73 problems

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…

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…

Coset modularity conjecture. All cosets $H\subset G_k$ satisfy the conditions of Theorem A.

Rehren's conjecture. The representation category of $\2F$ is non-degenerate.

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

Let $\2K$ be a pre-modular tensor category. A fully embedded subcategory is one included as a full tensor subcategory of a modular category $\2C$. Write $\2Z_2(\2K)$ for the center…

Equivalence conjecture. This construction is equivalent to the construction due to A. Ocneanu.

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…

Let $\2C_0$ be a finite non-modular braided tensor category, and let $\2Z(\2C_0)$ denote its center. A modular category $\2C$ is an ambient modular category containing $\2C_0$ as a…

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 $\mathcal{C}$ 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…

Completeness conjecture. The results presented in the source cover all unitary and modular cases among these RTCs.

Pseudo-modularity conjecture. Every pseudo-modular RTC associated with a simple affine Lie algebra is modular.

Let $\mathcal{A}$ be a non-semisimple modular category, viewed through its Ind-completion as an object of the even higher Morita category $\operatorname{Alg}_2(\Pr)$. Let…

Let $\mathcal C$ be a near-group category of type $\mathbb Z/4\mathbb Ztimes \mathbb Z/4\mathbb Z+16$, provided that such a category exists, and let $Z(\mathcal C)$ denote its Drin…

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…

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

Haioun's boundary-condition conjecture. Such a boundary condition on the invertible TQFT associated with $\mathcal{C}$ should recover the non-semisimple $(2+1)$-TQFT constructed in…

Let $W$ be the four-dimensional simple module, and let $\mathrm{I}_W$ denote the framed-link invariant obtained from the ambidextrous trace on $W$. Let the Links–Gould invariant be…

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

Let $V$ be a simple positive-energy self-contragredient $C_2$-cofinite vertex operator algebra, and let $\mathcal{C}$ be its category of grading-restricted generalized modules. Hua…

Minimal modular extension conjecture. Every slightly degenerate braided fusion 1-category admits a minimal modular extension.

Let $U_{\xi}^{H}({\mathfrak{sl}}(m|n))$ be the quantum special linear Lie superalgebra, let $\mathscr{D}^{\wp}$ be its category of perturbative modules, let…

Classification conjecture. The quantum group categories in Proposition 3.9 are all the s-simple non-split transitive super-modular categories, up to Galois conjugates and spherical…