The Geometry Underlying Mirror Symmetry
David R. Morrison
TL;DR
The paper addresses the geometric underpinnings of mirror symmetry by proposing that a Calabi–Yau manifold $X$ should be the compactified, complexified moduli space of special Lagrangian tori on its mirror $Y$, with T-duality supplying the dual torus fibration. It develops a framework for geometric mirror pairs that integrates moduli of special Lagrangian submanifolds, D-brane data, and compactifications, and links this to quantum moduli spaces and variations of Hodge structure through topological and Hodge-theoretic mirror tests. It introduces a geometric definition of mirror pairs via correspondences $Z_t$ and dual torus fibrations and explores how this translates into the classical and quantum data of the conformal field theories. The K3 surface case is treated in detail using Mukai theory and Fourier–Mukai transforms, illustrating how special Lagrangian fibrations, moduli of simple sheaves, and derived-equivalence frameworks realize geometric mirror symmetry in a concrete setting with full mathematical structure.
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.