The Geometry Underlying Mirror Symmetry
Abstract
The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of Tduality is reinterpreted as providing a way of geometrically characterizing which CalabiYau manifolds have mirror partners. The geometric descriptionthat one CalabiYau manifold should serve as a compactified, complexified moduli space for special Lagrangian tori on the other CalabiYau manifoldis 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' CalabiYau 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.
 Publication:

arXiv eprints
 Pub Date:
 August 1996
 arXiv:
 arXiv:alggeom/9608006
 Bibcode:
 1996alg.geom..8006M
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory
 EPrint:
 26 pages, AmSLaTeX. Final version, to appear in Proc. European Algebraic Geometry Conference (Warwick, 1996)