555 problems

Exact dimension formula. The dimension is

Verlinde's conjecture. The modular $S$-matrix governs the fusion rules through

Let $V$ be a holomorphic vertex operator algebra of central charge $24$, and let $V_1$ denote its dimension-one part. Schellekens classified $71$ possible choices for $V_1$, includ…

Higgs branch conjecture. The Higgs branch, viewed as a holomorphic symplectic variety, is canonically isomorphic to the associated variety of the corresponding VOA:

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 $\widehat{\mathscr{W}}_{3,2}$ be the module category of the $\mathfrak{sl}_3$ minimal model $\mathsf{A}_2(3,2)$. A rigid braided tensor category is a braided tensor category in…

Norton's generalized moonshine conjecture. For every commuting pair $(g,h)$ in $\mathbb M$, there exists such a genus-zero modular function $Z(g,h,\tau)$; explicitly, if…

Braided tensor structure conjecture. The quotient category $\mathcal{C}_{p_+,p_-}/\mathcal{S}$ has the structure of a braided tensor category.

C2-cofiniteness–rationality conjecture. If $V$ is $C_2$-cofinite, then $V$ is rational.

Let $V$ be a vertex operator algebra, let $G\leq\operatorname{Aut}(V)$ be a finite group, and let … be the fixed-point subalgebra. Assume that $V$ is $C_2$-cofinite and rational. O…

Let $W$ be a vertex operator algebra containing Heisenberg fields $\alpha_i$, and let $S_i=\operatorname{Res}(e^{\alpha_i})$. Assume that $W$ decomposes into integer eigenvalues un…

Let ${\cal V}$ be the boundary vertex operator algebra of a three-dimensional theory, and let $R_{\cal V}$ denote its Zhu $C_2$ algebra. Write $\operatorname{Specm}(R_{\cal V})$ fo…

Let $V_n$ be the graded pieces of the Monster module and, for an element $g$ of the Monster group, let … be the associated McKay–Thompson series. A hauptmodul is a generator of the…

Partition function subalgebra uniqueness conjecture. The equality

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…

Let $\mathsf{C}$ denote the extended vertex operator algebra obtained as the order-$2$ simple current extension of $\mathsf{A}_1\brac{5,2}$. Let $\mathfrak{g}_2$ be the simple Lie…

Let ${\mathcal H}(n)$ be the rank-$n$ Heisenberg vertex algebra, with orthogonal-group orbifold ${\mathcal H}(n)^{\operatorname{O}(n)}$. Its generators have a first relation at wei…

Boundary-level rationality conjecture. The vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(\mathcal{k},0)$ is rational in the category $\mathcal{O}$, and its irreducible w…

Consider an $(A_{k-1}, A_{N-1})$ theory and its associated vertex operator algebra. Zhu's $C_2$-algebra is a truncation of the vertex operator algebra, and a Jacobi algebra is the…

A unitary vertex operator algebra (VOA) is a unitary VOA in the sense used for chiral conformal field theory, and it is strongly local when the associated operator-valued distribut…

Creutzig–Milas conjecture. This Verlinde algebra is isomorphic, as a ring, to the Grothendieck ring of the singlet algebra.

Let $\mathcal{C}$ be a modular tensor category, and suppose there is a positive, strong-rational vertex operator algebra $V$ with $\operatorname{Rep}(V)\cong\mathcal{C}$. Here…

Let $\mathcal{T}^{A/B}$ be a three-dimensional theory with boundary VOA $V^A$ or $V^B$, and consider suitable modules for the boundary VOA. The boundary-module conjecture. Certain…

Let $A_{5+1/2}$ denote the intermediate Lie algebra between $A_5$ and $E_6$, and let $L_k(A_{5+1/2})$ be its affine VOA at positive integer level $k$. Write $r(k)$ for the number o…

Energy-boundedness conjecture. Every unitary simple vertex operator algebra is energy-bounded.