343 problems

A minimal simple $W$-algebra is one of the unitary minimal simple $W$-algebras under consideration, and an extremal (massless) module is an untwisted highest weight module of the e…

Conjecture on $L_{k_m}(D_4)$. For each $m\geq1$, $L_{k_m}(D_4)$ is a quasi-lisse vertex algebra, it has a unique irreducible ordinary module, and $KL_{k_m}(D_4)$ is a semisimple br…

Let $\mathcal V^{(N)}$ be the vertex algebra associated with the $N$th power of the monodromy operator of a four-dimensional ${\mathcal N}=2$ superconformal field theory. Let $r$ b…

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 $V$ be the vertex algebra under consideration, let $g$ be an automorphism of finite order $T$, and let $n,m\in (1/T)\mathbb{N}$. Denote by $\tilde{O}_{g,n,m}(V)$ and…

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…

Beem–Rastelli conjecture. The Higgs branch of the 4d SCFT should coincide with the associated variety of the corresponding 2d CFT.

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…

For each integer $p\,{\geqslant}\, 2$, let $\mathfrak{C}_p$ be the category of representations of the triplet $\boldsymbol{\mathcal{W}}(p)$ algebra whose adjoint $L_0$-eigenspaces…

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…

Ramond unitarity conjecture. The set of unitary highest weight modules over $W^k_{\min}(\mathfrak g)$ is the union of

Let $\mathfrak{g}$ be a semisimple finite-dimensional complex Lie algebra with simple roots $\alpha_1,\ldots,\alpha_n$ and Killing form $(-,-)$. For an integer $p\geq 2$ divisible…

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 $n\geq 1$ 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 $V^k(\mathfrak{sl}_{n+1})$ be the universal affine vertex algebra of $\mathfrak{sl}_{n+1}$ at level $k$, let $bc^{\otimes n}$ be the tensor product of $n$ copies of the $bc$-sy…

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

Let $\mathbb M$ be the Monster group and let $V^\natural$ be a graded $\mathbb M$-module. The McKay–Thompson series $T_g$ are the specified modular functions associated with elemen…

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 $L_k(\mathfrak g)$ be a non-admissible simple affine vertex algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak g$, and let…

Let $C_{k,n}$ be the simple graded quotient of $C^{k_1,k_2}$ at $(k_1,k_2)=(k,n)$, where $n$ is a positive integer and $k$ is a positive integer or an admissible level for…

Let $\ell$ be a level, let $\Gamma_q$ be the equivariance group, and let $L_{\widehat{\mathcal{A}}}(\ell,0)$ be the corresponding quasi vertex operator algebra. Consider an…

Cohomological complete-reducibility conjecture. If every left $V$-module of finite length is completely reducible, then

Defining relations conjecture. These relations form a defining set of relations for the coordinate ring of the semi-infinite flag variety.

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