Conjecture on Nichols algebras of screening operators and logarithmic Kazhdan–Lusztig correspondences

Let $\mathcal{V}$ be a vertex operator algebra (VOA) with a vertex tensor category $\mathcal{C}$ of $\mathcal{V}$-modules $M_1,\ldots,M_n$. Fix suitable elements $m_1,\ldots,m_n$ in these modules, and let $\mathfrak{Z}_1,\dots,\mathfrak{Z}_n$ be their screening operators. The joint kernel of these screening operators is denoted by $\mathcal{W}$.

Nichols-algebra and screening-operator conjecture. The screening operators generate a corresponding Nichols algebra $\mathfrak{N}$ in the braided tensor category $\mathcal{C}$; $\mathcal{W}$ is a subVOA, and if $\mathfrak{N}$ is finite-dimensional then $\mathcal{W}$ has representation theory similarly good to that of $\mathcal{V}$, for example, $C_2$-cofiniteness of $\mathcal{V}$ implies $C_2$-cofiniteness of $\mathcal{W}$. Moreover, the braided tensor category of representations of $\mathcal{W}$ is equivalent to the category of $\mathfrak{N}$-Yetter–Drinfeld modules in $\mathcal{C}$.

These assertions propose a connection between screening operators, Nichols algebras, and the representation theory of vertex operator subalgebras. Assertion a) was proved when $\mathcal{V}$ is a lattice vertex algebra, while in cases without proper rigidity, such as $\mathcal{W}_{p,q}$, a weaker relation between the categories is expected.