46 problems

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…

Let $S$ be a curve of bidegree $(k,k)$ in ${\mathbb P}_1\times {\mathbb P}_1$. Let $L^r(a,b)$ denote the line bundle used in the hyperbolic-monopole spectral-curve construction, re…

Let $N\geq 4$, let $\mathbf{T}_{m;n_1,n_2}(z)$ be the monodromy matrix, and let $L$ be the parity parameter appearing in its construction. Write $\vec{e}_1$ and $\vec{e}_N$ for the…

Approximation property. Any smooth potential can be approximated by algebraic-geometrical ones with the same periods.

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…

Degree-eight distinct-diagonal test case. Every reducible connected degree-eight spectral curve is obtained from a constant branch or from reversal symmetry. In particular, there i…

Primitive codimension-growth conjecture. One has

No-connected-hypersurfaces conjecture. For every $n\ge4$, every codimension-one component of the reducible locus is one of the cutting hyperplanes $b_i=0$. Equivalently, the reduci…

Amended reducibility conjecture. Every reducible factorization of $\chi_n$ is expected to be obtained by iterating the following operations on connected blocks: cutting the chain a…

Let $(A,\phi,\psi)\in\mathcal{M}_\alpha^G$ be a solution and let $b=\mathrm{Hit}_\alpha^G(A,\phi,\psi)$. Denote by $\Sigma_b$ the associated spectral curve and by $L$ the correspon…

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…

Real spectral-curve component conjecture. The following assertions hold:

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 $X$ be a compact Riemann surface of genus $g_X$, and let $L\to X$ be a holomorphic line bundle of positive degree. Fix a generic section … with distinct zeros, and consider the…

Let $U({\bf p};\hbar)=\partial_{p_1}^2\log\bigl(\mathcal D^{\rm NP}_{\mu,\nu}(\hbar\cdot{\bf t}({\bf p});\hbar;w)\bigr)$ be the stated solution of the KdV equations. Let $G_2$,…

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…

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…

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…

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…

For each $k\geq 1$, let $\Sigma_{k,m}$, for $m\geq 1$, be the weight-$k$ spectral curve … Here $\mathscr{P}_k$ denotes the set of partitions of weight $k$, and $\ell(\lambda)$ deno…

Fix $\ell\in\mathbb Z_{\neq 0}$ and $\omega>0$. Let $\Gamma_\ell\subset\mathbb R^2_{(\lambda,\mu)}$ be the real spectral curve of the special double confluent Heun equations, and i…