Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

Abstract

We prove that the wrapped Fukaya category of any 2ndimensional Weinstein manifold (or, more generally, Weinstein sector) W is generated by the unstable manifolds of the index n critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer homology and the fairly simple observation that Floer homology vanishes for Lagrangian submanifolds that can be disjoined from the isotropic skeleton of the Weinstein manifold. Note that we do not need any additional assumptions on this skeleton. By applying our generation result to the diagonal in the product W × W , we obtain as a corollary that the open-closed map from the Hochschild homology of the wrapped Fukaya category of W to its symplectic cohomology is an isomorphism, proving a conjecture of Seidel. We work mainly in the "linear setup" for the wrapped Fukaya category, but we also sketch the minor modifications which we need to extend the proofs to the "quadratic setup" and to the "localisation setup". This is necessary for dealing with Weinstein sectors and for the applications.

H-principles for regular Lagrangians

“…A positive answer would also imply that any regular Lagrangian disk in T * S n std can be disjoined from some cotangent fiber. Corollary 1.3 already shows that for any regular disk D n ⊂ T * S n std , there exists another regular disk C n , namely the co-core of the handle H n Λ , which is disjoint from D n and generates the wrapped Fukaya category of T * S n std ; see [4]. We also point out that there is another natural decomposition of regular Lagrangian disks.…”

“…As noted before, Theorem 1.1 does not hold in dimension 4 since there are Lagrangian formal classes not realized by any genuine Lagrangians. However an analogous construction in dimension four was considered by Yasui [18], who constructed many Lagrangians disks in B 4 std by trivially extending the Lagrangian unknot…”

Homological Mirror Symmetry for the universal centralizers I: The adjoint group case

“…To be more precise, the skeleton depends on a choice of the Weinstein structure. We further show that the cocores to some of the critical handles, which are the Kostant sections, generate the partially wrapped Fukaya category of J G (using general results from [GPS1,GPS2,CDGG]).…”

Symplectic cohomology of compound Du Val singularities

“…where W(V ) is the wrapped Fukaya category of V . An early version of this result based on Legendrian surgery is due to Bourgeois-Ekholm-Eliashberg ( [6], elaborated in [13]) which concerned Hochschild homology; a definitive version based on duality appeared in [20, Theorem 1.1] (see also the more recent [10]).…”