Topological Mirror Symmetry

Abstract

This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special Lagrangian condition. In this setting, natural questions include: does there exist a nice class of T^3-fibrations for which the dual fibration can be constructed? Do such fibrations exist on a manifold such as the quintic threefold? If so, is the dual fibration the mirror? We answer these questions affirmatively. We introduce a class of topological T^3-fibrations for which duals can be constructed, including over the singular fibres. We then construct such a fibration on the quintic threefold, and show that by applying this general dualizing construction to this particular case, one obtains the mirror quintic. Thus we have constructed the mirror quintic topologically with no a priori knowledge of the mirror. This shows that in a non-degenerate (and representative) case, the Strominger-Yau-Zaslow conjecture correctly explains mirror symmetry.