87 problems

Mathieu moonshine conjecture. Each $\mathcal{D}_i$ is, in some natural but unexplained way, a representation of the finite simple group $M_{24}$. The conjecture concerns the appear…

Thompson's conjecture. There is an infinite-dimensional graded $\mathbb{M}$-module $V$ whose graded dimension is $J(\tau)$ and whose McKay–Thompson series $T_g(\tau)$ are distingui…

Frenkel–Lepowsky–Meurman uniqueness conjecture. The moonshine VOA $V^\natural$ is the unique holomorphic VOA of central charge $24$ with $\dim V_1=0$.

Let $f=q^{-1}+\sum_{n=1}^\infty a_nq^n$ be a formal $q$-series with rational coefficients that is replicable. Norton's conjecture. The series $f$ is either a Hauptmodul for a genus…

A vertex operator algebra is strong-rational if it satisfies the paper's strong rationality conditions, and it is holomorphic if its representation category is trivial. The moonshi…

Strong graded locality theorem. Every nice holomorphic VOSA $V$ with central charge $c\leq 24$ is strongly graded-local. In particular, it gives rise to a holomorphic graded-local…

Let $\mathbb M$ be the Monster group and let $V^\natural$ be a graded $\mathbb M$-module. The McKay–Thompson series $T_g$ are the specified modular functions associated with elemen…

Let $\Pi=P_X\oplus P_Z$ be the positive plane determining a supersymmetric non-linear K3 sigma model, and let $G_\Pi$ be the corresponding group of automorphisms. For $g\in G_\Pi$,…

Let $V^{f\natural}$ be the super vertex operator algebra constructed from the Leech lattice, and let $\operatorname{Aut}(V^{f\natural})$ denote its automorphism group. Conway group…

Norton's hauptmodul conjecture. Each $Z(g,h,\tau)$ is either constant or a hauptmodul.

Let $V\!B^{\natural}$ denote the shorter moonshine module, also called the Baby Monster SVOA. A self-dual unitary rational SVOA is called “very nice” in the sense used in the sourc…

Let ${\frak{M}}^*$ be the Moonshine cohomology functor from conformal quantum operator algebras to Batalin–Vilkovisky algebras. The Monster finite group acts naturally on…

A self-dual meromorphic conformal field theory (MCFT) here means a conformal field theory of central charge $C=24$ whose genus-one partition function is modular invariant and whose…

Let $z(q)$ be the mirror map for a one-parameter deformation of an algebraic K3 surface arising from an orbifold construction whose Picard–Fuchs equation is third order. A Lian–Yau…

Let $V\!B^{\natural}$ denote the shorter Moonshine module, a super vertex operator algebra of central charge $23\frac{1}{2}$. Consider super vertex operator algebras satisfying the…

Frenkel–Lepowsky–Meurman's characterization conjecture. The vertex operator algebra $V^{\natural}$ can be characterized by these three conditions: it is the only irreducible ordina…

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…

Let $A_{{\operatorname{\textsl{Ru}}}}$ be the self-dual enhanced $\operatorname{\textsl{U}}(1)$--VOA of rank $28$ whose full automorphism group is the direct product of a cyclic gr…

Dong–Mason–Nagatomo conjecture. There are no nontrivial obstructions to attaining the expected modular forms as graded traces of highest-weight vectors.

Uniqueness conjecture of the moonshine vertex operator algebra. $V^\natural$ is the unique holomorphic vertex operator algebra with central charge $24$ whose weight-one subspace is…

Let a generalised Moonshine function (GMF) be associated with a commuting pair of elements, and let the method described in the paper be the method based on orbifold considerations…

Let $g$ and $h$ be commuting elements with orders $o(g)$ and $o(h)$, and let … be a generalised Moonshine function (GMF). Suppose this GMF is rational. Generalised power-conjugatio…

Let $\mathcal{V}^{\natural}$ be the Moonshine module, a self-dual $C=24$ meromorphic conformal field theory, and let $J(\tau)$ denote its characteristic function. Moonshine-module…

Let $g$ and $h$ be commuting elements such that every $g^{a}h^{b}$ is a normal class, and suppose the $g$-twisted vacuum is one dimensional, so that $D_g=1$. Let $\phi_g(h)$ denote…

Monster orbifold conjecture.