24 problems

Unique-norm integral Fourier matrix conjecture. Then

Modular-data conjecture. Consistent genus-one couplings should be governed by the modular data of the chiral categories $\mathcal{C}$ and $\widetilde{\mathcal{C}}$, for example thr…

Let $\mathcal{C}$ be a modular tensor category, let $a,b,c\in\operatorname{Irr}(\mathcal{C})$, and let … where $N^{kl}_a$ denotes the relevant fusion multiplicity. BHS nonnegativit…

PSS integrality conjecture. For all $a,b,c\in\operatorname{Irr}(\mathcal{C})$, one has

Three-solution conjecture. The zero conditions $$ have only the following three sets of solutions:

Twelve-solution conjecture. The zero conditions $$ have only the following twelve sets of solutions:

Nonexistence conjecture for conductor 5. None of the three combined $d$-numbers can satisfy both sets of conditions … .

Kitaev–Wen conjecture. Modular data satisfy

Let a topological phase of matter be described by its modular data, including the modular $S$ and $T$ matrices. Modular-data determination conjecture. The modular data determines t…

Let $A$ be an abelian group with $A_e\cong\mathbb{Z}_{2^n}\times\mathbb{Z}_2$ for $n\geq2$. Mixed-even-part conjecture. The modular data of the Drinfeld center of the…

Let $A$ be an abelian group with $A_e\cong\mathbb{Z}_{2^{n+1}}$. Cyclic-even-part de-equivariantization conjecture. The modular data of the Drinfeld center of the $\mathbb{Z}_2$-de…

Let $A$ be a finite abelian group of odd order, and let $\mathcal{C}$ be a quadratic category of type $(\mathbb{Z}_2,A,1)$ whose $\operatorname{Vec}_{\mathbb{Z}_2}$ subcategory has…

Let $G_o$ be an odd metric group and let $n\geq1$. Cyclic 2-group near-group conjecture. The pair of tensor-product modular data displayed in the source is the modular data of the…

Let the fifth family of potential modular data be obtained from the fourth family by switching the roles of $G$ and $\Gamma$. Fifth-family realization conjecture. Examples of this…

Let the third family of potential modular data be defined by involutive metric groups whose fixed-point subgroups have order $4$, with the stated compatibility conditions. Generali…

Let $A$ be an odd finite abelian group and let $G=\mathbb{Z}_{2^n}\times A$ with $n\geq1$. Inverse-involution near-group conjecture. The constructed modular data are the modular da…

Let $A$ be a finite abelian group, let $G=A\times A$, and let $\langle\langle\cdot,\cdot\rangle\rangle$ be a non-degenerate symmetric bicharacter on $A$. Define…

Let a pair of metric groups with involutions and compatible quadratic forms be chosen as input to the construction of potential modular data. For certain particular choices of the…

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…

Modular-data conjecture. When $|G|=n$ is odd, the modular data for the near-group category of type $G+n$ is ${\mathcal{M}}{\mathcal{D}}_{G,G'}(Q,Q')$.

Rational VOA realization conjecture. There is a rational VOA realization of each ${\mathcal{D}}^{\omega}{\rm Hg}_{\nu}$ for any sufficiently large central charge $c$, where $c$ is…

Twisted realization conjecture. For any $nu=2n+1$ and any set $P$ consisting of an even number of distinct prime divisors of $mu$, possibly empty, the modular data…

Izumi-type realization conjecture. Subfactor realizations of ${\mathcal{D}}^{0}{\rm Hg}_{2n+1}$ should be provided by Izumi's hypothetical generalization of the Haagerup: the const…

Let a projective congruence datum that is also Galois have exponent $N$ and global dimension $n$. Stronger congruence conjecture. If $n\equiv 2\pmod{4}$, then … and … The conjectur…