Categorification of invariants in gauge theory and symplectic geometry

@article{Fukaya2017CategorificationOI,
  title={Categorification of invariants in gauge theory and symplectic geometry},
  author={Kenji Fukaya},
  journal={Japanese Journal of Mathematics},
  year={2017},
  volume={13},
  pages={1-65}
}
  • K. Fukaya
  • Published 2 March 2017
  • Mathematics
  • Japanese Journal of Mathematics
This is a mixture of survey article and research announcement. We discuss instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998–2012, those problems have been studied emphasizing the ideas from analysis such as degeneration and adiabatic limit (instanton Floer homology) and strip shrinking… 
5 Citations

Categorification of invariants in gauge theory and symplectic geometry

  • K. Fukaya
  • Mathematics
    Japanese Journal of Mathematics
  • 2017
This is a mixture of survey article and research announcement. We discuss instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory

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

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic

T-equivariant disc potentials for toric Calabi-Yau manifolds

We study the equivariant disc potentials for immersed SYZ fibers in toric Calabi-Yau manifolds. The immersed Lagrangians play a crucial role in the partial compactification of the SYZ mirrors.

Unobstructed immersed Lagrangian correspondence and filtered A infinity functor

In this paper we construct a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered A infinity categories. We consider arbitrary (compact) symplectic manifolds

References

SHOWING 1-10 OF 77 REFERENCES

Instanton Floer homology with Lagrangian boundary conditions

In this paper we define instanton Floer homology groups for a pair consisting of a compact oriented 3–manifold with boundary and a Lagrangian submanifold of the moduli space of flat SU.2/–connections

$SO(3)$-Floer homology of 3-manifolds with boundary 1

In this paper the author discuss the relation between Lagrangian Floer homology and Gauge-theory (Donaldson theory) Floer homology. It can be regarded as a version of Atiyah-Floer type conjecture in

Lagrangian Floer theory over integers: spherically positive symplectic manifolds

In this paper we study the Lagrangian Floer theory over $\Z$ or $\Z_2$. Under an appropriate assumption on ambient symplectic manifold, we show that the whole story of Lagrangian Floer theory in

Pseudoholomorphic Quilts

Lagrangian Floer cohomology associates to a pair of Lagrangian manifolds a chain complex whose differential counts pseudoholomorphic strips with boundary values in the given Lagrangians. These form a

Intersection Theory for Lagrangian Immersions

We introduce Floer homology for transversely intersecting Lagrangian immersions L and L′ in a symplectic manifold (X, ω). By using this homology, if π2(X, L) = 0 and L′ is the image of L under a

Anti-Self-Dual Instantons with Lagrangian Boundary Conditions I: Elliptic Theory

We study nonlocal Lagrangian boundary conditions for anti-self-dual instantons on 4-manifolds with a space-time splitting of the boundary. We establish the basic regularity and compactness properties

Floer Homology on the Extended Moduli Space

Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant in the form of a relatively \(\mathbb{Z}/8\mathbb{Z}\)-graded abelian

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

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic

Legendrian cobordism and Chern–Simons theory on 3-manifolds with boundary

In Floer's instanton homology theory for homology 3-spheres, the ChernSimons function (or a suitable perturbation of it) is used as a sort of Morse function on the space of SU(2) connections modulo

Floer homologies for Lagrangian intersections and instantons

In 1985 lectures at MSRI, A. Casson introduced an interesting integer valued invariant for any oriented integral homology 3-sphere Y via beautiful constructions on representation spaces (see [1] for
...