22 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…

Let $\mathrm{L}=\bigoplus_{i\in\mathbb{Z}}\mathrm{L}_i$ be a simple $\mathbb{Z}$-graded Lie superalgebra of growth one whose even part contains the Witt algebra $\mathrm{Vir}$. Suc…

Figueroa-O'Farrill's conjecture. The BRST cohomology of any string theory and, more generally, of any two-dimensional topological conformal field theory is isomorphic to the chiral…

Subsingular-vector conjecture. Outside the submodules built on the level-zero singular vectors, there are generically only subsingular vectors, which are singular only for some dis…

Let $r$ and $s$ index a positive-parity singular vector at level $\frac{rs}{2}$, and let $\Delta_{r,s}$ be the corresponding highest weight. Write this vector as … Here $a$ and $b$…

Dör zapf-mapping conjecture. In all cases, the two partners of a given Dör zapf pair are connected by these mappings, and hence belong to the same family.

Chiral-Verma subsingular-vector conjecture. This result holds at any level.

Dör zapf-pair connectivity conjecture. The two partners in every Dör zapf pair of these four types are connected by mappings made from ${\cal A}$, ${\cal U}_{\pm1}$, ${\cal G}_0$,…

Dör zapf-family conjecture. The two partners in every Dör zapf pair belong to the same family under these mappings.

Singular-vector isospin conjecture. The singular vectors in the admissible representations are conjectured to be precisely those with isospin satisfying the displayed relation. Thi…

Charge-restriction conjecture. Singular vectors in chiral Neveu–Schwarz Verma modules ${\cal V}_{h/2,h}^{NS,ch}$ should exist only for

Let a superconformal algebra be a simple $\mathbb{Z}$-graded Lie superalgebra of growth one whose even part contains the Witt algebra $\mathrm{Vir}$. The known examples consist of…

Let $\mathcal{O}_{c^{\mathfrak{ns}}(t)}^{\rm fin}$ denote the category of finite-weight modules at the $N=1$ super Virasoro central charge $c^{\mathfrak{ns}}(t)$, where…

Let $M_{h,u}$ be a Verma module in the NS sector, and let ${\rm det}^{\rm NS}_{2,0}$ denote the Kac determinant on the subspace of $p$-degree $2$ and $x$-degree $0$. Let $q$, $k$,…

Let $m,m'\in\mathbf N$ and $j,j'\in\mathbf Z$. For $m\in\mathbf N$ and $j\in\mathbf Z$, let $\overset{N=3}{CH}{}^{(+)[K(m),j]}$ denote the $\mathbf C((q^{\frac12}))$-linear span of…

BBT conjecture. After this special bosonization, the singular vectors in the Neveu–Schwarz sector are identified with Uglov symmetric functions with rectangular Young diagrams.

Let $V=\mathrm E_{8,2}/\mathbb Z_2$, whose spin-$\frac32$ fields form the real representation $\mathbf{3875}$ of $\mathrm E_8$. Let $\star$ denote the $\mathrm E_8$-invariant commu…

Let $V=\mathrm E_{7,1}^2/\mathbb Z_2$, and consider the three $N{=}1$ structures constructed on $V$. The automorphism group is…

Let $V$ be an $N{=}1$ SVOA and let $V_{\mathrm{ev}}$ denote its even subalgebra. Consider the three exceptional even subalgebras $\mathrm E_{7,1}^2$, $\mathrm E_{7,2}$, and…

Drinfeld–Sokolov decomposition conjecture. The reduced vertex operator algebra decomposes as

Shatashvili–Vafa algebra conjecture. The section pairs $\{\Phi_{+},K_{+}\}$ and $\{\Phi_{-},K_{-}\}$ generate two commuting copies of the Shatashvili–Vafa $G_{2}$ superconformal al…

Strong-generation conjecture. In this case, ${\mathcal V}^{(p)}$ is strongly generated by $e,f,h$ and $\tau_{(p)}^+,\overline{\tau}_{(p)}^+,\tau_{(p)}^-,\overline{\tau}_{(p)}^-$. T…