Ray Singer Analytic Torsion of CY Manifolds II

In this paper we construct the analogue of Dedekind eta function for odd dimensional CY manifolds. We use the theory of determinant line bundles. We constructed a canonical holomorphic section $\eta^{N}$ of some power of the determinant line bundle on the moduli space of odd dimensional CY manifolds. According to Viehweg the moduli space of moduli space of polarized odd dimensional CY manifolds $\mathcal{M}(M)$ is quasi projective. According to a Theorem due to Hironaka we can find a projective smooth variety $\bar {\mathcal{M}(M)}$ such that $\bar{\mathcal{M}(M)}\backslash$ $\mathcal{M}(M)=\mathcal{D}_{\infty}$ is a divisor of normal crossings. We also showed by using Mumford's theory of metrics with logarithmic growths that the determinant line bundle can be canonically prolonged to $\bar {\mathcal{M}(M)}$ $.$ We also showed that there exists section $\eta$ of some power of the determinant line bundle which vanishes on $\mathcal{D}_{\infty}$ and has a Quillen norm the Ray Singer Analytic Torsion$.$ This section is that analogue of the Dedekind eta Function $\eta.$