56 problems

A unitary vertex operator algebra (VOA) is a unitary VOA in the sense used for chiral conformal field theory, and it is strongly local when the associated operator-valued distribut…

Strong graded locality theorem. Every nice holomorphic VOSA $V$ with central charge $c\leq 24$ is strongly graded-local. In particular, it gives rise to a holomorphic graded-local…

Compactness claim. The group $\pi_{0}\bigl(G\text{--}\operatorname{Rep}^{I_0}(\mathcal{A})^{\times}\bigr)$ is compact.

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…

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

Lattice strong locality conjecture. The vertex operator algebra $V_\Lambda$ is strongly local.

The constructions above produce mirror extensions of diagonal cosets. For each such extension, let $\tilde {\cal A}\subset \tilde {\cal C}$ be the corresponding inclusion, and let…

Let $M$ be a von Neumann algebra with a cyclic and separating unit vector $\Omega$, let $\omega(\cdot)=(\Omega,\cdot\Omega)$ be the associated faithful normal state, and let…

Let a completely rational local conformal net have a system of irreducible DHR sectors equipped with a braiding. The associated $S$-matrix is obtained from this braiding, while the…

Let $G=SU(N_1)\times SU(N_2)\times\cdots\times SU(N_k)$, and let ${\cal A}_G$ denote the associated net. Complete rationality conjecture. The net ${\cal A}_G$ is completely rationa…

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…

Let $\mathcal A$ be a chiral conformal net with a finite gauge-group action by $\Gamma$, and let $\mathcal A^\Gamma$ denote its fixed-point orbifold net. Consider the irreducible D…

Let ${\cal B}$ be a nontrivial local extension of the $c=1$ Virasoro net ${\cal A}_{(\rm Vir,1)}$. The spectrum condition is that the representation of ${\cal A}_{(\rm Vir,1)}$ on…

Strong locality conjecture. Every simple unitary VOA is strongly local.

A unitary vertex operator algebra is a unitary VOA, and AQFT-locality is the locality condition introduced in the paper that is sufficient to generate an associated chiral conforma…

Unitary vertex operator algebras (VOAs) and chiral conformal nets are the two principal mathematical axiomatizations of two-dimensional unitary chiral conformal field theories. Equ…

Let $W_k(\mathfrak{g},f)$ be a unitary minimal $W$-algebra, with corresponding conformal net denoted by $\mathcal{A}_{W_k(\mathfrak{g},f)}$. The algebra is strongly rational when i…

Let $W$ be a unitary minimal $W$-algebra, and let its corresponding conformal net be denoted by $d49c_W$. The algebra $W$ is strongly rational when it has the strong rationality pr…

Let $\mathcal{A}$ be an irreducible graded-local conformal net, meaning a graded-local conformal net satisfying the irreducibility condition. Converse conformal-net conjecture. For…

A unitary vertex operator algebra (VOA) is a vertex operator algebra equipped with a compatible positive-definite Hermitian structure. A strongly local VOA is one whose smeared ver…

A unitary VOA is a vertex operator algebra equipped with a compatible positive-definite invariant Hermitian structure. A VOA is strongly local when its vertex operators generate a…

Compression conjecture. Any intertwining operator of $V_{\mathfrak g}^l$ can be realized as the compression of an intertwining operator of…

Central-charge divisibility conjecture. The central charge satisfies

Unitary VOA–conformal net correspondence conjecture. The map gives a one-to-one correspondence between simple unitary VOAs and irreducible conformal nets.

Let $G$ be a simple and simply connected compact group, and let $\Omega G$ denote its based loop group. A positive-energy representation of $\Omega G$ is a representation whose ene…