Abstract: We study spectral invariants of torsion type on Calabi-Yau manifolds of dimension three. We define a new holomorphic invariant modifying the so-called BCOV torsion and explore its appications in the study of moduli space of Calabi-Yau manifolds. We determine the asymptotic behavior of the new invariant near a singular Calabi-Yau variety with only one ODP singularity, which leads to a current curvature formula. As an application, we verify a conjecture of Beshadsky-Ceccotti-Ooguri-Vafa. This is recent joint work with H. Fang and K.-I. Yoshikawa.