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 $D_4$, $E_6$, $E_7$, and $E_8$, as well as the level $-2$ case for $D_\ell$ with $\ell\geq 5$. We prove all the remaining cases: the level $-1$ case for $D_\ell$ with $\ell\geq 5$, 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 $D_\ell$, 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.