11 problems

Feigin–Jimbo–Miwa conjecture. Free field realizations of these algebras are labelled by the Dynkin diagrams of $\mathfrak{gl}(1|N)$.

Critical-level conjecture. For each $1\leq l\leq m-1$, the $\sigma^l$-twisted intertwining-operator construction realizes the Type II bracket of $\widetilde{G}$ exactly at the crit…

Let $\sigma=[l^{w_l},\ldots,1^{w_1}]$ be a partition, let $q_i=\sum_{j=i}^l w_j$, and set $n=q_1+\cdots+q_l$. Consider the Miura map and Wakimoto realization … and, assuming the Ka…

Beem–Nair homomorphism conjecture. The immersion gives rise to a vertex operator algebra homomorphism between the chiralization of $\mathrm{KT}_G$ and the chiralization of…

For a simple linear algebraic group $G$, let $\mathfrak{Z}_G$ be the universal centralizer, and let $\mathrm{KT}_G$ be the Kostant–Toda lattice associated to $G$. Let…

Let $\mathcal{L}^{(x)},\mathcal{L}^{(y)},\mathcal{L}^{(z)},\mathcal{L}^{(w)}$ be the elementary pseudo-differential operators associated with the four coordinate functions on…

Let $\mathcal{L}^{(\alpha)}$ be the elementary Miura operator for $\alpha=x,y,z,w$, with parameter $N_\alpha$, and let $x^{N_3}y^{N_2}z^{N_4}w^{N_1}$ denote the algebra obtained fr…

For a four-dimensional theory with associated vertex operator algebra $\mathcal{V}$, suppose a free field construction equips $\mathcal{V}$ with a natural filtration, and let the u…

Let $R=\mathbb C[t^{\pm 1},u\mid u^2=t^2+4t]$, let $\mathfrak V$ be the three-point Virasoro algebra, and let the three-point current algebra act in the free-field realization on a…

Let $H_k$ be the operators in the Heisenberg algebra corresponding to the operators $\hat H_k$ defined by … Let $(t\Psi)_k$ denote the mode obtained from the product of the generat…

Wakimoto–Verma isomorphism conjecture. Generically, the Wakimoto type module constructed in Theorem at $r=0$ and the corresponding Verma module are isomorphic.