292 problems

Let $\mathcal{W}^k(\mathfrak{sl}_n,f_{\mathrm{sub}})$ be the subregular $\mathcal{W}$-algebra of $\mathfrak{sl}_n$, where $f_{\mathrm{sub}}$ is in the nilpotent orbit corresponding…

Frenkel–Kac–Wakimoto conjecture. Drinfeld–Sokolov reduction sends every admissible $\widehat{\mathfrak{g}}_\kappa$-module to either zero or a cohomological shift of a minimal serie…

Let ${\mathcal W}^k({\mathfrak s}{\mathfrak l}_{n+1|n})$ be the principal ${\mathcal W}$-algebra, a conformal extension of the ${\mathcal N}=2$ superconformal algebra with central…

Let $L_k(\mathfrak{sl}_N)$ be the simple affine vertex algebra at level $k$, let $f_\lambda$ be the nilpotent indexed by a partition $\lambda\vdash N$, and let $H_{f_\lambda}$ deno…

Let $\mathfrak{g}$ be a simple Lie algebra, $f$ a nilpotent element, and $k$ an admissible level. Let $\mathbb{O}_k$ denote the corresponding nilpotent orbit. Rationality conjectur…

For $p\geq 4$, let $\mathcal B_p$ be the algebras from the logarithmic conformal field theory construction, and let $W_{p-1}^{(2)}$ denote the affine $W$-algebra introduced by Feig…

For $n\geq2$ and $m\geq3$, let $\mathcal F_k(n)$ denote the simple quotient of the symplectic minimal coset family, and let…

Let $\varphi^{(i)}$ be the $N$ bosonic fields with vanishing sum, and define … The operators $W_n^{(m)}$ denote the modes of the $\mathcal{W}_{1+\infty}$ generators. Since…

Let $\fg$ be a simple Lie algebra, let $f\in\cN$ be nilpotent with orbit $\BO_f$, and let $L_k(\fg)$ be a simple affine vertex algebra with associated variety $X_{L_k}$. Let…

Let $A$ be a $W$-algebra over a field of characteristic zero. A class of algebras has the Specht property if every $T$-ideal of the class is finitely generated. The Specht conjectu…

Coset conjecture. The commutant $\operatorname{Com}(W,V)$ is also rational and lisse. This is presented as a widely believed assertion and is used in the paper to deduce rationalit…

Let the affine Laumon space be the moduli space whose equivariant homology carries the relevant geometric representation, and let an iterated $W$-algebra of type $A$ and an affine…

Let an iterated $W$-algebra of type $A$ act on the equivariant homology space of an affine Laumon space, and let the shifted affine Yangian be the algebra constructed in the cited…

Let $n\geq 1$ and let … The simple subregular algebra $\mathcal{W}_k(\mathfrak{so}_{2n+1},{f_{\mathrm{sub}}})$ at such a level is called exceptional. Kac–Wakimoto–Arakawa rationali…

Let $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{u}(1)$, where $\mathfrak{h}$ is an ADE algebra or the super Lie algebra $\mathfrak{sl}_{n|m}$. The refined vacuum character of the aff…

Let the affine Laumon space be the moduli space whose equivariant homology carries the proposed representation, and let an iterated $W$-algebra of type $A$ and an affine shifted Ya…

Coset conjecture. For all $n\geq 2$, there is an isomorphism

Let … , and $k$ an admissible level of the form $k=-h^\vee+p/q$. Let $(f,q)$ be an exceptional pair in the broader sense used in the paper, namely $f$ belongs to the nilpotent orbi…

Let $\mathfrak g$ be a simply-laced Lie algebra, let $k$ be a level, and let $W_k(\mathfrak g)$ denote the simple regular (principal) W-algebra associated with $\mathfrak g$ at lev…

Bershadsky-Polyakov coset conjecture. For every positive integer $\ell$, the simple coset ${\mathcal C}_{\ell}$ should be the principal…

Let $Q$ be a connected reductive algebraic group over a field of characteristic zero, let $e\in\mathcal{N}(\mathfrak{q})$ be nonzero, and let $U(\mathfrak{q},e)$ be the associated…

Fix the vertex algebra $\mathcal{W}_{1+\infty,-1}$ and its irreducible modules. Theorem gives a two-parameter family of irreducible modules, including modules parametrized by…

The reduction ${\mathcal W}(sl_N)$ is obtained by imposing … for $k\in\mathbb Z$ in the defining relation for the reduced algebra; the modes of the resulting fields generate the…

Let $V_c^+$ be the vertex algebra considered in the paper, with generating fields $W^i(z)$, and let $c$ be a positive half-integer, so that $c\in\frac12\mathbb Z_+$. Minimal genera…

Let $V_c$ be the vertex algebra in the paper, with generating fields $J^i(z)$, and let $l$ be a positive integer so that $c=-l$. Minimal generation conjecture. A minimal set of gen…