35 problems

Verlinde's conjecture. The modular $S$-matrix governs the fusion rules through

Braided tensor structure conjecture. The quotient category $\mathcal{C}_{p_+,p_-}/\mathcal{S}$ has the structure of a braided tensor category.

Creutzig–Milas conjecture. This Verlinde algebra is isomorphic, as a ring, to the Grothendieck ring of the singlet algebra.

Let $\bar K_{\lambda\mu}(q)$ denote the restricted Kostka polynomial, namely the coefficient of $\bar s_\lambda$ in the restricted modified Hall–Littlewood function $\bar Q'_\mu$,…

Let restricted $q$-analogues be obtained by applying the $t$-reduction algorithm to $q$-analogues of unrestricted products of Schur functions, with $t=q$ for spin, charge, and ener…

Let $V_{\mu_1},\ldots,V_{\mu_\ell}$ be self-conjugate irreducible representations of a rational vertex algebra, and let $\nu_{\mu_i}\in\{1,-1\}$ denote their Frobenius–Schur indica…

Twist-symmetry conjecture. The representations ${\cal R}_{\mp 1/3}$ should be related by $\pi_{\pm1}$, so that the twist symmetry holds for the fusion rule denoted by…

Twist-symmetry conjecture. The actual fusion rules are symmetric under the twist symmetry; in particular, the representations related by $\pi_s$ have correspondingly related fusion…

Creutzig's conjecture. The actual fusion products are obtained by applying $P$ to the Verlinde products in the Grothendieck ring:

Let $\mathscr{D}$ be the category of weight $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$-modules with finite-dimensional weight spaces and real weights. Let $\mathbin{\boxtimes}$ denote its…

Let $\mathsf{k}$ be admissible-nondegenerate. For $\ell,\ell'\in\mathbb{Z}$, $[j],[j']\in\mathbb{R}/\mathbb{Z}$, and…

Let $\widehat{\mathscr{W}}_{3,2}$ denote the category of modules for the $\mathfrak{sl}_3$ minimal model $\mathsf{A}_2(3,2)$, and let its Grothendieck fusion multiplicities be the…

Contour-integral and fusion-rule conjecture. Contour integrals should describe the characters of certain families of RCFTs, and integrality of the resulting fusion rules may provid…

Let $\mathsf{A}_1\brac*{u,v}$ be a non-unitary affine minimal model, let $\mathsf{E}_{\lambda,\Delta_{1,1}^{\textup{aff}}}$ and $\mathsf{E}_{\lambda',\Delta_{r,s}^{\textup{aff}}}$…

Let $\mathsf{A}_1\brac*{u,v}$ be a non-unitary affine minimal model. Its remaining fusion rules are expected to involve reducible but indecomposable staggered modules with four com…

Let $\mathsf{M}\brac*{u,v}$ be a non-unitary $N=2$ minimal model, and let ${}^{[i]}\mathsf{P}_{p;r,s}$ denote one of the conjectural staggered modules, with…

Let $V_k(sl(2))$ be the affine vertex algebra at level $k$, and let $\pi_\ell$ denote spectral flow by $\ell$. For relaxed $V_k(sl(2))$-modules $\mathcal E_{\mu,\Delta_{r,s}}$ and…

$(3,2)$ fusion-rule conjecture.

Verlinde-coefficient conjecture. The Verlinde coefficient $N^{\mu_2}_{\lambda,\mu_1}$ can be expressed as a certain delta function with a non-negative integer coefficient, and this…

Let $k$ be a finite level, and let $\lambda$ and $\mu$ be integrable weights of $\widehat{su}(4)_k$. Write $\bar\mu$ for the conjugate of $\mu$, and let $\lambda\star_k\mu$ and…

Let $p\in\mathbb{Z}_{\ge 2}$. For $j\in\mathbb{Z}$, let $\pi_j$ be the irreducible $\overline{M(1)}$-module defined in the surrounding construction, and let $M_t$ denote the corres…

Let $p'$ be the minimal-model parameter, and let $\langle\mathcal{K}_{1,3}\rangle$ be the unital fusion subring generated by $\mathcal{K}_{1,3}$. Let $U_n$ and $T_n$ denote the Che…

Let $p$ be the minimal-model parameter, and let $\langle\mathcal{K}_{3,1}\rangle$ denote the fusion subring generated by $\mathcal{K}_{3,1}$. Write $\mathcal{S}_{kp,1}^{a,0}$ for t…

Let $p$ be the minimal-model parameter and let $\mathcal{K}_{3,1}^{\mathbin{\times} n}$ denote the $n$-fold iterated fusion product of $\mathcal{K}_{3,1}$. Iterated-fusion decompos…

Let $p$ and $p'$ be the minimal-model parameters, and consider the Kac modules $\mathcal{K}_{r,s}$ appearing in the decompositions of…