Geometry of Mirror Symmetry

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AbstractAward: DMS-0405859Principal Investigator: Eric ZaslowThis project addresses geometric aspects of mirror symmetry. ThePI will study the geometry of the Strominger-Yau-Zaslowconjecture by looking at the conjectural picture of a Calabi-Yauthreefold near the large complex structure limit, where thespecial Lagrangian torus fibration collapses. The resultingmanifold is a real integer affine manifold with singularities.The first project will produce a real Monge-Ampere metric on anopen ball outside a Y-shaped singular locus, then study itsmonodromy. A knowledge of this region is essential to gaining aglobal understanding of the Calabi-Yau, including the role ofdisc instanton corrections. Related projects are to study thegeometry of flat or Ricci-flat torus fibrations in Ricci-flatmanifolds in general, and to study the counting problem ofspecial Lagrangian and other calibrated submanifolds by varyingthe metric away from the Ricci flat one. The final project isinvestigate integrable hierarchies underlying the topologicalvertex.The project studies the mathematics behind the physicalphenomenon of mirror symmetry. Mirror symmetry is when twodifferent physical theories give rise to the same predictionsabout nature. Mathematically, there are deep connections behindthe equivalence, and this project looks at the geometry of theequivalence. Understanding the geometry of mirror symmetry willlead to a broader comprehension of the structure of mathematics,and conceivably physics. For instance, mirror symmetry aids inthe calculation of topological field theories, which have beenshown to have computational relevance to the conventional gaugetheories that underly particle physics. At a more basic,mathematical level, understanding the geometry of mirror symmetrycan lead to a concrete description of the special structuresconjectured by Calabi to exist in 1957 and proved to exist by Yauin 1979. Still, we know very little about this important aspectof geometry. The rich interplay of mathematics and physics hasmuch to teach us.