The toric orbifold mirror theorem

Let $\mathcal{X}$ be a toric orbifold with extended Kähler cone $\widetilde{C}_\mathcal{X}$, and let $\hat{\rho}$ denote the relevant anticanonical parameter. Let $I(q,z)$ be the $I$-function, let $J(\tau,z)$ be the $J$-function, and let $\tau(q)$ be the mirror map defined by the asymptotics of $I(q,z)$. Assume

$$\hat{\rho}\in\operatorname{cl}(\widetilde{C}_\mathcal{X}).$$

The toric orbifold mirror theorem. The $I$-function and the $J$-function coincide under the coordinate change $\tau=\tau(q)$:

$$I(q,z)=J(\tau(q),z).$$

The statement is presented as a mirror theorem to be proved in subsequent work, so it identifies the hypergeometric $I$-function with the genus-zero Gromov–Witten $J$-function in the stated convergence region.