Homological Mirror Symmetry for Calabi-Yau Hypersurfaces

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AbstractAward: DMS-1104779Principal Investigator: Eric ZaslowMirror symmetry in string theory has, quite famously, linkeddisparate fields of mathematics. The central theorem in thesubject is Kontsevich's homological mirror symmetry (HMS)conjecture. The principal investigator proposes to prove thisconjecture using his recent work with collaborators. Connectionsbetween mathematical physics and topology, representation theory,and combinatorics have been revealed through the PI's researchwith Nadler and with Fang, Liu and Treumann. What emerges is alanguage for studying mirror symmetry and other advances inmathematical physics from a simple, geometric viewpoint whicheasily lends itself to computation. This perspective renderssome formidable hurdles of HMS rather tractable. The PI will,with collaborators and graduate students, aim to prove HMS inseveral stages. First, by defining a category, the ConstructiblePlumbing Model (CPM), which models the Fukaya category of aCalabi-Yau manifold at its large radius limit by defining aformal Lagrangian skeleton and gluing together categories ofconstructible sheaves made from pieces of the skeleton. Second,by proving that CPM is equivalent to the category of perfectcomplexes on the mirror Calabi-Yau at its large complex limitpoint. After these steps, a deformation-of-categories argumentcan be made to establish a mirror map and prove HMS.The aim of string theory is to merge the two pillars of modernphysics: Einstein's theory of gravity and the quantum theory ofparticles. Models of the universe from string theory rely on aclass of geometric spaces called Calabi-Yau manifolds.Calculations in these models are quite formidable, but are oftenmade tractable through the phenomenon of mirror symmetry. Theidea of mirror symmetry is that one theory can look totallydifferent from another theory, but the two lead to the samepredictions. Hard calculations using one Calabi-Yau manifold canbecome easy calculations in the completely different "mirror"Calabi-Yau manifold. But to be truly useful, one must havecomplete confidence in the equivalence, namely that thecalculations in the mirror theory can be trusted. This requiresa rigorous mathematical formulation of the model, a rigorousstatement of how to apply the equivalence, and a rigorous proofthat the equivalence is, in fact, true. The statements in mirrorsymmetry have been made rigorous by Fields Medal laureate MaximKontsevich. What is still lacking is a general proof ofKontsevich's conjecture. The principal investigator proposes toprove this conjecture in several steps, using a simple geometricmodel which easily lends itself to calculations. The model canalso serve as a framework for exploring other predications andphenomena of modern theoretical physics.