The integral asymptotic expansion conjecture for irreducible Calabi–Yau Picard–Fuchs systems

Let $\mathcal{P}$ be a Picard–Fuchs operator of a family of Calabi–Yau threefolds with a MUM point at $0$, and suppose that its associated local system is irreducible. For a loop $\gamma\in\pi_1(\mathbb{P}^1\setminus\mathcal{S},t_0)$, write $N_\gamma:=M_\gamma-\operatorname{Id}$, where $M_\gamma$ is the corresponding monodromy matrix, and let $d_\gamma$, $a_\gamma$, and $p_\gamma$ be integers. Integral asymptotic expansion conjecture. There exists $\gamma\in\pi_1(\mathbb{P}^1\setminus\mathcal{S},t_0)$ with $d_\gamma\neq0$ such that

$$N_\gamma(-v)=\frac{d_\gamma}{6(2\pi i)^{3}}\log^{3}(t)+\frac{p_\gamma}{48\pi i}\log(t)+\frac{a_\gamma}{(2\pi i)^3}\zeta(3)+o(1),$$

where $d_\gamma,a_\gamma,p_\gamma\in\mathbb{Z}$. This is motivated by mirror symmetry and is a stronger form of the expected arithmetic constraints on the transition matrix between generalized Doran–Morgan and normalized Frobenius bases. The source does not provide evidence of a resolution, so the conjecture remains open.