183 problems

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…

Let $p$ and $q$ be parameters, define … and define … Let ${\mathcal L}_s$ be the Lie superalgebra introduced above, and let $L(c_{p,q},h_{p,q}^{m,n})$ denote the corresponding irre…

Exact dimension formula. The dimension is

Let $\mathbb{F}$ be a field of characteristic $p>2$, let $c\in\mathbb{F}$, and let $L_{\operatorname{Vir}}(c,0)_{\mathbb{F}}$ denote the simple quotient Virasoro vertex operator al…

Zhu's conjecture. If $V$ is rational, then $R_V$ is finite-dimensional.

Orbifold rationality conjecture. The conformal subalgebra $V^G$ is also strongly rational.

Holomorphic VOA unitarity conjecture. Every holomorphic VOA is unitary.

Compact orbifold uniqueness conjecture. The unitary $V^G$-module $V$ has a unique unitary VOA structure up to isomorphism.

Geometrization conjecture. If $k$ is a positive integer divisible by $4$, every group-like element of

Reconstruction conjecture. If

Let $q\geq 2$ with $3\nmid q$, and define the generalized vertex operator algebras … … Duality means that each simple vertex (super)algebra can be equipped with a module structure…