Quantum Correction and the Moduli Spaces of Calabi-Yau Manifolds

TL;DR

This paper explores the geometric structure of the Teichmüller space of Calabi-Yau manifolds, showing how quantum corrections influence its symmetry properties and the behavior of period maps, with implications for hyperkähler manifolds.

Contribution

It introduces the concept of quantum correction in the Teichmüller space of Calabi-Yau manifolds and establishes conditions under which this space is locally symmetric or maps into Hermitian symmetric spaces.

Findings

Teichmüller space is locally symmetric under no weak quantum correction.

The image of the Teichmüller space under the period map is an open submanifold of a Hermitian symmetric space for Calabi-Yau threefolds.

Families of special classes are constructed over the Teichmüller space of Hyperkähler manifolds.

Abstract

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.