211 problems

Etingof's pivotal structure conjecture. Every fusion category admits a pivotal structure.

Kashina's conjecture. The exponent of $H$ divides its dimension:

A modular category is a braided fusion category whose braiding is non-degenerate, and its rank is the number of isomorphism classes of simple objects. Wang's rank-finiteness conjec…

Pivotality conjecture. Every fusion category is pivotal.

Let $\mathcal{E}$ be a symmetric fusion category and let $\mathcal{C}$ be a non-degenerate braided fusion category over $\mathcal{E}$. For a fusion category $\mathcal{D}$ containin…

Let $G$ be an abelian group of odd order, and let $\mathcal{A}$ be a near-group fusion category of type $(G,|G|)$. Write $\mathcal{Z}(\mathcal{A})$ for its Drinfeld center, let…

Let $p$ be a positive integer and let $\mathrm{Ver}_p$ denote the Verlinde category. A tensor category has subexponential growth if its growth is subexponential in the sense used i…

Let $\mathcal{C}$ be a finite tensor category over an algebraically closed field $\mathbf{k}$ of characteristic $q\ge 0$, with Frobenius–Perron dimension $p>2$ prime. A tensor cate…

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…

Vertex algebra–conformal net correspondence conjecture. There is a bijective correspondence between completely rational local conformal nets and unitary $C_2$-cofinite vertex opera…

A fusion category of rank $3$ is a fusion category with three isomorphism classes of simple objects, and a set of fusion rules records the tensor-product decompositions of those si…

Rigidity conjecture. (i) Any fusion ring has at most finitely many realizations over ${\bf k}$, up to equivalence (possibly none). (ii) The number of tensor functors between two fi…

Let $\mathcal A$ be the conformal net under consideration, let $f_1$ be the distinguished twisted sector used in the preceding lemma, and let $\tau$ be the automorphism label obtai…

Cosemisimplicity conjecture. Then $H$ is cosemisimple.

In a higher-dimensional setting, consider a separable algebra and the unitarity and Frobenius conditions on an algebraic structure of this kind. Higher-dimensional structure conjec…

Let a generalized fusion categorical symmetry emerge in the infrared, and let its simple objects have quantum dimensions $d_i$. Weak integrality conjecture. The squares of the dime…

A unitary modular fusion category is a unitary modular fusion category with central charge $\frac{110}{29}\mod 8$. For a modular fusion category $\mathcal C$ of rank $r$, define it…

Let $l\geq 7$ be an integer not divisible by $3$, and let $q$ be the parameter defining the braided finite tensor category $\mathcal{C}(G_2,G_2(a_1),l,q)$. Write $T(\omega_1)$ and…

Let $\mathcal{C}$ be a premodular fusion category. An embedding of $\mathcal{C}$ as a Galois-closed subcategory of a modular category is called a Galois-modular extension. Schopier…

Let $N$ and $k$ range over positive integers, and consider all non-pointed étale algebra objects in the categories $\mathcal{C}(\mathfrak{sl}_N,k)$. Their super-transitivity is the…

Let $N$ and $k$ range over positive integers, and let $\mathcal{C}(\mathfrak{sl}_N,k)$ denote the corresponding categories. Let $\mathcal{A}$ be the set of all non-pointed étale al…

Let $a$ be a self-dual, non-invertible element of a fusion category, and let $p^{(a,a)}$ denote the projection defining the anyonic chain from the fusion of two copies of $a$. Corc…

A braided weakly fusion category is the braided categorical object associated to a geometric braided functor; a braided fusion category is the corresponding rigid finite semisimple…

Bundles with flat connections associated to braided fusion and modular fusion categories are defined over $\overline{\mathbb{Q}}$. Etingof–Varchenko's conjecture. For any braided f…

Let $\repM_1,\repM_2,\dots,\repM_m$ be families of modules over $\eptl_N(\beta)$, with $m\geqslant 2$, and let $\widehat N_i=\sum_{k=1}^iN_k$. For vectors $u_i\in\repM_i(N_i)$, wri…