发表机构
Sichuan University（四川大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明荒川 - 莫罗猜想中关于仿射顶点代数极大理想的剩余情况，通过确定奇异向量在最小约化下的像，结合多种论证确立约化商单性并提升到仿射商，还制定极大性原理及给出相关证明，使猜想中候选商均为单仿射顶点代数。
AI 中文摘要
荒川和莫罗在\(D\)型和\(E\)型的一族负级通用仿射顶点代数中构造了明确的奇异向量，并猜想由这些向量生成的理想是极大的。先前工作已确立了\(D_4\)、\(E_6\)、\(E_7\)和\(E_8\)的\(n = 0\)情况以及\(\ell\geq5\)时\(D_\ell\)的\(-2\)级情况。本文证明了其余情况：\(\ell\geq5\)时\(D_\ell\)的\(-1\)级情况以及\(D_4\)、\(E_6\)、\(E_7\)和\(E_8\)的\(n>0\)的负级情况。通过结合多种论证方法，确定了规定奇异向量在最小德里菲尔德 - 索科洛夫约化下的像，并确立了约化商的单性，进而提升到相应仿射商的单性。还基于最小约化制定了一般极大性原理，给出了\(D_\ell\)已知\(-2\)级结果的另一种约化理论证明，并获得了坍缩族\(V^{2 - 2r}(D_{2r})\)的极大理想定理的秩约化证明。因此，荒川 - 莫罗猜想1中出现的每个候选商都是相应的单仿射顶点代数。
英文摘要
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.
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