Principal Investigator: Eric Zaslow
Mirror symmetry in string theory has, quite famously, linked disparate fields of mathematics. The central theorem in the subject is Kontsevich's homological mirror symmetry (HMS) conjecture. The principal investigator proposes to prove this conjecture using his recent work with collaborators. Connections between mathematical physics and topology, representation theory, and combinatorics have been revealed through the PI's research with Nadler and with Fang, Liu and Treumann. What emerges is a language for studying mirror symmetry and other advances in mathematical physics from a simple, geometric viewpoint which easily lends itself to computation. This perspective renders some formidable hurdles of HMS rather tractable. The PI will, with collaborators and graduate students, aim to prove HMS in several stages. First, by defining a category, the Constructible Plumbing Model (CPM), which models the Fukaya category of a Calabi-Yau manifold at its large radius limit by defining a formal Lagrangian skeleton and gluing together categories of constructible sheaves made from pieces of the skeleton. Second, by proving that CPM is equivalent to the category of perfect complexes on the mirror Calabi-Yau at its large complex limit point. After these steps, a deformation-of-categories argument can be made to establish a mirror map and prove HMS.
The aim of string theory is to merge the two pillars of modern physics: Einstein's theory of gravity and the quantum theory of particles. Models of the universe from string theory rely on a class of geometric spaces called Calabi-Yau manifolds. Calculations in these models are quite formidable, but are often made tractable through the phenomenon of mirror symmetry. The idea of mirror symmetry is that one theory can look totally different from another theory, but the two lead to the same predictions. Hard calculations using one Calabi-Yau manifold can become easy calculations in the completely different "mirror" Calabi-Yau manifold. But to be truly useful, one must have complete confidence in the equivalence, namely that the calculations in the mirror theory can be trusted. This requires a rigorous mathematical formulation of the model, a rigorous statement of how to apply the equivalence, and a rigorous proof that the equivalence is, in fact, true. The statements in mirror symmetry have been made rigorous by Fields Medal laureate Maxim Kontsevich. What is still lacking is a general proof of Kontsevich's conjecture. The principal investigator proposes to prove this conjecture in several steps, using a simple geometric model which easily lends itself to calculations. The model can also serve as a framework for exploring other predications and phenomena of modern theoretical physics.