184 problems

Vertex algebra–conformal net correspondence conjecture. There is a bijective correspondence between completely rational local conformal nets and unitary $C_2$-cofinite vertex opera…

For every smooth projective surface $S$ with numerically trivial canonical class, and for all tautological insertions (in particular, line bundles $L_1,\ldots,L_N$ and indices…

Exact dimension formula. The dimension is

Tensor-product equivalence conjecture. The map above induces an equivalence of monoidal categories

Let $V$ be a strongly rational, holomorphic vertex operator algebra of central charge $24$, and let $V_1$ denote its weight-one Lie algebra. Uniqueness conjecture. The Lie algebra…

Norton's generalized moonshine conjecture. There is an assignment of holomorphic functions $T_{(g,\tilde h)}:\mathbb{H}\to\mathbb{C}$ to every commuting pair $(g,h)$ in the monster…

G-crossed tensor-category conjecture. This category has a natural structure of a $G$-crossed tensor category satisfying additional properties.

Witt-realization conjecture. There is at least one unitary vertex operator algebra (VOA) $V$ of central charge $c$ such that $\operatorname{Mod}_V$ is a UMTC Witt equivalent to…

Let $V$) be a simple vertex operator algebra satisfying the following conditions: $V_{(0)}=\mathbb{C}\mathbf{1}$, $V_{(n)}=0$ for $n<0$, and the contragredient $V'$ is equivalent t…

Let $V$ be a finitely strongly generated simple vertex operator algebra of CFT type. Write … Here $X_V$ denotes the associated variety of $V$. The Zhu algebra variety conjecture. W…

Let $V$ be a vertex operator algebra satisfying the conditions in the earlier conjecture on products of twisted intertwining operators, and let $G$ be a finite group of automorphis…

Let $V^\bullet[\cdot]$ be a topological vertex operator algebra (TVOA), with dot product and bracket product defined by the vertex-operator operations above. The homotopy Gerstenha…

Let $V$ be a vertex operator algebra and let $G$ be a group of automorphisms of finite order of $V$. A $G$-simple current is an irreducible weak $g$-twisted $V$-module whose tensor…

Let $V$ be a rational vertex operator algebra, and let $W_{1},\dots,W_{r}$ be a complete set of representatives of equivalence classes of irreducible $V$-modules. Here $U(V)$ is th…

Let $V$ be a topological vertex operator algebra, and let an appropriate topological completion mean the completion of $V$ required in the cited construction. A genus-zero topologi…

A topological vertex operator algebra is the algebraic structure considered in the paper, and a homotopy Gerstenhaber algebra is the corresponding higher-algebraic structure introd…

Let $(U,Y,{\bf 1},\Omega)$ be a nice rational enhanced VOA with local algebra $\mathcal{A}(\Omega)\cong\mathcal{A}(\Omega_{\operatorname{\textsl{Ru}}})$. Assume that $U$ has rank…

Zero-location conjecture. For every $k\neq 2i^2-2$, $i\geq 2$, the zeros of $G(\mathcal{F}_{L(k\Lambda_0)},j)$ are simple and lie inside the interval $[0,1728]$. This is a numerica…

Let $L(k\Lambda_0)$ denote the level-$k$ vacuum module, let $\mathcal{F}_{L(k\Lambda_0)}(q)$ be its normalized generating function, and let $p=2k+3$ be a prime with $p\geq 5$. Cong…

Rationality–semisimplicity conjecture. $V$ is rational if and only if $A(V)$ is semisimple.

Let $Q_1\in SK(m)$ and $Q_2\in SK(n)$ be supergeometric data of the forms … and … Here $SK(k)$ denotes the space of supergeometric configurations with $k$ tubes, and $\Gamma$ and…

Finiteness conjecture. The set $\operatorname{gen}(V)$ is finite.

Let $V$ be a rational vertex operator algebra of CFT type, let $M$ be an irreducible $V$-module, and let ${\mathfrak h}$ and $L^{\circ}$ be the spaces defined in the source, with…

Order-three conjecture. Under this assumption, $\tau_e\tau_f$ has order three.

Let $p,q$ be positive integers, set … and for integers $m,n$ set … For the Lie superalgebra ${\mathcal L}_s$ and the irreducible module $L(c_{p,q},h_{p,q}^{m,n})$, finite-dimension…