21 problems

Let $Z_{PT}(q)$ denote the stable-pairs partition function, and let … be the weighted generating function of labelled box configurations, with boundary partitions $(,,)$. Pandharip…

Let $\mu^1,\mu^2,\mu^3$ be partitions, and let $W_{\mu^1,\mu^2,\mu^3}(q)$ denote the topological vertex. For $i,j\in\{1,2,3\}$ and $m,n\geq 0$, let $A_{mn}^{ij}(q)$ be the explicit…

Let $G^{\bullet}(\lambda;x,y,-x-y;p^+,p^-)$ be the exponential generating function built from the two-partition Hodge integrals $G_{\mu^+,\mu^-}$, and let…

For partitions $[2001~\lambda_1$, $[2001~\lambda_2$, and $[2001~\lambda_3$, let $\tilde{\mathcal{W}}_{\lambda_1,\lambda_2,\lambda_3}(q)$ be the rational function in $q^{1/2}$ used…

Let $\mu^1$ and $\mu^2$ be partitions, let $Q$ and $q$ be formal variables, and define … Write … Here $\mathcal W_{\mu^1}$ and $\mathcal W_{\mu^2}$ denote the corresponding one-leg…

All-orders trivalent-vertex conjecture. The two generating functionals agree to all orders. The claim is motivated by agreement of the expansions checked in the paper, including se…

Fractional-framing conjecture. The localization generating functional and the topological vertex agree to all orders when the topological vertex is evaluated at the corresponding f…

Membrane-index conjecture. The series of equivariant Gromov–Witten invariants equals the $\hat{A}$-genus of the mathematically yet-to-be-constructed moduli space of M2-branes on…

Let the topological vertex be regarded in the fermionic picture, and let the fermionic vacuum vector belong to the $3$-component fermionic Fock space. A Bogoliubov transform is the…

Let $U$ be an unknot in $\mathbb{R}P^3$ colored by an $n$-dimensional skew-symmetric or symmetric representation $R$ of $su(2)$. Let $Z_{R\emptyset}(U;Q,q)$ be its invariant, and l…

Let $\boldsymbol\mu=\{\mu_a\}_{a=1}^{4}$ be plane partitions with no PT$_0$ moduli, and let $\mathsf V^{\mathrm{DT}}_{\mu_1\mu_2\mu_3\mu_4}(q)$ and…

Fix plane partitions $\boldsymbol{\mu}=\{\mu_a\}_{a=1}^{4}$ corresponding to $(\mathbb C^*)^3$-fixed subschemes of $\mathbb C^3$ of dimension at most one, viewed as asymptotic cros…

Framed ADKMV conjecture. The framed topological vertex is given by

Termination conjecture. The expansion of $\widetilde{Z}_\nu(P,Q)$ by $Q$ terminates at the order $Q^{|\nu|}$. This finite-order behavior has been observed for various choices of…

Termination conjecture. The expansion by $Q$ of $Z_\nu(P,Q)$ terminates at the order $Q^{|\nu|}$ for any $\nu$. The observed finite termination removes the apparent divergences in…

Let $V_{,,}$ be the DT topological vertex and let $W_{,,}$ be the weighted labelled-box vertex. Let … be the plane-partition generating function. Pandharipande–Thomas topological v…

The topological vertex is a combinatorial object used to compute all-genus A-model topological string amplitudes on smooth non-compact toric Calabi–Yau threefolds. Li, Liu, Liu and…

Let $\lambda$, $\nu$, and $\mu$ be representations, let $\lambda^t$, $\nu^t$, and $\mu^t$ denote their transposes, and let $\langle\lambda,\nu,\mu\rangle$ denote the topological ve…

The topological vertex is denoted by $\langle \lambda,\nu,\mu\rangle$, and Kodaira–Spencer theory refers to the deformation theory associated with the topological vertex. Fermionic…

Consider the topological vertex generating functions in open Gromov–Witten theory, together with the framed mirror curve to $\mathbb{C}^3$. Remodeling conjecture. The topological v…

Let $V^{(a_1,a_2,a_3)}_0\in \mathcal{F}^{(0)}_1\otimes\mathcal{F}^{(0)}_2\otimes\mathcal{F}^{(0)}_3$ be the charge-zero vector corresponding under the boson–fermion correspondence…