Floer Homology Groups in Yang-Mills Theory

Donaldson, Simon; Furuta, Mikio; Kotschick, D.

doi:10.1017/cbo9780511543098

Cited by 218 publications

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“…This discussion ensures that the moduli spaces of index 1 and 2 are compact up to the breaking of trajectories as in Morse theory. This is proven by combining the local elliptic estimates with the exponential decay on long strips; see Floer [8] and Donaldson [5]. Finally, part (c) requires a gluing theorem identifying the ends of the moduli space with broken trajectories.…”

Section: Remark 523mentioning

confidence: 98%

“…(For even r we can insert a diagonal into the sequence L, then the quilted holomorphic strips of widths ı can be identified with quilted holomorphic strips for the new sequence with widths . With these remarks in mind, we can follow the standard construction of Floer theory (which is currently probably best outlined by Salamon [29] for the case of holomorphic cylinders, fully executed by Donaldson [5] for a gauge theoretic setting, and hopefully soon available in Oh [20] for holomorphic strips). The moduli space (before quotienting by the R-action) is described as the zero set of the scaled Cauchy-Riemann operator x @ J ı ;H , which is a Fredholm section of a Banach bundle over the usual Banach manifold of maps wW R OE0; 1 !…”

Section: Remark 523mentioning

confidence: 99%

“…The Fredholm property of the linearized operators follows as in eg Floer [8] from the general theory of Lockhart and McOwen [17] (also see Donaldson [5]) for operators of the form @ s C D s , where the operators D s converge to invertible operators as s !˙1. After a Hamiltonian transformation (moving the perturbation onto the Lagrangian L .0/ and replacing J ı with H J ı , which retains the same properties) these operators take the form .…”

Section: Remark 523mentioning

confidence: 99%

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Quilted Floer cohomology

2010

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We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. This provides the foundation for the construction of a symplectic 2-category as well as for the definition of topological invariants via decomposition and representation in the symplectic category. Here we give some first direct symplectic applications: Calculations of Floer cohomology, displaceability of Lagrangian correspondences and transfer of displaceability under geometric composition. 53D40; 57R56

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Section: Remark 523mentioning

confidence: 98%

Section: Remark 523mentioning

confidence: 99%

Section: Remark 523mentioning

confidence: 99%

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Quilted Floer cohomology

2010

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“…The associated instanton Floer homology group, which Kronheimer and Mrowka denote by I * (Y ) w , is the Z/8Z-graded C-module arising from the Morse homology of the Chern-Simons functional on C/G (cf. [3]). Given any closed, embedded surface R ⊂ Y there is a natural operator…”

Section: Instanton Floer Homologymentioning

confidence: 99%

Instanton Floer homology and contact structures

Baldwin

Sivek

2015

Sel. Math. New Ser.

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Abstract. We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds-with or without boundary-defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by the authors in [1]. IntroductionFloer-theoretic invariants of contact manifolds have been responsible for many important results in low-dimensional topology. Notable examples include the invariants of closed contact 3-manifolds defined by Kronheimer and Mrowka [10] and by Ozsváth and Szabó [18] in monopole and Heegaard Floer homology, respectively. Also important is the work in [8], where Honda, Kazez, and Matić extend Ozsváth and Szabó's construction, using sutured Heegaard Floer homology to define an invariant of sutured contact manifolds, which are triples of the form (M, Γ, ξ) where (M, ξ) is a contact 3-manifold with convex boundary and Γ ⊂ ∂M is a multicurve dividing the characteristic foliation of ξ on ∂M . Recently, we defined an analogous invariant of sutured contact manifolds in Kronheimer and Mrowka's sutured monopole Floer homology theory [1].The goal of this paper is to define an invariant of sutured contact manifolds in Kronheimer and Mrowka's sutured instanton Floer homology (SHI). To the best of our knowledge, this is the first invariant of contact manifolds-with or without boundary-defined in the instanton Floer setting. Like the Heegaard Floer invariants but in contrast with the monopole invariants, our instanton Floer contact invariant is defined using the full relative Giroux correspondence. Its construction is inspired by a reformulation of the monopole Floer invariant in [1] which was used there to prove that the monopole invariant is well-defined.A unique feature of the instanton Floer viewpoint is the central role played by the fundamental group. Along these lines, we conjecture a means by which our contact invariant in SHI might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings, a relationship which has been largely unexplored to this point. Below, we sketch the construction of our contact invariant, describe some of its most important properties, state some conjectures, and discuss plans for future work which include using the constructions in this paper to define invariants of bordered manifolds in the instanton Floer setting.1.1. A contact invariant in SHI. Suppose (M, Γ) is a balanced sutured manifold. Roughly speaking, a closure of (M, Γ) is formed by gluing on some auxiliary piece a...

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“…In particular, instantons can be characterised as critical points of a Chern-Simons functional, hence zeroes of its gradient 1-form [3]. The explicit case of G 2 -manifolds, which we now describe, was first examined in the author's thesis [11].…”

Section: Def Inition Of the Chern-simons Functional ϑmentioning

confidence: 99%

Generalised Chern-Simons Theory and G₂-Instantons over Associative Fibrations

Earp

Henrique²

2014

SIGMA

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Abstract. Adjusting conventional Chern-Simons theory to G 2 -manifolds, one describes G 2 -instantons on bundles over a certain class of 7-dimensional flat tori which fiber nontrivially over T 4 , by a pullback argument. Moreover, if c 2 = 0, any (generic) deformation of the G 2 -structure away from such a fibred structure causes all instantons to vanish. A brief investigation in the general context of (conformally compatible) associative fibrations f : Y 7 → X 4 relates G 2 -instantons on pullback bundles f * E → Y and self-dual connections on the bundle E → X over the base, a fact which may be of independent interest.

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Floer Homology Groups in Yang-Mills Theory

Cited by 218 publications

References 0 publications

Quilted Floer cohomology

Quilted Floer cohomology

Instanton Floer homology and contact structures

Generalised Chern-Simons Theory and G₂-Instantons over Associative Fibrations

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Floer Homology Groups in Yang-Mills Theory

Cited by 218 publications

References 0 publications

Quilted Floer cohomology

Quilted Floer cohomology

Instanton Floer homology and contact structures

Generalised Chern-Simons Theory and G2-Instantons over Associative Fibrations

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Generalised Chern-Simons Theory and G₂-Instantons over Associative Fibrations