On singular vectors of simply-laced universal affine vertex operator algebras
Cuipo Jiang, Jingtian Song
TL;DR
The paper resolves the weights of singular vectors of the universal affine vertex algebra $V^{\kappa}(\mathfrak{g})$ at minimal conformal weight for simply-laced $\mathfrak{g}$ when non-simplicity occurs, building on the Gorelik–Kac determinant and Jantzen filtration. It introduces and analyzes the graded pieces $M_{p,D}$ and the minimal index $D_p$, proving that the minimal conformal weight is $D_p q$ and describing the corresponding vector weights as $\kappa\Lambda_0 - D_p q\delta + \lambda_i$. The authors provide explicit results for Type $\mathsf{A}$ and Type $\mathsf{D}$, including parity and divisibility cases, and compile $\mathsf{E}$-type data in Section 6. This yields concrete descriptions of the maximal ideals of $V^{\kappa}(\mathfrak{g})$ in the simply-laced setting and advances the understanding of non-simple affine vertex algebras and their representations. The work thus offers a precise, combinatorial characterization of minimal-weight singular vectors across the simply-laced families, with broad implications for the structure and modules of these vertex algebras.
Abstract
Given a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ and a complex number $κ$, let $V^κ(\mathfrak{g})$ be the associated universal affine vertex algebra. In [GK07], the authors gave a sufficient and necessary condition for $V^κ(\mathfrak{g})$ to be simple. In this paper, we determine the weights of singular vectors of $V^κ(\mathfrak{g})$ with minimal conformal weights, when $\mathfrak{g}$ is simply-laced and $V^κ(\mathfrak{g})$ is not simple.