Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras}},
author = {Sihai Jin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.25249},
eprint = {2607.25249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.25249},
}
RIS
TY - UNPB
TI - Completing the Arakawa--Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras
AU - Sihai Jin
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2607.25249
UR - https://arxiv.org/abs/2607.25249
AB - Arakawa and Moreau constructed explicit singular vectors in a family of negative-level universal affine vertex algebras of types D and E and conjectured that the ideals generated by these vectors are maximal. Previous work established the n=0 cases for D_4, E_6, E_7, and E_8, as well as the level −2 case for D_ℓ with ℓ≥ 5. We prove all the remaining cases: the level −1 case for D_ℓ with ℓ≥ 5, and the negative-level cases with n>0 for D_4, E_6, E_7, and E_8. Together with the previously known results, this completes Arakawa--Moreau Conjecture 1. The proof determines the images of the prescribed singular vectors under minimal Drinfeld--Sokolov reduction and establishes simplicity of the reduced quotients by combining a Ramond--Zhu algebra argument, a Casimir-gap argument, and Li's spectral flow. Exactness and a nonvanishing theorem for the reduction functor then lift simplicity to the corresponding affine quotients. We also formulate a general maximality principle based on minimal reduction, give an alternative reduction-theoretic proof of the known level −2 result for D_ℓ, and obtain a rank-reduction proof of the maximal-ideal theorem for the collapsing family V^2−2r(D_2r). Consequently, every candidate quotient appearing in Arakawa--Moreau Conjecture 1 is the corresponding simple affine vertex algebra.
ER -