Atiyah-Floer conjecture: A formulation, a
                    strategy of proof and generalizations

Daemi, Aliakbar; Fukaya, Kenji

doi:10.1090/pspum/099/01736

Cited by 12 publications

(11 citation statements)

References 28 publications

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“…He then started working on Lagrangian correspondence and its relation to Lagrangian Floer theory. (The main motivation of the author's study is its application to gauge theory ( [DF,Fu8,Fu9]). ) This paper is an outcome of that study.…”

Section: Introductionmentioning

confidence: 99%

“…Some of the applications are now on the way being worked out and being written or already available as a preprint. (See [DF,EL,Fu8,Fu9,Fu11] and etc..) In this paper we concentrate in defining the basic object in as much general form as possible, leaving applications to other papers. A generalization of the story to the non-compact case is now studied by Yuan-Gao [Yu].…”

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confidence: 99%

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Unobstructed immersed Lagrangian correspondence and filtered A infinity functor

Fukaya¹

2017

Preprint

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In this paper we construct a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered A∞ categories. We consider arbitrary (compact) symplectic manifolds and its arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered A∞ category associated to (X, ω) is defined by using Lagrangian Floer theory in such generality. ([FOOO1,AJ].) The morphism of unobstructed immersed Weinstein category (from (X 1 , ω 1 ) to (X 2 , ω 2 )) is by definition a pair of immersed Lagrangian submanifold of the direct product and its bounding cochain (in the sense of [FOOO1, AJ]). Such a morphism transforms an (immersed) Lagrangian submanifold of (X 1 , ω 1 ) to one of (X 2 , ω 2 ). The key new result proved in this paper shows that this geometric transformation preserves unobstructed-ness of the Lagrangian Floer theory.Thus, this paper generalizes earlier results by Wehrheim-Woodward and Mau's-Wehrheim-Woodward so that it works in complete generality in the compact case.The main idea of the proofs are based on Lekili-Lipiyansky's Y diagram and a lemma from homological algebra, together with systematic use of Yoneda functor. In other words the proofs are based on a different idea from those which are studied by Bottmann-Mau's-Wehrheim-Woodward, where strip shrinking and figure 8 bubble plays the central role.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Unobstructed immersed Lagrangian correspondence and filtered A infinity functor

Fukaya¹

2017

Preprint

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“…If the reader is only interested in the case of Lagrangian Floer homology for Lagrangians with minimal Maslov number greater than two, then this section can be skipped. This turns out to be the case for Lagrangians coming from Yang-Mills gauge theory [DF1].…”

Section: Compatibility Of Kuranishi Structures With Forgetful Mapsmentioning

confidence: 88%

Monotone Lagrangian Floer theory in smooth divisor complements: II

Daemi¹,

Fukaya²

2018

Preprint

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In [DF2], the first part of the present paper, we study the moduli spaces of holomorphic discs and strips into an open symplectic, isomorphic to the complement of a smooth divisor in a closed symplectic manifold. In particular, we introduce a compactification of this moduli space, which is called the relative Gromov-Witten compactification. The goal of this paper is to show that the RGW compactifications admit Kuranishi structures which are compatible with each other. This result provides the remaining ingredient for the main construction of [DF2]: Floer homology for monotone Lagrangians in a smooth divisor complement.

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“…Roughly speaking, the conjecture tells us that symplectic Floer homology and instanton Floer homology are in congruence. A proof is still missing due to some obstructions related to the singularity of the spaces of flat connections and the resulting problems in defining the right side of (124) in a rigorous way [37,38]. On the level of our membrane field theory this conjecture is restated by saying that the stable (∞, 1)-category of L with Lagrangian cobordisms as morphisms M is identified with the partially wrapped Fukaya category.…”

Section: Homological Mirror Symmetrymentioning

confidence: 99%

General relativity and topological string duality through Penrose–Ward transform

Hristov

2022

Eur. Phys. J. C

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This paper discusses the relation between topological M-theory, self-dual Yang–Mills and general relativity. We construct a topological membrane field action from Witten’s cubic string field theory, which reduces to topological Yang–Mills on a one-parameter family of conifolds. It turns out that this can be interpreted as the twistor space of the four-dimensional Lagrangian submanifold M for large momenta. From the viewpoint of the target, we find that A-model and B-model on M unify in the topological membrane theory through the Penrose–Ward transform. The partition function is constructed and it is shown that, in the weak-coupling regime, it is equal to the partition function of Donaldson-Witten theory. Additionally, homological mirror symmetry, background independence as well as role of knot cobordisms as topological two-branes is discussed. It is outlined that all types of Floer homology are part of the topological membrane theory. Additionally, we find evidence that in the non-perturbative regime, the partition function of the membrane field action and that of the partially twisted (2,0) SU(N) superconformal field theory on the worldvolume of N topological fivebranes must coincide.

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Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations

Cited by 12 publications

References 28 publications

Unobstructed immersed Lagrangian correspondence and filtered A infinity functor

Unobstructed immersed Lagrangian correspondence and filtered A infinity functor

Monotone Lagrangian Floer theory in smooth divisor complements: II

General relativity and topological string duality through Penrose–Ward transform

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