The integral mirror-map conjecture for toric Calabi–Yau complete intersections

Under the hypotheses for the toric Calabi–Yau complete intersection above, let $z_1,\ldots,z_t$ be $\nabla_{AP}$-flat coordinates on $\tilde H^2$. For logarithmic solutions $\Phi_i(z)$ of ${\cal D}$ satisfying

$$\Phi_i(z)=(\log z_i)\Phi_0(z)+\Psi_i(z),\qquad \Psi_i(0)=0,$$

with each $\Psi_i$ regular at $z=0$, define

$$q_i=\exp\!\left(\frac{\Phi_i(z)}{\Phi_0(z)}\right),\qquad i=1,\ldots,t.$$

Integral mirror-map conjecture. These are the $\nabla_{AV}$-flat coordinates on $\tilde H^2$, and every coefficient in their expansions as power series in the $\nabla_{AP}$-flat coordinates $z_i$ is an integer.

The conjecture gives a canonical normalization of logarithmic solutions and generalizes the integral mirror maps observed in known complete-intersection examples.