Quantum Correction and the Moduli Spaces of Calabi-Yau Manifolds

We define the quantum correction of the Teichm"uller space $\mathcal{T}$ of Calabi-Yau manifolds. Under the assumption of no weak quantum correction, we prove that the Teichm"uller space $\mathcal{T}$ is a locally symmetric space with the Weil-Petersson metric. For Calabi-Yau threefolds, we show that no strong quantum correction is equivalent to that, with the Hodge metric, the image $\mathrm{\Phi }(\mathcal{T})$ of the Teichm"uller space $\mathcal{T}$ under the period map $\mathrm{\Phi }$ is an open submanifold of a globally Hermitian symmetric space $W$ of the same dimension as $\mathcal{T}$. Finally, for Hyperk"ahler manifold of dimension $2n\ge 4$, we find both locally and globally defined families of $(2,0)$ and $(2n,0)$-classes over the Teichm"uller space of polarized Hyperk"ahler manifolds.