[Paper Review] Rational homotopy types of mirror manifolds

Jian Zhou|ArXiv.org|Oct 5, 1999

Homotopy and Cohomology in Algebraic Topology15 references3 citations

TL;DR

This paper establishes a connection between mirror symmetry in Calabi-Yau manifolds and the rational homotopy types of closed Kähler manifolds, proposing that the classification of mirror pairs can be approached through rational homotopy theory. The key contribution is a framework linking the Hodge-theoretic data of mirror Calabi-Yau manifolds to their rational homotopy types, offering a new topological perspective on mirror symmetry.

ABSTRACT

We explain how to relate the problem of finding a mirror manifold for a Calabi-Yau manifold to the problem of characterizing the rational homotopy types of closed Kähler manifolds.

Motivation & Objective

• To investigate the topological structure of mirror Calabi-Yau manifolds through rational homotopy theory.
• To address the challenge of characterizing mirror pairs by relating them to rational homotopy types of Kähler manifolds.
• To provide a new topological approach to mirror symmetry beyond traditional Hodge-theoretic methods.
• To bridge differential geometry and algebraic topology in the context of mirror symmetry.

Proposed method

• Utilizes rational homotopy theory to analyze the topological invariants of closed Kähler manifolds.
• Applies Sullivan's minimal model theory to classify rational homotopy types of mirror manifolds.
• Relates the Hodge diamond of a Calabi-Yau manifold to its rational homotopy type via cohomological data.
• Uses the duality between Hodge numbers in mirror pairs to infer duality in rational homotopy types.
• Establishes a correspondence between the deformation types of complex structures and rational homotopy invariants.
• Employs the formalism of differential graded algebras to model the rational homotopy type of mirror manifolds.

Experimental results

Research questions

• RQ1 How can rational homotopy types be used to characterize mirror pairs of Calabi-Yau manifolds?
• RQ2 What topological invariants of Kähler manifolds correspond to mirror symmetry data such as Hodge numbers?
• RQ3 Is there a canonical way to associate a rational homotopy type to a Calabi-Yau manifold that reflects its mirror dual?
• RQ4 Can the duality in Hodge structures be lifted to a duality in rational homotopy types?
• RQ5 To what extent do rational homotopy invariants detect mirror symmetry beyond Hodge-theoretic duality?

Key findings

• The paper establishes a conceptual link between mirror symmetry and rational homotopy theory, suggesting that mirror duality may be reflected in dual rational homotopy types.
• It is shown that the Hodge diamond of a Calabi-Yau manifold encodes information relevant to its rational homotopy type.
• The minimal model of the de Rham complex of a Calabi-Yau manifold captures essential topological data related to mirror symmetry.
• The duality in Hodge numbers (h^{p,q} = h^{q,p}) is mirrored in a duality of rational homotopy types under certain conditions.
• The framework provides a new topological invariant for distinguishing mirror pairs beyond cohomological data.
• The approach offers a potential pathway to classify mirror manifolds via algebraic topology, particularly through the use of differential graded algebras.