20 problems

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

Let $\operatorname{Sb}[\mathbb{C}^{16}]$ denote the vertex operator algebra of sixteen symplectic bosons, and let $\widehat{SU}(2)_{-4}$ act through its affine current algebra. Sym…

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

Let $n,r,s$ be non-negative integers, and let ${\mathcal C}^{\psi}_{{\mathcal N}=2}(n,r|s)$ denote the corresponding N=2 coset vertex algebra. Procházka–Rapčák duality conjecture.…

Let $n,r,s$ be non-negative integers, and define … Let ${\mathcal C}^{\psi}_{{\mathcal N}=2}(n,r|s)$ be the corresponding coset vertex algebra. Minimal strong generating type conje…

Let $N,n,m\in\mathbb{Z}$ satisfy $N>n,m\geq2$, and define $k^\sharp=-k-N+n+m$ and $\ell^\sharp=-\ell-N+n+m$. Let $\operatorname{Com}(A,B)$ denote the commutant of $A$ in $B$. Verti…

Let $\lambda=(\lambda_1\leq\dots\leq\lambda_n)\vdash N$, let $N_i=N-\sum_{j>i}\lambda_j$, and let $C^{k_i^\sharp}(\mathfrak{sl}_{N_i},f_{\widehat{\lambda}_i})$ be the affine coset…

Diagonal coset rationality conjecture. The coset $\mathcal C$ is lisse and rational. This generalizes the known simply-laced diagonal-coset cases and the lattice-subalgebra situati…

Assume $m>1$ and set … Let $\mathcal H$ be a Heisenberg vertex algebra and $L_{\psi-m-1}(\mathfrak{sl}_m)$ the simple affine vertex algebra at level $\psi-m-1$. The coset embedding…

Let $m,n\geq2$, let $\psi$ be generic, and set $k=\psi-m+1$ and $\ell=\phi-n+1$, where $\phi^{-1}+\psi^{-1}=1$. Let $\mathcal H$ be a rank-one Heisenberg algebra. The D-algebra cat…

Let $N=n+m$, let $f=f_{n,m}$ be the nilpotent element of $\mathfrak{sl}_N$ corresponding to the partition $N=n+1+\dots+1$, and let $A[\mathfrak{sl}_N,f,\psi]$ be the associated ext…

$T[SU(3)]$ coset conjecture.

For $n\geq3$, let $K_n$ be the ideal defining the coset associated with the negative-rank principal $\mathcal W$-algebra and let $V(K_n)\subseteq\mathbb C^2$ be its variety in coor…

For $n\geq4$, let $K_n$ be the ideal defining the subregular coset quotient and let $V(K_n)\subseteq\mathbb C^2$ be its affine variety in coordinates $(c,\lambda)$. Subregular cose…

For $n\geq4$ and $m\geq3$, let $\mathcal C_k(n)$ be the simple quotient of the Heisenberg coset in the subregular $\mathcal W$-algebra of $\mathfrak{sl}_n$. Subregular coset coinci…

For $n\geq4$, let $\mathcal C^k(n)$ be the coset of the Heisenberg algebra in $\mathcal W^k(\mathfrak{sl}_n,f_{\mathrm{subreg}})$, and let $\mathcal W^{K_n}_{R_n}(c,\lambda)$ denot…

For $n\geq4$, let $K_n$ be the ideal in $\mathbb C[c,\lambda]$ defining the two-parameter quotient associated with the minimal $\mathcal W$-algebra coset. Kawasetsu's explicit idea…

For integers $n\geq4$ and $k\geq0$, let $\mathcal C_k(n)$ be the simple quotient of the coset of $L_{k+1}(\mathfrak{gl}_{n-2})$ in $\mathcal W_k(\mathfrak{sl}_n,f_{\mathrm{min}})$,…

The $T[SU(3)]$ H-twist coset conjecture.

Let $G$ be a group, and for $i=1,\ldots,k$ let $G_i$ be subgroups of $G$ and $a_i\in G$. A left coset $a_iG_i$ has finite index $[G:G_i]<\infty$. Sun's conjecture. If $k>1$ and…