16 problems

The $(A_1,A_4)$ Gröbner-basis conjecture. The monomials in $\mathcal{G}(\ell,m)$ constitute a Gröbner basis for $\langle J(x)^2\rangle$, with the recursive description of…

The $(A_1,D_{2n+1})$ Zhu-algebra conjecture. The Zhu algebra of $\mathfrak{su}(2)_{-\frac{4n}{2n+1}}$ is isomorphic to $R$:

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…

For the polynomial variables ordered by … use degree-lexicographic ordering, and let $G_n$ be the Gröbner basis of the relevant ideal. For $I_{\ell,m}$, define … with $F_z$ and $F$…

Let ${\cal V}$ be a finitely strongly generated vertex operator algebra with generators $U=\{u^1,\ldots,u^N\}$, and write … where $I$ records relations among the generator fields.…

Let $\mathfrak{g}$ be a Lie algebra, let $\mathfrak{p}$ be a parabolic subalgebra, let $\kappa$ be an invariant symmetric bilinear form on $\mathfrak{g}$, and let…

Higher level Zhu algebra conjecture. One has

Let $M_a(1)$ be the Heisenberg vertex operator algebra, let $A_n(M_a(1))$ denote its level-$n$ Zhu algebra, and let $p(n)$ denote the number of unordered partitions of $n$ into non…

Inner-automorphism invariance conjecture. There is a one-to-one correspondence between the simple objects in the category of lower bounded logarithmic $g$-twisted modules and the s…

Polynomial formula. There is a nonzero constant $A_p$ such that

All-p conjecture. The theorem asserting that, for $p\leq 20$, $A_{\Psi}(\overline{M(1)})$ is semisimple and isomorphic to

Dimension conjecture. The Zhu algebra $A(\mathcal{SW}(m))$ is $(6m+1)$-dimensional.

Twisted Zhu-algebra dimension conjecture.

Minimal-polynomial conjecture. The polynomial $f^{\mathfrak R}_m(x)$ is the minimal polynomial of $[\omega]$ in $A_{\sigma}(\mathcal{SW}(m))$.

Center-and-trace dimension conjecture.

Let $p\geq 2$, let $A(\mathcal{W}(p))$ be the Zhu algebra of the triplet vertex algebra, and let $h_{i,1}$ denote the relevant conformal weights. For each $i$, let…