CAREER: Fukaya categories, mirror symmetry, and low-dimensional topology

University Of Texas At Austin, Austin TX

Investigators

Abstract

Around 1989, a number of mathematical physicists, studying hypothetical string-theoretical models of the universe, discovered that certain such models come in pairs, with versions "A" and "B" of the theory based on different equations yet sharing the same physically observable quantities. This phenomenon, which acquired the metaphorical name "mirror symmetry," led to extraordinarily prescient mathematical predictions about the geometries ("Calabi-Yau manifolds") involved in such theories. In 1994, Maxim Kontsevich proposed an organizing framework for multiple mathematical aspects of mirror symmetry in his "homological mirror symmetry" (HMS) conjecture, which, he said, would "unveil the mystery of mirror symmetry." At the time, one of the two main mathematical notions invoked by HMS, the "Fukaya category of a symplectic manifold," was new and undeveloped, and it was very difficult both to prove instances of Kontsevich's conjecture and to deduce consequences of it. Recent developments in the mathematics of Fukaya categories have changed that. At the center of this CAREER project is a program to realize Kontsevich's vision. The PI and collaborators have formulated a notion of "core homological mirror symmetry," and the program aims to show that if core HMS holds for a particular mirror pair then many other facets of mirror symmetry naturally follow as logical consequences. The project includes support for graduate students working with the PI, which will assist them at the beginning of their research careers, and support for the development of a coherent honors program for mathematics undergraduates. The main strand of the research program concerns mirror symmetry, primarily for Calabi-Yau (CY) manifolds. The program (underway in work of the PI and collaborators Sheridan and Ganatra) studies the consequences of "core homological mirror symmetry" for a given pair of polarized CY manifolds, which roughly says that one can match the tensor powers of the polarizing line bundle over one of these two with some collection of Lagrangian submanifolds of the other in a way that respects categorical structures. This hypothesis is quite natural in light of geometric approaches to mirror pairs based on the Strominger-Yau-Zaslow philosophy, though currently only proven in a few cases. We aim to show that core HMS implies full HMS and several aspects of "closed-string" mirror symmetry, notably the normalization of the holomorphic volume form and the precise form of the mirror map. This has become feasible in light of recents developments, due principally to Abouzaid, relating the geometric "open-closed string map," from quantum cohomology to Hochschild homology of the Fukaya category, to the categorical property of "homological smoothness" of the Fukaya category. A key aim is to give a "conceptual" proof that the generating function for the numbers of rational curves on a quintic 3-fold equals the Yukawa coupling of its mirror -- a proof that does not rely on calculating the two sides. This research gives key roles to symplectic topology, pseudo-holomorphic curves, and symplectic Floer cohomology. Those techniques are equally important in a second strand, which is to develop Floer-thoeretic 3-manifold invariants on similar lines to Heegaard Floer theory but based not on symmetric products of a Heegaard surface (viewed as a complex curve), but rather on moduli spaces of rank 2 holomorphic bundles with section over such curves. Such invariants may serve to mediate between Heegaard and instanton versions of Floer theory for 3-manifolds.