On the quantum differential equations for a family of non-Kähler monotone symplectic manifolds

In this paper we prove Gamma Conjecture $1$ for twistor bundles of hyperbolic $6$ manifolds, which are monotone symplectic manifolds which admit no Kähler structure. The proof involves a direct computation of the $J$-function, and a version of Laplace's method for estimating power series (as opposed to integrals). This method allows us to rephrase Gamma Conjecture $1$ in certain situations to an Apéry-like discrete limit. We use this to give a simple proof of Gamma Conjecture $1$ for projective spaces. Additionally we show that the quantum connections of the twistor bundles we consider have unramified exponential type.