125 research outputs found
Duality between Lagrangian and Legendrian invariants
Consider a pair , of a Weinstein manifold with an exact Lagrangian
submanifold , with ideal contact boundary , where is a
contact manifold and is a Legendrian submanifold. We
introduce the Chekanov-Eliashberg DG-algebra, , with
coefficients in chains of the based loop space of and study its
relation to the Floer cohomology of . Using the augmentation
induced by , can be expressed as the Adams cobar
construction applied to a Legendrian coalgebra, .
We define a twisting cochain:via holomorphic curve counts, where
denotes the bar construction and the graded linear dual. We show under
simply-connectedness assumptions that the corresponding Koszul complex is
acyclic which then implies that and are Koszul
dual. In particular, induces a quasi-isomorphism between
and the cobar of the Floer homology of , .
We use the duality result to show that under certain connectivity and locally
finiteness assumptions, is quasi-isomorphic to for any Lagrangian filling of . Our constructions have
interpretations in terms of wrapped Floer cohomology after versions of
Lagrangian handle attachments. In particular, we outline a proof that
is quasi-isomorphic to the wrapped Floer cohomology of a
fiber disk in the Weinstein domain obtained by attaching
to along (or, in the
terminology of arXiv:1604.02540 the wrapped Floer cohomology of in with
wrapping stopped by ). Along the way, we give a definition of wrapped
Floer cohomology without Hamiltonian perturbations.Comment: 126 pages, 20 figures. Substantial overall revision based on
referee's comments. The main results remain the same but the exposition has
been improve
Koszul duality patterns in Floer theory
We study symplectic invariants of the open symplectic manifolds
obtained by plumbing cotangent bundles of 2-spheres according to a plumbing
tree . For any tree , we calculate (DG-)algebra models of the
Fukaya category of closed exact Lagrangians in
and the wrapped Fukaya category . When
is a Dynkin tree of type or (and conjecturally also for
), we prove that these models for the Fukaya category
and are related by (derived)
Koszul duality. As an application, we give explicit computations of symplectic
cohomology of for , based on the Legendrian surgery
formula of Bourgeois, Ekholm and Eliashberg.Comment: 72 pages, 20 figures/tables. Minor corrections and improvements. To
appear in Geometry & Topolog
Arithmetic mirror symmetry for genus 1 curves with marked points
We establish a -linear derived equivalence
between the relative Fukaya category of the 2-torus with distinct marked
points and the derived category of perfect complexes on the -Tate curve.
Specialising to gives a -linear derived
equivalence between the Fukaya category of the -punctured torus and the
derived category of perfect complexes on the standard (N\'eron) -gon. We
prove that this equivalence extends to a -linear derived
equivalence between the wrapped Fukaya category of the -punctured torus and
the derived category of coherent sheaves on the standard -gon.Comment: 53 pages, 9 figures. Minor revision. To appear in Selecta Mathematic
Fukaya categories of the torus and Dehn surgery
This paper is a companion to the authors' forthcoming work extending Heegaard
Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary
components via quilted Floer cohomology. We describe the first interesting case
of this theory: the invariants of 3-manifolds bounding S^2 union T^2, regarded
as modules over the Fukaya category of the punctured 2-torus. We extract a
short proof of exactness of the Dehn surgery triangle in Heegaard Floer
homology. We show that A-infinity structures on the graded algebra A formed by
the cohomology of two basic objects in the Fukaya category of the punctured
2-torus are governed by just two parameters (m^6,m^8), extracted from the
Hochschild cohomology of A. For the Fukaya category itself, m^6 is nonzero.Comment: 29 pages, 2 figures, a footnote adde
Floer cohomology of the Chiang Lagrangian
We study holomorphic discs with boundary on a Lagrangian submanifold in a
Kaehler manifold admitting a Hamiltonian action of a group which has as
an orbit. We prove various transversality and classification results for such
discs which we then apply to the case of a particular Lagrangian in
first noticed by Chiang. We prove that this Lagrangian has
non-vanishing Floer cohomology if and only if the coefficient ring has
characteristic 5, in which case it generates the split-closed derived Fukaya
category as a triangulated category.Comment: 40 pages, 13 figures; v2 added computation of module structure and
strong generation result, v3 incorporated referee's comments to agree with
accepted version. To appear in Selecta Mathematic
Associative Yang-Baxter equation and Fukaya categories of square-tiled surfaces
We show that all strongly non-degenerate trigonometric solutions of the
associative Yang-Baxter equation (AYBE) can be obtained from triple Massey
products in the Fukaya category of square-tiled surfaces. Along the way, we
give a classification result for cyclic -algebra structures on a
certain Frobenius algebra associated with a pair of 1-spherical objects in
terms of the equivalence classes of the corresponding solutions of the AYBE. As
an application, combining our results with homological mirror symmetry for
punctured tori (cf. arXiv:1601.06141), we prove that any two simple vector
bundles on a cycle of projective lines are related by a sequence of 1-spherical
twists and their inverses.Comment: 37 pages, 9 figures. Minor revision after a referee's comments. To
appear in Advances in Mathematic
Examples of planar tight contact structures with support norm one
We exhibit an infinite family of tight contact structures with the property
that none of the supporting open books minimizes the genus and maximizes the
Euler characteristic of the page simultaneously, answering a question of
Baldwin and Etnyre in arXiv:0910.5021 .Comment: 5 pages, 5 figures. Final version. Minor corrections and
clarification
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