The prepotential reconstruction conjecture for noncompact toric Calabi–Yau threefolds

Let $X$ be a noncompact toric Calabi–Yau threefold with $\dim H_4(X,\mathbb{Z})=0$, and let $\{l^1,\dots,l^m\}$ define $X$ via symplectic quotient. Define

$$\omega=\sum_{n>0}c(n,\rho)z^{n+\rho},$$

where

$$\Pi_{ij}^{\mathrm{inst}}=\sum_n\left(\partial_{\rho_i}\partial_{\rho_j}c(n,\rho)\right)\big|_{\rho=0}z^n$$

and

$$c(n,\rho)^{-1}=\prod_i\Gamma\left(1+\sum_k l_i^k(n_k+\rho_k)\right).$$

If $\mathcal F$ is the prepotential, and $s_i=\partial_{\rho_i}\omega\big|_{\rho=0}$ for each $i$ such that

$$l^i=\begin{pmatrix}1&1&-1&-1&0&\dots&0\end{pmatrix}$$

up to a permutation of the columns of $l^i$, prepotential reconstruction conjecture. There are rational numbers $m_{ij}\in\mathbb Q$ such that

$$\sum_{i,j}m_{ij}\Pi_{ij}^{\mathrm{inst}}=\sum_i(-1)^{i-1}\frac{\partial\mathcal F^{\mathrm{inst}}}{\partial s_i}.$$

The claim expresses the principle, supported by the examples discussed in the paper, that suitable combinations of instanton Picard–Fuchs solutions recover derivatives of the instanton prepotential for this class of local Calabi–Yau geometries.