21 problems

Associated-variety conjecture. For any integral dominant $\lambda\in\frak h^*$ with degree of atypicality $k$,

Let $\fg$ be simple of simply-laced type, let $k\in\BQ$ satisfy $k+\check{\operatorname{\mathbbm{h}}}=m/u$ in lowest terms with $m\geq1$ and $\operatorname{gcd}(m,u)=1$, and let…

Let $\fg$ be simple of simply-laced type and let $k\in\BQ$ satisfy $k+\check{\operatorname{\mathbbm{h}}}=m/u$ in lowest terms with $m\geq1$. Let $\check{\fg}^{(u)}$ be the metaplec…

Let $\fg$ be a simple simply-laced Lie algebra, let $k\in\BQ$ satisfy $k+\check{\operatorname{\mathbbm{h}}}=m\geq1$, and let $L_k(\fg)$ be the simple affine vertex algebra. Let…

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{E_7(a_5)}$ be the nilpotent orbit of Bala–Carter type $E_7(a_5)$ in $E_7$. Associated-variety…

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{F_4(a_3)}$ be the nilpotent orbit of Bala–Carter type $F_4(a_3)$ in $F_4$. Level $-6$ $F_4$ co…

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{E_6(a_1)}$ be the nilpotent orbit of Bala–Carter type $E_6(a_1)$ in $E_6$. Associated-variety…

Let $W$ be a vertex algebra that is a finite extension of a vertex algebra $L$, and let $X_W$ and $X_L$ denote their associated Poisson varieties. Dominance conjecture. The morphis…

Let a three-dimensional abelian gauge theory be equipped with the A-twist and additional boundary free fermions, and let $V^A$ be the resulting boundary VOA. Write $X_{V^A}$ for it…

Let $k=-3+\frac{2}{2m+1}$ with $m\geqslant 0$, let $V^k(sl_3)$ be the universal affine vertex operator algebra, and let $\mathcal I_k$ be its maximal ideal. Let $\mathcal V(I_k)$ d…

Let $V$ be a simple quasi-lisse conformal vertex algebra, and let $X_V$ denote its associated variety. Arakawa–Moreau's conjecture. … The paper explains that the theorem proved the…

Let $L_{-2}(G_2)$ be the simple vertex algebra at level $-2$, and let $X_{L_{-2}(G_2)}$ denote its associated variety. Let $\mathbb{O}_{\mathrm{subreg}}$ be the subregular nilpoten…

Let ${\mathcal V}$ be a simple, strongly finitely generated VOA that is $\mathbb N$-graded by conformal weight, with ${\mathcal V}[0]\cong\mathbb C$. Let $G$ be a finite group of a…

Let $\lambda=[\lambda_1,\ldots,\lambda_n]$ be a partition, let $m_1(\lambda)$ denote the multiplicity of its parts equal to $1$, and let $\mathcal{F}_j(\lambda)$ be the family of d…

Let $\mathcal{O}$ be a special nilpotent orbit, let $\mathscr{C}$ be the corresponding double cell, and let $\mathscr{C}/ {\small \stackrel{R}{\sim}}$ be the set of right cells con…

Irreducibility conjecture. The associated variety of a quasi-lisse conical vertex algebra is irreducible.

Let $V$ be a simple, finitely strongly generated, positively graded conformal vertex operator algebra with $V_0=\mathbb{C}$, and let $X_V=\operatorname{Specm}(R_V)$ be its associat…

Let $r$ be an even integer, let $\mathfrak{g}$ be the type D Lie algebra used in the source, let $V_{2-r}(\mathfrak{g})$ be the simple affine vertex algebra at level $2-r$, and let…

Let $G$ be a real reductive group, let $G'$ be a reductive subgroup, and choose a maximal compact subgroup $K$ of $G$ such that $K'=G'\cap K$ is a maximal compact subgroup of $G'$.…

Let $V$ be a finitely strongly generated simple vertex operator algebra of CFT type. Write … Here $X_V$ denotes the associated variety of $V$. The Zhu algebra variety conjecture. W…

Associated-variety equality conjecture. The inclusion in that theorem is equality: