8 problems

For integers $n,g\geq 0$ and positive degree $d_E>0$, let $\mathcal{F}^{(n,g)}_{d_E}$ denote the refined free energy in degree $d_E$, and let the conifold and orbifold points be th…

For fixed $N\in\mathbb{Z}_{>0}$, let $\mathfrak{B}_{\phi_N}=\{\Phi_{\sigma,n;N}(\hbar)\}_{\sigma=0,\dots,l_0;\,n\in\mathbb{Z}}$ be the minimal resurgent structure, with Stokes cons…

For the local $\mathbb{P}^2$, let $\widetilde{n}_d^{i,j}$ denote the combinatorial refined BPS invariants obtained from the Nekrasov partition function, and let $n_d^{i,j}$ denote…

Let $t_{\mathrm{con}}$ be the flat coordinate near the conifold point, and let $F_{g,\mathrm{con}}^{\mathbb{P}^2\!/\!E}$ denote the genus-$g$ free energy expressed as a function of…

Let $P_{\operatorname{con}}$ be the conifold polarization, and let $F^g_{\operatorname{con}}$ denote the genus-$g$ correlation function with respect to this polarization. The sourc…

Let $Y$ be the local resolution and let $\mathcal{X}$ be the corresponding orbifold, with genus-zero free energies $F_Y^0$ and $F_{\mathcal{X}}^0$. Their differentials define graph…

Discrete-spectrum and trace-class property. The operator $\mathsf{O}_{\mathbb{P}^2}$ has a discrete spectrum, and its inverse $\rho$ is trace class.

Coefficient conjecture. The invariants satisfy