In this talk, a compact connected Kahler manifold of even dimension is called simple Enriques if it is not simply connected and its universal covering is either Calabi-Yau or hyperkahler. These manifolds were introduced and studied independently by
Boissiere-Nieper-Weisskirchen-Sarti and Oguiso-Schroer. We introduce a holomorphic torsion invariant of simple Enriques 2n-folds and study the corresponding function on the moduli space of such manifolds. We report its basic properties such as the strong plurisubharmonicity and the automorphy, as well as possible (conjectural) applications. If time allows, we will also report the explicit formula for the invariant as an automorphic function on the moduli space in some cases.