31 problems

Let $\theta$, $\theta_t$, $\beta$, and $s$ be parameters, let $G$ denote the Barnes $G$-function, and let $\Phi$ and the matrix elements appearing below be the conformal-block data…

Let $q=\exp(\pi i/\ell)$ with $\ell\ge3$, and let $\operatorname{Conf}_{\vec\Lambda}^{(\ell)}$ be the space spanned by the lattice restrictions of the Weyl anti-symmetrized hyperge…

Let $\vec\Lambda\in\mathbb Z_{\ge0}^n[2{\mathfrak m}]$, and let $\mathbb M_\Theta$ be the Macdonald–Ruijsenaars operator acting on $L_{\vec\Lambda}[0]$-valued trigonometric polynom…

Let $Q=b+b^{-1}$, and let $\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_{23}$ be complex Liouville momenta. The fusion coefficients are denoted by…

Let $\mathbb S$ be the contour of intermediate Liouville momenta, and let ${{\mathcal F}}^s_{\alpha_{21}}$ and ${{\mathcal F}}^t_{\alpha_{32}}$ denote the s- and t-channel conformal…

Let $X$ be a smooth projective curve, let $S\subseteq X$ be a nonempty set containing the bad points $\mathcal{R}$ of a parahoric Bruhat–Tits group scheme $\mathcal{G}$, and let…

Let $\lambda$ be a Young diagram, and let Theorem denote the theorem proved in the paper for Young diagrams with two columns, identifying the fused Specht polynomials indexed by…

Let $n$ be the rank of the gauge-origami construction, and let the conformal blocks … be the blocks defined in the source. The corresponding multi-leg Pandharipande–Thomas (PT) ver…

Let $K_N(\Delta,\Delta_1,\beta)$ be defined by the recursion in the source, and let $P_N$ be the quantity occurring there, $Q_N:=P_N-N$, and $B_s$ the Laurent-polynomial basis defi…

Let $H^{(N)}_{\Delta}$ denote the contribution at order $N$ to the torus one-point conformal block, expressed in terms of the central charge $c$. Singularity-free conjecture. At ea…

Let $\mathfrak{g}$ be a Lie algebra with a standard automorphism $\sigma$, and let $D_{c,\sigma}$ and $D_c$ be the corresponding sets of dominant weights. For…

General dessin conformal-block conjecture. For a dessin satisfying the proposition's conditions, it corresponds to a family of 4-point CBs whose states follow the stated table.

Bohr–Sommerfeld conjecture for type $\mathcal G$.

Confluent exponentiation conjecture for type $\mathcal G$. The confluent conformal block exponentiates semiclassically:

Let $f_k(x)$ be the function defined in equation (a5), let $N$ be the parameter appearing there, and let $\mathcal{B}_{\lambda,\mu}$ denote the conformal blocks defined in equation…

Let $\mathfrak{g}$ be a simple Lie algebra, and let $X$ be a stable curve in $\overline{\mathcal{M}}_{g,n}$. Write $\mathbb{V}_{X,\mathfrak{g}}^{\dagger}$ for the algebra of confor…

Let $G$ be the Barnes $G$-function, and let $a$ be a complex number satisfying $a^{3/2}=-\sqrt{-2}$. Let $\Phi^{\Delta,\beta}_{\Lambda',\Lambda}$ denote the indicated ramified irre…

Let $\mathrm{M}_{\mathbf{p}}(r,0)$ be the moduli space of parabolic bundles, let $\mathrm{Eff}(\mathrm{M}_{\mathbf{p}}(r,0))$ denote its effective cone, and let…

Let ${\mc C}$ be a smooth projective algebraic curve, let $p_i$ be marked points, let $L_{\lambda_i,{}^Lk}$ be the relevant modules for the affine algebra at level ${}^Lk$, and let…

Generic bilinear relation.

Algebraic-solution relation. For $s=\pm1$,

Bilinear relations. The $q$-deformed conformal blocks satisfy

AGT conjecture. One has

Geometric interpretation conjecture. There are polarized pairs $(\mathcal{X}_x,\mathcal{L}_x)$ such that the isomorphism in Equation holds for every $x\in Z$.

Let $W_{0,3}(SL_m(\mathbb{C}))$ be the algebra associated with the three-pointed genus-zero construction, and let extended conformal blocks of level $1$ be the generators under con…