Cyclic symmetry and adic convergence in Lagrangian Floer theory

By K. Fukaya. 2009

153 citations recorded by the source corpus.

In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infinity algebra to every relatively spin Lagrangian submanifold. We use the same trick to construct a local rigid analytic family of filtered A infinity structure associated to a (family of) Lagrangian submanifolds. We include the study of homological algebra of pseudo-isotopy of cyclic (filtered) A infinity algebra.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c55c437023bcc25215c2e5a23fa13a1337e19c62

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Floer cohomology and pencils of quadrics (2010, background): “over Care cyclic, in the sense that the A∞-operations µk satisfy the cyclic relation hµk(an,...,a1),a0i= hµk(a0,an,...,a2),a1i. Thishighly non-trivial result (which we shall not need)is due to Fukaya [32]; although the A∞-operations morally count disks for which the boundary marked points are cyclic”
Fukaya A∞-structures associated to Lefschetz fibrations. I (2009, background): “Techniques from [17] (see also [20, Section 10] for the general algebraic framework) allow us to define a cyclic (and graded) A∞-algebra over R = K m whose cohomology is ⊕”
The monotone wrapped Fukaya category and the open-closed string map (2012, methodology): “Using the cyclicity of the Fukaya category [17], and recent work of AbouzaidFukaya-Oh-Ohta-Ono [7], it is equivalent to the injectivity of the closed-open string map, a natural ring map COB : QH (B) → HH(F(B)).”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, unknown): “ith a trivialization of its canonical bundle, and that the homology spheres Vi are Spin and have zero Maslov index, hence can be equipped with gradings. Take the coeﬃcient ﬁeld K = R. Techniques from [17] (see also [20, Section 10] for the general algebraic framework) allow us to deﬁne a cyclic (and”
Suspending Lefschetz Fibrations, with an Application to Local Mirror Symmetry (2009, background): “uces the maps (5.5). This is a weak version of the Calabi-Yau property of Fukaya categories. For coefﬁcient ﬁelds Kcontaining R, the strong form of that property (cyclicity) has been proved by Fukaya [5]. This would be enough for the applications to mirror symmetry, where K = C. There is actually an”
Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces (2012, methodology): “We therefore obtain an identification of the moduli space YL of simple objects of the Fukaya category supported on L with its first cohomology with coefficients in UΛ: YL ≡ H 1(L;UΛ).”