Partially Wrapped Fukaya Categories and Functoriality in Mirror Symmetry

Harvard University, Cambridge MA

Investigators

Abstract

In modern geometry it is often useful to think of spaces in non-local terms rather than relying on classical concepts such as position. This is especially true of symplectic geometry -- the geometry of phase spaces of classical mechanics, where points are "too small" to be relevant, and it is more natural to consider objects called Lagrangian submanifolds. These objects and their interactions are encoded by an algebraic structure (or "non-commutative space") called the Fukaya category. A remarkable mathematical conjecture inspired by ideas from theoretical physics, "homological mirror symmetry", asserts that Fukaya categories are in fact often equivalent to honest (commutative) spaces of the sort studied in algebraic geometry. This research project studies versions of Fukaya categories for symplectic manifolds equipped with one or more (commuting) functions and/or relative to certain directions at infinity. The rich structure of these categories, coming from the additional data, yields new ways of understanding the effect of various geometric constructions on the Fukaya category of a symplectic manifold; this in turn should greatly extend the range of settings in which homological mirror symmetry can be verified. The project will also provide research opportunities for several graduate students and generally aim to make this research area more accessible to the broader mathematical community. From a technical standpoint, the first goal of this project is to develop a better geometric setup for partially wrapped Fukaya categories as they arise in mirror symmetry, to arrive at a formulation where computations are possible and the expected structural features are manifest. The other main goal is to use these foundations to give a general proof of homological mirror symmetry for (not necessarily Calabi-Yau) complete intersections in toric varieties, and to study canonical bases of their coordinate rings. Finally, this project will also bring conceptual clarity to the field by unifying different proposed constructions of partially wrapped Fukaya categories, finding new instances of functoriality in mirror symmetry, and studying the interplay between geometric constructions and categorical ones. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.