22 problems

Let $V$ be a holomorphic vertex operator algebra and let $G$ be a finite automorphism group of $V$. Write $V^G$ for the vertex operator subalgebra of $G$-invariants, and let…

Let $V$ be a holomorphic vertex operator algebra and let $G$ be a finite subgroup of the automorphism group of $V$. Write $V^G$ for the fixed-point vertex operator subalgebra. For…

Noncompact extension conjecture. The isomorphism of Theorem also holds for such noncompact $X$ and $G$.

Let $V$ be a $C_{2}$-cofinite vertex operator algebra of positive energy, let $G$ be a finite group of automorphisms of $V$, and let $g\in G$. Consider the category of grading-rest…

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…

In orbifold conformal field theory, let $V$ be a suitable vertex operator algebra and consider a category of suitable generalized twisted $V$-modules. Crossed tensor category conje…

A simple normal crossings pair $(X|D)$ has naive Gromov–Witten invariants obtained from stable maps to $X$ equipped with logarithmic lifts to each pair $(X|D_i)$. Let the constant-…

Let $\mathbb U=\mathbb V^{\otimes E}$ be the tensor product vertex operator algebra appearing above, and let $G$ be a finite group acting on $\mathbb U$ by permutations of the tens…

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…

Orbifold module conjecture. Under some conditions on $V$, every irreducible $V^G$-module is a submodule of some irreducible $g$-twisted $V$-module for some $g\in G$.

Let $M(1)$ be the rank-$\ell$ Heisenberg vertex operator algebra, let $S_{\ell}$ act on it by permuting a basis $h_1,\dots,h_{\ell}$ of the underlying space, and let…

Let a twisted sector be associated with a conjugacy class of the orbifold group, and let the corresponding resolution have an associated irreducible blow-up divisor. Quarter-BPS lo…

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

Orbifold theory conjecture. The vertex operator algebra $V^G$ is rational, and every irreducible $V^G$-module occurs in an irreducible $g$-twisted $V$-module for some $g\in G$.

Let $p$ be the parameter of the $D_2$ identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,x_2,x_3,x_4}$ denote the iterated residue in the displayed variabl…

Let $m>2$, let $p$ be the parameter of the $D_m$ identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,\ldots,x_{m+2}}$ denote the iterated residue in the dis…

Let $m$ and $p$ be the parameters of the $A_m$ constant-term identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,\ldots,x_{m+1}}$ denote the iterated residu…

Let $h_{p,m}(t)$ and $\tilde{h}_{p,m}(t)$ be the polynomials defined by the factorizations of the residue polynomials $H_{p,m}(t)$ and $\tilde{H}_{p,m}(t)$ for the $D_m$ orbifold v…

Let $V$ be a vertex operator algebra with only finitely many irreducible modules, and let $\operatorname{Irr}(V)$ denote the set of irreducible $V$-modules. For…

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 $G$ 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…