Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 018, 47 pages      arXiv:1109.4848      https://doi.org/10.3842/SIGMA.2014.018
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

Fukaya Categories as Categorical Morse Homology

David Nadler
Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720-3840, USA

Received May 16, 2012, in final form February 21, 2014; Published online March 01, 2014

Abstract
The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to that for constructible sheaves. Assuming this structure, we exhibit the Fukaya category as the global sections of a sheaf on the conic topology of the Weinstein manifold. This can be viewed as a symplectic analogue of the well-known algebraic and topological theories of (micro)localization.

Key words: Fukaya category; microlocalization.

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