20 problems

The explicit projector formula. The idempotent projecting $\lambda\otimes S$ onto the simple object indexed by $b$ is

Let $\2C$ be a braided tensor $C^*$-category of finite dimension, and let $\2Z(\2C)$ denote its center. A modular extension of $\2C$ is a modular category $\2M$ containing $\2C$ as…

Congruence kernel conjecture. The projective representation of $\operatorname{SL}(2,\mathbb Z)$ given by $\rho$ has kernel that is a congruence subgroup of level…

Let $\mathcal C$ be a modular tensor category, and let $\operatorname{FPdim}(\mathcal C)$ denote its Frobenius–Perron dimension. Solvability conjecture. If…

Complete anisotropy conjecture. The category $\mathcal{C}$ is completely anisotropic: the only rigid Frobenius algebra in $\mathcal{C}$ is the unit object.

The 16-fold way conjecture. Every super-modular category $\mathcal{B}$ admits a minimal modular extension, and it has exactly 16 minimal modular extensions up to braided monoidal e…

Schellekens' conjecture. There are exactly $71$ holomorphic tame vertex algebras of central charge $24$.

Let $\mathfrak g$ be a simple Lie algebra, with dual Coxeter number $h^\vee$ and lacety $d$. Let $\ell=-h^\vee+\frac{a}{b}$ be an admissible level, where $a,b$ are coprime positive…

Let $\mathcal{C}$ be a unitary modular category, and let $(\mathcal{C},c)$ denote an admissible genus consisting of $\mathcal{C}$ together with a compatible central charge $c$. Adm…

Let $\mathcal{C}$ be a braided fusion category, and let $X,Y\in\mathcal{C}$ be simple objects. For each $n$, let … be the associated braid group representation. Property $F$ conjec…

Let $\mathcal{C}$ be a modular category, and let $(S,T)$ be its modular data. Modular-data determination conjecture. The $S$ and $T$ matrices of $\mathcal{C}$ determine it up to ri…

Let a $2D$ topological phase of matter (TPM) be an equivalence class of connected stable topological Hamiltonian schemas, and let its topological order be the unitary modular categ…

Splitness conjecture. Then $\mathcal{B}$ is split super-modular.

A super-modular category is a premodular category whose Müger center is equivalent to $\sVec$. A spin modular category is a modular category containing a fermion. For a fusion cate…

Let $\mathcal{V}$ be a strongly-finite VOA, and let $\mathrm{Mod}^{g.r}(\mathcal{V})$ be its category of grading-restricted generalized modules. Its semi-simplification is obtained…

Let $\mathcal{V}$ be a strongly-finite VOA, and let $\mathrm{Mod}^{g.r}(\mathcal{V})$ denote its category of grading-restricted generalized modules. A log-modular tensor category i…

Let $G$ be a finite abelian group of odd order, let $q$ be a quadratic form on $G$, and let $\mathcal{NG}(G,q,b,c)$ be the corresponding near-group fusion category. Let $E_j$ be th…

Fix a set of fusion rules, and consider all modular categories having those fusion rules. For a modular category $\mathcal C$, let $\operatorname{ID}(\mathcal C)$ denote its fusion…

A modular category is a braided fusion category over $\mathbb C$ equipped with a compatible ribbon structure whose braiding is nondegenerate; a defining number field is a number fi…

Wang's finiteness conjecture. There are finitely many modular categories of rank $n$ for any fixed $n\in\mathbb{N}$.