Integral Affine Structures and Calabi-Yau Metric Degenerations

Harvard University, Cambridge MA

Investigators

Abstract

Abstract Award: DMS-0405939 Principal Investigator: Ilia Zharkov The proposed research lies in the subject of mirror symmetry, of which there are several mathematical interpretations. The investigator will address a possible link between two of them -- the homological mirror symmetry of Kontsevich and the duality of Strominger-Yau-Zaslow torus fibrations -- by studying the limiting metric behavior of maximally degenerating families of Calabi-Yau manifolds. The main objective of this research is to prove the metric collapse conjecture for the toric Calabi-Yau families. The key ingredient here is to understand the behavior at the discriminant. Another direction is to study the Kaehler affine structures on the limiting space. Finally, the tropical geometry approach is proposed for the limiting counting of holomorphic curves and understanding the degeneration of the Fukaya side of the homological mirror symmetry. In recent years there has been a remarkable renaissance in the interaction between mathematics and physics. After a long period during which mathematicians and physicists pursued seemingly independent paths, their interests have now converged in a striking manner. Deep connections have been established between quantum field theory and string theory on one side and topology, algebraic and differential geometry on the other. Mirror symmetry is considered by many to be the mostly responsible for bridging the gap between the two fields created in previous decades. On a broader scale, as physicists nowadays believe that a Calabi-Yau 3-fold together with 3+1 Minkowski space constitute the space-time of 10-dimensional string theory, a better understanding of these manifolds will help to uncover mysteries of the Universe.