25 problems

Let $\mathfrak{g}$ be a semisimple finite-dimensional complex Lie algebra with simple roots $\alpha_1,\ldots,\alpha_n$ and Killing form $(-,-)$. For an integer $p\geq 2$ divisible…

Consider the line with $i_{13}=24$, which is a singlet level, and its finite-size amplitude fit. Let the Jordan cell referred to in the source contain two fields. Jordan-cell ident…

Let $p_+$ and $p_-$ be the parameters of the logarithmic $(p_+,p_-)$ model. Let ${\bar{\mathscr{C}}}$ be the space of generalized characters, let $\mathscr{G}$ be the subspace desc…

Let $p,q\in\mathbb{Z}_{\geq 2}$ be coprime, let $c_{p,q}=1-\frac{6(p-q)^2}{pq}$, and let $\mathcal{O}_{c_{p,q}}^0$ be the full subcategory of Virasoro modules at central charge…

Logarithmic W-algebra conjecture. The logarithmic $\mathbb{W}$-algebras have module categories equivalent to those of the corresponding small quantum group $\operatorname{FK}{G}_q$…

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…

Projective-character submodule conjecture. The subspace $\mathcal{P}$ is an $\mathcal{L}_{1,0}^{\mathrm{inv}}(\bar U_q)$-submodule of $\operatorname{SLF}(\bar U_q)$. The subspace i…

Extension-group conjecture. The extension groups

For $p\geq 3$, let ${\mathcal V}^{(p)}$ be the generalized vertex algebra introduced by … Consider its modules in the category $\mathcal{O}$. Module-category conjecture. For every…

Classification conjecture. All indecomposable Virasoro modules appearing in physical LCFTs for which Virasoro is the maximal chiral algebra should be obtained as scaling limits of…

Let $r,s\in\mathbb{Z}_+$, set $n=rs$, and define … For the non-chiral logarithmic coupling associated with $h=h_{r,s}$, write it as $\beta_{r,s}$. The $\beta_{1,s}$ coupling conjec…

Fix $s$ with $1\,{\leqslant}\, s\,{\leqslant}\, p-1$. Let $\mathrm{Ext}^\bullet_s$ be the associative Yoneda algebra formed from the extension spaces between the specified irreduci…

Let $p\,{\geqslant}\, 2$ be an integer. Let $\mathscr{C}^+_p$ be the specified full tensor subcategory of $\mathscr{C}_p$, and let $\mathcal D_p$ be the category of representations…

For an integer $p\,{\geqslant}\, 2$, let $\mathscr{C}^+_p$ be the indicated tensor subcategory of modules for the Lusztig-limit quantum group, and let ${\mathcal{LM}}(1,p)$ denote…

Let $p\,{\geqslant}\, 2$ be an integer. Consider the “long” screening extension of $\overline{\mathscr{U}}_{\mathfrak{q}}s\ell(2)$, namely the Lusztig limit…

Logarithmic two-point correlation conjecture. Given the conjectured field identifications, for every $h\geq 2$ the dominant term is

Coset–triplet symmetry conjecture. The coset model $\mathcal{C}$ has the same extended chiral symmetry algebra as the triplet model.

Parafermion conjecture. For general $p$, $\mathbb{C}_{\mathfrak{q}}[z,\partial]$ should allow expressing the relations among the fields of the $(p,1)$ model with an explicit quantu…

A finite tensor category is a rigid tensor category in which every object has finite length, every morphism space is finite-dimensional, there are only finitely many inequivalent s…

Let $p,p'\in\mathbb{Z}_{>0}$ with $p\ne p'$ and $p\geq 2$. Let $L=\mathbb{Z}\alpha$ satisfy $\langle\alpha,\alpha\rangle=2pp'$, and let $\mathcal{W}(p,p')$ be the vertex operator a…

Extended minimal-model coexistence conjecture. The coexistence of incompatible modules on the bulk and boundary may also apply to other extended minimal models, such as the $O(1)$…

Sector-separation conjecture. The apparent contradiction is resolved if the two incompatible modules belong to different sectors, whether bulk or boundary, and all bulk-boundary fu…

Let $p$ and $q$ specify logarithmic $(p,q)$ models, and let Cardy states mean boundary states satisfying the Cardy condition. Cardy-state count conjecture. There are … Cardy states…

The $(1,p)$ models are logarithmic conformal field theories whose Verlinde algebra is nonsemisimple; denote this algebra by the Verlinde algebra of the $(1,p)$ model, and let a DAH…

Flohr–Grabow–Koehn conjecture. The characters of the $c_{k,1}$ models admit fermionic representations labelled by the Lie algebra $\textup{D}_k$.