29 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…

Coset conjecture. The commutant $\operatorname{Com}(W,V)$ is also rational and lisse. This is presented as a widely believed assertion and is used in the paper to deduce rationalit…

Let $V^k(\mathfrak{sl}_{n+1})$ be the universal affine vertex algebra of $\mathfrak{sl}_{n+1}$ at level $k$, let $bc^{\otimes n}$ be the tensor product of $n$ copies of the $bc$-sy…

Let $\mathfrak g$ be a simply-laced Lie algebra, let $k$ be a level, and let $W_k(\mathfrak g)$ denote the simple regular (principal) W-algebra associated with $\mathfrak g$ at lev…

Rationality conjecture. The covariant representations $\pi_{i,\alpha}$ can be decomposed into a direct sum of a finite number of irreducible representations, and the localized sect…

Coset $W_N$-generation conjecture. For a more general affine Lie superalgebra, the coset space

Let the two-dimensional conformal field theory under discussion be the theory realizing the symmetry associated with the unitary Haagerup fusion category $H_3$. Let a Haagerup rati…

Let $n\in\frac12\mathbb Z$ with $n\neq0,-\frac12$, and let $\Psi_n$ be the map from the preceding theorem between the relevant one-parameter vertex algebras. Half-integral extensio…

Simple-quotient embedding conjecture. The map descends to the simple quotient and induces

Let $\mathcal C_\ell(n)$ be the coset defined in the source, with $\ell\in\mathbb Z_{\geq1}$. Strong-unitarity conjecture. The vertex algebra $\mathcal C_\ell(n)$ is strongly unita…

Let $\mathcal P^k(n)$ be the affine coset considered in the source, and suppose the relation in minimal weight is decoupling. Minimal strong-generation conjecture. Then…

Let $A$ be the commutant constructed earlier as … and let $B$ be the commutant of the embedded minimal W-algebra obtained from the translation construction. Commutant identificatio…

Consider the coset … which has central charge $57/40$. Here $W_k(E_7,f_\theta)$ denotes the indicated affine $W$-algebra. Effective minimal-model conjecture. This coset is isomorph…

Let $(E_{7+1/2})_k$ denote the affine VOA at level $k$, and let $M_{\rm eff}(13,4)$ be the effective description of the minimal model $M(13,4)$ with effective central charge…

Let $p\in\mathbb{N}\cup\{0\}$, let $c^{\cal H}=8(2p+3)$, and let $m$ be the parameter labelling the series. Define $(n_1,n_2)=(p+2,p+m+2)$. Conjecture on infinite coset series. Sta…

Let $\mathcal{FT}_k(\mathfrak g)$ be the Feigin–Tipunin algebra, let ${\text{FF}}$ denote the ambient free-fermion VOA, and let $\widetilde{{\mathcal N}}_{G,k}$ be the modified log…

Let $\widetilde{{\mathcal N}}_{n,k}$ be the slight modification of the boundary VOA ${\mathcal N}_{n,k}$ obtained by successive extension and orbifold, let…

Let $Y_{r_1,r_2,r_3}$ be the coset algebra associated to the labels $r_1,r_2,r_3$, and let $\mathcal{W}_{r_1,r_2,r_3}$ be the algebra defined by the corresponding screening operato…

Let $\mathfrak{g}$ be a reductive simply-laced Lie algebra, let $n$ be a positive integer, and let $P^+_n$ be the set of dominant integral weights of $\mathfrak{g}$ such that…

Let $\mathfrak{g}$ be a simply-laced simple Lie algebra over $\mathbb{C}$, let $h$ be its Coxeter number, and let $p\in\mathbb{Z}$ satisfy $p\geq h$. Set … For a positive integer l…

Let $n,m$ be positive integers and let $p$ satisfy … Let $\mathcal{W}_{n,m}$ be the family of W-algebras defined by quantum Hamiltonian reduction, let…

Type A coset conjecture. There is an isomorphism

Type C coset conjecture. The vertex algebra ${\mathcal C}_k(n)$ is isomorphic to the $C_2$-cofinite, rational principal ${\mathcal W}$-algebra…

Let $\mathcal{V}$ be a VOA with mutually commuting subVOAs $\mathcal{W}$ and $\mathcal{C}$, and suppose … is its decomposition as a $\mathcal{W}\,\,\boxit{\times}\,\,\,\mathcal{C}$…

Kac–Wakimoto's coset conjecture. The $(p,q)$-minimal model of $\mathscr{W}^k(\mathfrak{g})$ is isomorphic to the commutant of $V_{l+1}(\mathfrak{g})$ inside…