Some speculations on pairs-of-pants decompositions and Fukaya categories

Abstract

This is partly a survey and partly a speculative article, concerning a particular question about Fukaya categories of symplectic manifolds. Namely, can we decompose a symplectic manifold into standard pieces, and then reconstruct its Fukaya category by gluing together categories depending only on the geometry of each piece, in a (loosely understood) sheaf-theoretic way? For this to work, some degree of control over pseudo-holomorphic curves is required, an issue which depends on the geometry of the decomposition under consideration. Sheaf-theoretic ideas have been successfully applied to the symplectic geometry of cotangent bundles, starting with the work of Fukaya-Oh [17], and followed by Kasturirangan-Oh [26, 39] and Nadler-Zaslow [35,36] (for a survey of the last-mentioned work and related ideas of Fukaya-Smith, see [19]). Recently, Kontsevich [28] has proposed a generalization to Stein manifolds whose Lagrangian skeleta have certain singularities. However, that is not quite the direction we wish to take here.