59 problems

Let $g\in\frac12\mathbb{Z}_{\geq0}$ and $n\in\mathbb{Z}_{\geq1}$ satisfy $2g-2+n\geq1$, and let $\omega_{g,n}$ be the multidifferential defined by Definition RTR. Refined topologic…

The Chekhov–Eynard–Orantin topological recursion is a recursive procedure associated with data on a spectral curve; for simple ramification points, it can be related to cut-and-joi…

The large-$n$ expansion of the Toeplitz determinant is organized by the remainder $m$ of dividing $n$ by $2r+1$, and let $F^{\{0\}}$ denote the corresponding $O(1)$ coefficient as…

Artemev's conjecture. Up to a suitable normalization, $A^{(g),\vee}_n(k_1,\dots,k_n)$ coincides with $A^{(g),\mathsf{sing}}_n(k_1,\dots,k_n)$ for $g\geq 0$, $n\geq 1$, and…

Bouquet-graph conjecture. For any topological recursion relation

Let $g$ be a non-negative integer and consider a degree $g$ topological recursion relation of the shape … where $\Gamma^\mathrm{loop}_{0,g,[n]}$ is the stable graph with one genus-…

Let $g\in\frac12\mathbb{Z}_{\geq0}$ and $n\in\mathbb{Z}_{\geq1}$ satisfy $2-2g-n<0$. Let $\omega_{g,n}^{\mathfrak b}$ be the refined correlator on the refined spectral curve, let…

Let $\mathcal{S}$ be an admissible spectral curve with $x,y$ satisfying … A quantisation of $P$ is an operator $\hat{P}(\hat{x},\hat{y};\hbar)$ of the form … where…

Let $r>2$ and let $Q_1,\ldots,Q_r$ and $\Lambda$ be the parameters of the half Seiberg–Witten curve, with $B_{2g}$ denoting the Bernoulli number. Topological free-energy conjecture…

Let ${\cal S}$ be a spectral curve with topological-recursion invariants $F_g({\cal S})$ and $\omega_{g,n}({\cal S})$, and let $\hbar$ be a small parameter. Define the formal tau f…

Let … is a consequence of the degree-$g$ topological recursion relations proposed in the cited work. This proposed connection relates universal equations for higher-genus Gromov–Wi…

Let $\Sigma=\mathbb{C}\mathrm{P}^1$. In Case 1, let $y(z)=z$ be a global affine coordinate and let $x=\log R(z)+Az$, where $R(z)$ is a rational function and $A\in\mathbb{C}$. In Ca…

Refined BPS free-energy conjecture. The refined topological recursion free energy satisfies

Let $\mathcal C=(\Sigma,x,y)$ be a spectral curve with $m$ boundaries. Let $\mathcal D({\bf t};\hbar)$ be its geometric total descendent potential, and for…

Let $\mathcal C=(\Sigma,x,y)$ be a spectral curve and let $\gamma\subset\Sigma\setminus{\bf b}$ be an admissible path associated with $e^{-x/\mathfrak u}$. Let $\eta$ be the CohFT…

Let $\mathcal C=(\Sigma,x,y)$ be a spectral curve with boundaries ${\bf b}=\{b_1,\ldots,b_m\}$, and let $\Lambda=(\lambda_1,\ldots,\lambda_m)$ be local coordinates near the boundar…

Variational extension conjecture. Theorem $$ holds for any $\Sigma$.

Extension conjecture. Theorem $$ holds for any $\Sigma$.

Let $\mathcal{S}_{\bm\mu,\bm\kappa}$ be a hyperelliptic refined spectral curve, and let $\omega_{g,n}$ be the multidifferentials constructed by refined topological recursion. Exist…

Let $g\geq 0$ and $n\geq 1$. Let $\omega^{(g)}_{0,n}$ be the $n$-differentials associated with the original variables, let $\omega^{(h)}_{m,0}$ denote the corresponding differentia…

Higher BGW topological-recursion conjecture. The generating functions of the higher BGW models $\tau^{(m)}$ are given by a version of the Chekhov--Eynard--Orantin topological recur…

Functional-relations conjecture. For every $2g-2+n\geq 0$, these differentials satisfy the functional relations of Theorem, after those relations are expressed as relations between…

Bipartite quadrangulation conjecture. For $g>0$, the topological recursion for this spectral curve gives

Let $4\mathcal W_{g,n}4$ be the abstract $n$-point functions and let the abstract quantum field theory have spectral curve $4\mathcal C4$. Wang–Zhou's residue-reformulation conject…

Let $4\mathcal W_{g,n}4$ be the abstract $n$-point functions of the abstract quantum field theory of fat graphs. Under a suitable Feynman-rule realization, let $4\mathcal C4$ be th…