137 problems

Simple Hurwitz numbers count branched coverings of the sphere with prescribed ramification over one branch point and simple ramification over the remaining branch points. The Chekh…

The quartic Kontsevich model is a model whose correlation functions are studied through topological recursion and intersection numbers on moduli spaces of curves. Blobbed topologic…

Symplectic invariance conjecture. The free energies $\mathcal{F}^g$ are invariant under any symplectic transformation of $(x,y)$.

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…

The remodeling conjecture concerns a toric Calabi–Yau threefold and its mirror curve, on which the Eynard–Orantin topological recursion is defined. Remodeling conjecture. The open…

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 the spectral curve be … Here the stationary Gromov–Witten theory of ${\mathbb P}^1$ refers to Gromov–Witten invariants with stationary insertions, and topological recursion is…

Let $\omega_{g,n}$ be the multidifferentials produced by topological recursion from a spectral curve with functions $x$ and $y$, and let $F_g$ denote the corresponding free energy.…

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…

Fully simple maps conjecture. The generating functions for the weighted numbers of fully simple maps satisfy topological recursion.

Let double Hurwitz numbers be the enumerative invariants counting degree-$d$ ramified covers of $\mathbb{C}\mathbb{P}^1$ with prescribed ramification profiles over $0$ and $\infty$…

Let $\mathfrak{X}$ be a toric Calabi–Yau $3$-fold, with its mirror curve used as the initial data for topological recursion. Bouchard–Klemm–Mariño–Pasquetti conjecture. Taking the…

Pole structure conjecture. All $\omega_{g,n}$ are meromorphic differentials on the spectral curve and have poles of finite degree only at the branch point $z_j=0$.

Let $\mathfrak{K}$ be a knot embedded in $S^3$, let $J_N(q)$ be its colored-Jones polynomial, set $\hbar=\ln q$ and $x=N\ln q$, and let ${\cal S}$ be the character variety of…

BKMP remodeling conjecture. The generating series of Gromov–Witten invariants of $\mathfrak{X}$ are given by topological recursion applied to $\mathcal{S}_{\mathfrak{X}}$; in parti…

BKMP remodeling conjecture. The free energies $F^{(g)}$ are mapped by the mirror map to the genus $g$ generating functions of Gromov–Witten invariants of $X$, and the correlation f…

Let a local Calabi–Yau background have geometry reducing to a complex curve, and let its B-model holomorphic topological-string amplitudes be described by genus-$g$ free energies a…

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…

Artemev's conjecture. The resonance transformation is unnecessary if one swaps $x$ and $y$ in the spectral curve.

Let $(\bar C,x,y)$ be a genus $g=0$ admissible spectral curve on $\bar C=\mathbb{P}^1$, with $dx$ and $dy$ having at most simple poles. Let the free energy be the quantity defined…

Bouquet-graph conjecture. For any topological recursion relation

Let $W_1$ and $W_2$ be vector fields on the big phase space, with quantum product … Define $T$ by … where $\tau_+$ raises descendant level by one. Semisimple reconstruction conject…

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 $F^{\text{ZZ}}_{(n_+,n_-)}(\hbar)$ denote the free-energy coefficient in the $(n_+,n_-)$ ZZ-instanton sector, let $W^{\text{ZZ}}_{(n_+,n_-)}(E)$ denote the corresponding non-pe…

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…