25 problems

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…

Existence conjecture for ordinary twisted modules. Every such pair $(V,g)$ has at least one ordinary $g$-twisted $V$-module. A weaker conjecture is that

Existence conjecture for twisted modules. Every vertex operator algebra $V$ and finite-order automorphism $g$ have a nonzero $g$-twisted $V$-module.

Let $\widehat{\mathcal D}^-$ be the algebra occurring in the duality decomposition, let $c\in\frac12\mathbb Z_+$, and let $V_c^+$ be the associated vertex algebra. Consider the irr…

Permutation-twisted module counting conjecture. The number of isomorphism classes of irreducible $g$-twisted $V^{\otimes k}$-modules is equal to the number of isomorphism classes o…

Stable-module conjecture. If $V$ is rational, then the number of isomorphism classes of irreducible $g$-stable $V$-modules is finite and equal to the number of isomorphism classes…

Let $V$ be a vertex operator algebra and let $\sigma$ be a finite-order automorphism of $V$. A quasi $V$-module is a vector space equipped with the vertex-algebra module structure…

Equivalence conjecture. Any stack of $\mathbb{C}_X$-twisted $\mathcal{D}_X$-modules is $\mathbb{C}$-equivalent to a stack of the form

Let $V$ be a vertex operator algebra, let $G<\operatorname{Aut}(V)$ be a finite automorphism group, and let $V^G$ denote the fixed-point subVOA. A $V$-module is strongly rational w…

Let $V$ be a simple affine vertex algebra associated with one of the Lie algebras in the $\mathrm{DC}$ series … at level $-h^{\vee}/6-1$. Let $\chi^{\mathrm{twi}}(q)$ denote a norm…

Twisted-module realization conjecture. Under some conditions on $V$, every irreducible $V^G$-module is contained in some irreducible $g$-twisted $V$-module for some $g\in G$.

Let $V$ be a vertex operator algebra and let $G$ be a finite subgroup of $\operatorname{Aut}(V)$. For $g\in G$, let $V^{G}$ be the fixed-point subalgebra of $V$ under $G$, and let…

Let $V$ be a grading-restricted vertex (super)algebra and let $g$ be an automorphism of $V$. Denote by the twisted Zhu algebra and twisted zero-mode algebra the associative algebra…

Let $V$ be a Möbius vertex (super)algebra or quasi-vertex (super)algebra, and let $g$ be an automorphism of $V$. Define … Assume that $P(V)$ has no accumulation point in…

Let $V$ be a grading-restricted vertex superalgebra or Möbius vertex algebra, and let $g$ be an automorphism of $V$. A lower-bounded generalized $g$-twisted $V$-module is a module…

Let $V$ be a suitable vertex operator algebra, let $G$ be a finite group of automorphisms of $V$, and for each $g\in G$ let a $g$-twisted $V$-module be given. The $G$-crossed tenso…

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…

Let $V$ be a vertex operator superalgebra and let $k$ be a positive even integer. The cyclic permutation $(1\;2\;\cdots\;k)$ acts on the tensor product vertex operator superalgebra…

Let $V$) be a vertex operator superalgebra and let $k$ be an even positive integer. Consider weak, weak admissible, ordinary, parity-stable, and parity-unstable modules for the cyc…

Let $V$ be a vertex operator superalgebra, let $k$ be even, and let $g=(1\;2\;\cdots\;k)$ act by permuting the tensor factors of $V^{\otimes k}$. A parity-twisted $V$-module is a m…

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

Orbifold module conjecture. Under some conditions on $V$, every simple $V^G$-module is contained in some simple $g$-twisted $V$-module for some $g\in G$.

Let $V$ be a vertex operator algebra and let $H$ be a finite automorphism group of $V$. Write $V^H$ for the fixed-point vertex operator subalgebra, and let an $h$-twisted $V$-modul…

Let $V$ be a vertex operator algebra and let $G$ be a finite automorphism group of $V$. Write $V^G$ for the fixed-point vertex operator subalgebra, and let a $g$-twisted $V$-module…

Let $V$ be a vertex operator algebra and let $G$ be a finite automorphism group of $V$. Write $V^G$ for the fixed-point vertex operator subalgebra, and let $g$-twisted $V$-modules…