Analytic torsion for Borcea-Voisin threefolds

In their study of genus-one string amplitude, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable identification between holomorphic Ray-Singer torsion and instanton numbers for Calabi-Yau threefolds. The holomorphic torsion invariant for Calabi-Yau threefolds corresponding to the genus-one string amplitude is called BCOV invariant. In this paper, we establish an identification between the BCOV invariants of Borcea-Voisin threefolds and another holomorphic torsion invariants for K3 surfaces with involution. We also introduce BCOV invariants for abelian Calabi-Yau orbifolds. Between Borcea-Voisin orbifold and its crepant resolution, we compare their BCOV invariants.