7 problems

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 $\mathcal{Z}_{\mathrm{Nek}}$ denote a Nekrasov partition function with generic equivariant parameters $\epsilon_1,\epsilon_2$, and let its Fourier series be formed by summing o…

Let $\mathcal{Z}_{\mathrm{inst}}(i,i,i,iq^{\pm1/2},u;q^{-1},q|z^{1/2})$ denote the five-dimensional Nekrasov instanton partition function with the displayed special values of $v_i$…

Let $u=q^{2\sigma}$ and let $\tau$ and $\uptau$ denote the $c=1$ and $c=-2$ Painlevé VI tau functions, respectively. Let $\td{u}=\td{u}(u)$ and $\td{s}=\td{s}(s,\sigma)$ be functio…

Let $\mathcal{Z}_{tv}(\boldsymbol{\theta},\sigma|t)$ denote the normalized five-dimensional Nekrasov partition function with four matter fields appearing in the $q$-Painlevé VI tau…

Let $Z_{\mathrm{inst}}$ denote the Nekrasov instanton partition function, and let a tau-function of the Toda lattice hierarchy be a function satisfying the hierarchy's tau-function…

Scalar-product conjecture. In the appropriate identification of the parameters,