97 problems

AGT conjecture. One has

Zamolodchikov's exponentiation conjecture. In this semiclassical limit, the conformal blocks exponentiate:

Zamolodchikov's conjecture. In the semi-classical limit, conformal blocks on a four-point sphere have an exponential structure.

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…

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

For an $N=2$ conformal field theory, let $\Delta(\alpha)$ be the conformal dimension of a highest-weight state $|\alpha\rangle$. At level $n$, suppose the Kac determinant has facto…

Let ${\Cal O^{int}_\varkappa}$ be the hermitian category constructed from the integrable representations under consideration, with inner products on its spaces of morphisms defined…

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…

Affine Hecke coinvariant module conjecture. The $\widetilde{\mathbf H}_\kappa$-module $\widetilde{\mathcal C}(L_\kappa(\mu),L_\kappa^\dagger(\lambda))$ should be isomorphic to…

Let $k={\mathbb C}$, let $\Sigma$ be a compact smooth oriented surface, let $I\subset\Sigma$ be a finite nonempty subset, and let $V_i$ be a representation of $\mathfrak{g}$ for ea…

Trace-realization conjecture. The function $u$ is obtained by taking a similar trace of products of intertwining operators for the quantum affine algebra.

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 $\mathfrak g$ be one of the Lie algebras covered by Theorems 1 and 2, and let $\iota_{\mathfrak h}$ be the map whose image is contained in the spaces of holomorphic functions d…

Existence and uniqueness conjecture. In each polygon, there exists a unique global $\textnormal{-}\mathrm{SLE}_{}$ for each $$ -valenced link pattern $\in$ and for each $ \in (0,…

Let $C$ be the three-punctured sphere and let $\mathcal{F}^{\mathrm{GS}}(\alpha_0,\alpha_1,\alpha_\infty)$ denote the Goncharov–Shen block associated with the configuration shown i…

Let $W_K$ denote the higher-rank free-field algebra associated with the Toda theory under consideration, and consider free-field correlators in the large $N$ limit whose integratio…

The paper studies an extension of the free-field formalism to smooth spectral coverings, using the Maulik–Okounkov R-matrix to incorporate wall-crossing effects in spectral-network…

Let $\tau_{\mathrm{VI}}^{(0)}(t)$ be the tau function of the sixth Painlevé equation near $t=0$. Let $\theta_0,\theta_1,\theta_t,\theta_{\infty}$ be its parameters, and let…

Let $b$ tend to infinity, let $\tilde h,\tilde h_2,\tilde h_3,\tilde h_4$ be fixed parameters, and write $x$ for the conformal-block expansion variable. In the one-sided special ca…

Let ${\mathcal V}_{\mathbf b}$ be the vertex algebra associated to the theory, let ${\mathcal A}^!$ be the Koszul-dual algebra, and let derived conformal blocks on $\mathbb{CP}^1$…

Let $X$ be a curve over a field $k$ of characteristic $0$, let $\mathcal{G}$ be the parahoric group scheme under consideration, and let $L$ be a line bundle on the moduli stack…

Rank-$5/2$ Whittaker-state conjecture. The state $|\widetilde\Psi\rangle$ exists and is uniquely determined by $\{\alpha_n\}_{n\in\mathbb N}$ and $c$. Moreover,

Beta-deformed Painlevé I conjecture. The large-$s$ asymptotic expansion is