97 problems

For $i\in I$ and $\alpha\in J_i$, let $b(i,\alpha)$ be the quantity defined by … where $M^j=a_{\beta\otimes 1}$. Kac–Wakimoto conjecture. One has … for $\alpha\in J_i$. The conject…

Picard 2-group conjecture. It is conjectured that this algebraic structure captures part of the homotopy type of $\mathcal{M}(\mathcal{B})$.

Let $\mathcal{C}$ be a modular tensor category, and suppose there is a positive, strong-rational vertex operator algebra $V$ with $\operatorname{Rep}(V)\cong\mathcal{C}$. Here…

Let $V$ be a unitary strongly rational vertex operator algebra, and let $\mathcal{A}_V$ be the conformal net naturally associated to $V$ through strong locality. A conformal net is…

Witt-realization conjecture. There is at least one unitary vertex operator algebra (VOA) $V$ of central charge $c$ such that $\operatorname{Mod}_V$ is a UMTC Witt equivalent to…

Let $\mathcal C$ be a modular tensor category, and let $\rho_{\mathcal C}^{\lambda,1}$ be its modular representation; equivalently, consider the corresponding representation for a…

Let ${\Cal O^{int}_\varkappa}$ be the hermitian category constructed from the integrable representations under consideration, with inner products on its spaces of morphisms defined…

Monodromy representation conjecture. These representations are isomorphic.

In conformal field theory, modular tensor categories arise as representation categories of rational vertex operator algebras. Modularity conjecture. The representation category of…

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 $q\in\mathbb{Z}$ with $q\geq 2$ and $3\nmid q$. The strongly rational vertex operator algebras $L_{-2+q}(\mathfrak{sl}_2)$ and $L_\mathrm{Vir}(c_{q,3},0)$ have modular tensor r…

Let the boundary CFT associated with $SU(N)_k$ Chern–Simons theory have an operator algebra of boundary topological defects. A modular tensor category is a braided fusion category…

Let $V$ be a vertex operator algebra satisfying the three conditions stated for the modular tensor category theorem, and let $G$ be a finite group of automorphisms of $V$. Define t…

Let $V$ and $V'$ be positive, strong-rational vertex operator algebras in the same bulk genus, and suppose their representation categories are anisotropic. Bulk-genus orbifold conj…

Let $V$ be a strongly finite vertex operator algebra, meaning that $V$ is $C_2$-cofinite and has the finiteness properties specified in the paper, but is not strongly rational. Wri…

Bulk genera finiteness conjecture. Every bulk genus contains only finitely many isomorphism classes of vertex operator algebras.

Reconstruction conjecture. For every modular tensor category $\mathcal{C}$, there is a sufficiently large $c\in\mathbb{Q}$ for which the bulk genus $(\mathcal{C},c)$ exists. Equiva…

Let $V$ be a strongly finite vertex operator algebra, and let $\operatorname{Rep}(V)$ denote its category of grading-restricted generalized $V$-modules. Huang's conjecture. The cat…

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

Center-realization conjecture. An indecomposable subalgebra

Generalized pull-open conjecture. Any stable gapped domain wall ${}_B\CM_B$ is equivalent to

Homological modularity conjecture. The category $\mathsf{RFC}(\mathfrak{sl}(N),M_3)$ is modular if and only if

Seifert-gauge invariance conjecture. The relative fusion categories $\mathsf{RFC}(\mathfrak{sl}(N),M_3)$ and $\mathsf{RFC}(\mathfrak{sl}(N),M'_3)$ have the same modular data.

Completeness conjecture. The construction may provide a comprehensive list of modular tensor categories.