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 \( \Phi(\mathcal{T})\) of the Teichm\"uller space \(\mathcal{T}\) under the period map \(\Phi\) 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 \(2n \geq 4\), 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.