The Geometry Underlying Mirror Symmetry

TL;DR

This paper explores a geometric interpretation of mirror symmetry for Calabi-Yau manifolds, linking it to T-duality and proposing conjectures on moduli space compactification and mirror pair definitions.

Contribution

It introduces a new geometric framework for understanding mirror symmetry, independent of existing conjectures, and investigates its implications for Calabi-Yau moduli spaces and homology relations.

Findings

Proposes a geometric characterization of mirror pairs.

Formulates conjectures on moduli space compactification.

Analyzes the case of K3 surfaces in detail.

Abstract

The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners. The geometric description---that one Calabi-Yau manifold should serve as a compactified, complexified moduli space for special Lagrangian tori on the other Calabi-Yau manifold---is rather surprising. We formulate some precise mathematical conjectures concerning how these moduli spaces are to be compactified and complexified, as well as a definition of geometric mirror pairs (in arbitrary dimension) which is independent of those conjectures. We investigate how this new geometric description ought to be related to the mathematical statements which have previously been extracted from mirror symmetry. In particular, we discuss how the moduli spaces of the `mirror' Calabi-Yau manifolds should be related to one another, and how appropriate subspaces of the homology groups of those manifolds could be related. We treat the case of K3 surfaces in some detail.