Analytic torsion of certain Calabi-Yau threefolds

Physicists Bershadsky-Cecotti-Ooguri-Vafa introduced a certain combination of analytic torsions as a counter part in B-model of elliptic Gromov-Witten invariants of Calabi-Yau threefolds. For Borcea-Voisin threefolds without mirrors, we give an expression of  the BCOV torsion as a nice Borcherds product on the Kaehler moduli of a Del Pezzo surface.