Lagrangian Floer theory, from geometry to algebra and back again
Denis Auroux
TL;DR
The work surveys Floer theory from its origins in Arnold’s conjectures to its modern algebraic formulation in the Fukaya category and beyond, clarifying how Hamiltonian and Lagrangian Floer theories underpin fixed-point and intersection results and connect to quantum cohomology via the PSS isomorphism. It emphasizes the $A_ abla_ ext{infty}$-structure of the Fukaya category, the role of bounding cochains in obstructed settings, and how open-closed maps link symplectic and Hochschild invariants, underpinning homological mirror symmetry and the deformation theory of Fukaya categories. The text then develops a geometric, local-to-global perspective, detailing how family Floer theory, microlocal sheaves, and SYZ mirror symmetry recast Floer-theoretic data as sheaves on a base or on a mirror, with sectorial decompositions and relative invariants providing robust decomposition tools. Overall, the article highlights a unifying view in which Floer theory serves as a bridge between symplectic geometry, noncommutative geometry, and algebraic geometry, enabling concrete computations and insights across mirror symmetry, deformation theory, and low-dimensional topology.
Abstract
We survey various aspects of Floer theory and its place in modern symplectic geometry, from its introduction to address classical conjectures of Arnold about Hamiltonian diffeomorphisms and Lagrangian submanifolds, to the rich algebraic structures captured by the Fukaya category, and finally to the idea, motivated by mirror symmetry, of a "geometry of Floer theory" centered around family Floer cohomology and local-to-global principles for Fukaya categories.