--- date: 2022-03-26 00:23 modification date: Saturday 26th March 2022 00:23:01 title: "Lagrangian Floer homology" aliases: [Lagrangian Floer cohomology, Floer homology, Floer cohomology, Floer, Maslov index] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #todo/untagged - Refs: - Comparisons of Floer homologies: - - Links: - [Novikov ring](Unsorted/Novikov%20ring.md) - [associahedra](associahedra.md) - [exact triangle](exact%20triangle) - [A_infty categories](Unsorted/A_infty%20categories.md) - [Liouville manifold](Liouville%20manifold.md) - [Fredholm operator](Unsorted/Fredholm%20operator.md) - Types of Floer homology: - Morse homology - Floer homology of (Hamiltonian) symplectomorphisms - Lagrangian Floer homology - Fukaya category - Heegaard Floer homology - Seiberg-Witten Floer homology - Embedded contact homology - Cylindrical contact homology - [monopole Floer homology](monopole%20Floer%20homology.md) --- # Floer homology # Summary ![](attachments/Pasted%20image%2020220512180255.png) ![](attachments/Pasted%20image%2020220512180407.png) ![](attachments/Pasted%20image%2020220512180509.png) ![](attachments/Pasted%20image%2020220430182902.png) ![](attachments/Pasted%20image%2020220430182936.png) Notes: - [Heegard-Floer homology](Unsorted/Heegard-Floer%20homology.md) (Osvath-Szabo) is Lagrangian-Floer theory for [Lagrangian](Lagrangian.md) torii in the symmetric product of a [Riemann surface](Riemann%20surface). - Expected to be equivalent to [Seiberg-Witten theory](Unsorted/Seiberg-Witten%20theory.md). - Major applications in [knot theory](knot%20theory) and [3-manifolds](Unsorted/Three-manifolds%20MOC.md), gives an algorithm for computing [knot genus](knot%20genus). - Defining HF is very difficult -- see 1000+ page Fukaya-Oh-Ohta-Ono monograph. The major issue is [transversality](transversality) for the moduli spaces, which requires [perturbation theory](perturbation%20theory) on the [almost complex structure](Unsorted/almost%20complex%20structure.md), the PDE operator, Hamiltonian isotopies of the Lagrangians, etc. Dealt with via [Kuranishi spaces](Kuranishi%20spaces.md) and [polyfolds](polyfolds.md). - Nice case: [exact symplectic manifolds](exact%20symplectic%20manifolds.md) which contain no holomorphic spheres and the Lagrangians bounds no holomorphic discs. - Useful theory precisely because it's computable, see Floer's [surgery exact triangle](surgery%20exact%20triangle), and there are some functorially defined maps coming from [cobordisms](Unsorted/cobordism.md). - Floer cohomology of the diagonal recovers [quantum cohomology](Unsorted/quantum%20cohomology.md) with its [quantum product](quantum%20product), ie $\HF(\Delta, \Delta) \cong \QH^*(M)$. The product counts holomorphic triangles. - Appears in [homological mirror symmetry](Unsorted/homological%20mirror%20symmetry.md) relating algebraic and symplectic geometry -- the symplectic geometry side involves the geometry of Lagrangian submanifolds and their Floer homology groups. See the [Fukaya category](Unsorted/Fukaya%20category.md). - There is no general prediction for $\rank \HF(L_0, L_1)$ when $L_0\neq L_1$; its definition involves an [elliptic operator](Unsorted/elliptic%20operator.md) corresponding to a PDE one has no expectation of being able to write down explicitly! - For closed varieties $X$, there are techniques due to Seidel for studying it via the theory on $Y = X\sm \Sigma_0$ and [deformation theory](Unsorted/deformation%20theory.md), which has been carried out for the [quartic surface](quartic%20surface) and [genus 2 curves](genus%202%20curves). - A [Fredholm operator](Fredholm%20operator.md) $D$ on a Banach space is one such that the [index](index) $\dim\ker D - \dim\coker D$ makes sense. # Warnings ![](attachments/Pasted%20image%2020220430184909.png) ![](attachments/Pasted%20image%2020220430184956.png) # Misc ![](attachments/Pasted%20image%2020220424200042.png) Isomorphism with [quantum cohomology](Unsorted/quantum%20cohomology.md): ![](attachments/Pasted%20image%2020220424200104.png) Relation to [string theory](Unsorted/string%20theory.md): ![](attachments/Pasted%20image%2020220424200210.png) See [Fukaya category](Unsorted/Fukaya%20category.md) # Lagrangian Floer homology ![](attachments/Pasted%20image%2020220517131052.png) ![](attachments/Pasted%20image%2020220422113447.png) ![](attachments/Pasted%20image%2020220422113453.png) ![](attachments/Pasted%20image%2020220422113503.png) ![](attachments/Pasted%20image%2020220326002348.png) ![](attachments/Pasted%20image%2020220326002355.png) ![](attachments/Pasted%20image%2020220326002303.png) ![](attachments/Pasted%20image%2020220326002423.png) ![](attachments/Pasted%20image%2020220326002540.png) ![](attachments/Pasted%20image%2020220326002647.png) ![](attachments/Pasted%20image%2020220326005316.png) ![](attachments/Pasted%20image%2020220326005448.png) ![](attachments/Pasted%20image%2020220326005502.png) ![](attachments/Pasted%20image%2020220326012027.png)![](attachments/Pasted%20image%2020220326012036.png) ## Maslov index ![](attachments/Pasted%20image%2020220326012102.png) ![](attachments/Pasted%20image%2020220326012149.png) # Grading ![](attachments/Pasted%20image%2020220326012232.png) ![](attachments/Pasted%20image%2020220326012259.png) ![](attachments/Pasted%20image%2020220326012306.png) ![](attachments/Pasted%20image%2020220326012346.png) # Transversality ![](attachments/Pasted%20image%2020220326012430.png) ![](attachments/Pasted%20image%2020220326012509.png) # Compactness and bubbling ![](attachments/Pasted%20image%2020220326012539.png) ![](attachments/Pasted%20image%2020220326012606.png) ![](attachments/Pasted%20image%2020220326012633.png) # Gluing ![](attachments/Pasted%20image%2020220326012720.png) # Homological mirror symmetry Uses in [homoloical mirror symmetry](homoloical%20mirror%20symmetry) ![](attachments/Pasted%20image%2020220326013758.png)