18 problems

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…

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…

Let ${\cal C}$ be a $k$-linear abelian braided rigid monoidal category with $\operatorname{End}_{\cal C} I=k$ and finite-dimensional Hom spaces, generated by an object $V$ and its…

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…

Rigidity-factor conjecture. The objects $X$ and $Y$ are rigid.

Property $\mathrm{F}_\mathrm{braid}$ conjecture. A braided fusion category $\mathcal{B}$ has property $\mathrm{F}_\mathrm{braid}$ if and only if it is weakly integral.

A smooth 2-group is a smooth monoidal groupoid whose multiplication, associator, unitors, and inverses are smooth, and a braided smooth 2-group is a braided monoid in smooth groupo…

Let $\mathfrak{g}$ be a simple Lie algebra, let $P$ be its weight lattice and $Q$ its root lattice, and let $Q_{\mathfrak{g}}^N:P/Q\to\mathbb{C}^\times$ be the quadratic form defin…

Let $V$ be a rational, $C_2$-cofinite vertex operator algebra, and let $coverline{\mathcal{C}_V}$ denote its modular tensor category equipped with the reverse braiding. Realization…

Minimal modular extension conjecture. Every slightly degenerate braided fusion category admits a minimal modular extension; equivalently,

Minimal modular extension conjecture. Every slightly degenerate braided fusion category $\mathcal C$ admits a minimal modular extension $\mathcal C'$.

Let $\mathcal{C}$ be a braided semigroupal polyadic category with an $n$-ary tensor product, an $n$-ary associator $\scalebox{1.15}{\textsf{\upshape A}}^{\left(2n-1\right)\otimes}$…

Let $\mathrm{A}\boldsymbol{\mathrm{Link}}^+$ be the category of consistently oriented annular links and cobordisms, and let $K^b(\mathrm{A}\boldsymbol{\mathrm{Foam}}^+)$ be the hom…

Braided equivalence conjecture. For $\beta=e^{-\pi i d/8}$, the functor $F$ is a $\mathbb{C}$-linear braided monoidal equivalence. This conjecture identifies the representation cat…

For a completely rational conformal net $\mathcal{A}$, write $\operatorname{Rep}(\mathcal{A})^{\mathrm{rev}}$ for the unitary modular tensor category obtained by replacing its brai…

Let $M$ be a simple current in the representation category of a Möbius vertex algebra, without assuming that $M$ has integral weight. The balancing isomorphism is the natural isomo…

Conjecture. (1) ${\mathcal M}^{perv}_e$ is a semisimple abelian category with finitely many simple objects. (2) There exists $n_e\in{\mathbb Z}$ such that…

Let $\mathcal{C}$ be a braided fusion category. For a simple object $X$, define its Frobenius–Perron dimension $\operatorname{FPdim}(X)$ as the largest eigenvalue of the matrix rep…