17 problems

Frenkel–Kac–Wakimoto conjecture. Drinfeld–Sokolov reduction sends every admissible $\widehat{\mathfrak{g}}_\kappa$-module to either zero or a cohomological shift of a minimal serie…

Let $\fg$ be a simple Lie algebra, let $f\in\cN$ be nilpotent with orbit $\BO_f$, and let $L_k(\fg)$ be a simple affine vertex algebra with associated variety $X_{L_k}$. Let…

Minus-reduction conjecture. For any $\ta{ kappa}\in\mathbb C$ and any V\in\operatorname{Obj}\ta{ O}_{\ta{ kappa}},

Let $\fg$ be a Lie algebra, let $f$ be a nilpotent element of $\fg$, and let $L_k(\fg)$ be the simple affine vertex algebra at level $k$. Denote by $H_f^0(L_k(\fg))$ the vertex alg…

Subregular reduction conjecture.

Let ${\cal V}$ be a VOA corresponding to a four-dimensional ${\cal N}=2$ SCFT, and let DS reduction produce another VOA corresponding to a four-dimensional ${\cal N}=2$ SCFT. Denot…

Kac–Roan–Wakimoto–Arakawa conjecture. The functor $\Psi_f^-$ is exact, and the image of every simple module is either simple or zero. Moreover,

Let a codimension-two surface defect of general type be described by a parabolic structure, and let $\widehat{\mathfrak{sl}}_N$ denote the affine current algebra. Drinfeld–Sokolov…

The Poisson structures on polygons obtained by reduction from difference operators include lattice analogues such as the Volterra, lattice Virasoro, and Belov–Chaltikian lattice…

Frenkel–Kac–Wakimoto minimal-series conjecture. The resulting $\EuScript{W}_\kappa$-module is always a minimal series representation concentrated in cohomological degree zero.

Frenkel–Kac–Wakimoto nonvanishing conjecture. The reduction of an admissible series representation under $\Psi^-$ does not vanish if and only if its highest weight is antidominant.

Let $\mathfrak{g}$ be a simple Lie algebra, $f$ a nilpotent element, and $k$ a level. Write $L_k(\mathfrak{g})$ for the simple affine vertex algebra and $H_{DS,f}^0$ for degree-zer…

Drinfeld–Sokolov decomposition conjecture. The reduced vertex operator algebra decomposes as

Let $\mathfrak{g}$ be a Lie algebra, let $\widehat{\mathfrak g}$ be its affine algebra, let $d_{st}$ be the standard semi-infinite cohomology differential, let $\sigma^\lambda$ be…

Free-field realization conjecture. In the Lie-algebra case, $\mathrm{W}$ should be isomorphic to the corresponding W-algebra, namely the Hamiltonian or quantum Drinfeld–Sokolov red…

qDS reduction conjecture. The chiral algebra associated to the class $\mathcal{S}$ theory with a puncture of type $\Lambda$ is obtained by performing qDS reduction with respect to…

Consider $\mathcal{N}=2$ $\operatorname{SU}(N)$ gauge theories with a surface defect belonging to the class classified by partitions of $N$. Their dual conformal-block description…