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Equivalence of the pearly tree immersed Lagrangian Floer theory and the Hamiltonian immersed Lagrangian Floer theory

Zuyi Zhang

TL;DR

This work resolves an equivalence between immersed Lagrangian Floer theory formulated via pearly-tree discs and the Hamiltonian-perturbed immersed Floer theory in Weinstein neighborhoods. By extending the Alston–Bao framework to obstructed chain complexes, the authors show a canonical identification between the pearly-tree chain complex $ (\mathrm{CF}^P(L),\partial^P) $ defined by a Morse function on an immersion and the Hamiltonian chain complex $ (\mathrm{CF}^H(L,L_{\phi_\epsilon}),\partial^H) $ induced by a small local Hamiltonian flow, in both directions. The proof leverages a detailed analysis of $J$-holomorphic discs, a four-type classification of pearly-tree discs, and a key intermediate Hamiltonian $H_1$ to transport generators and boundary data across the two models, supported by a transversality/regularity framework and a Sard–Smale argument for families of Lagrangian immersions. The results unify two perspectives on immersed Lagrangian Floer theory, enabling computations via either pearly-trees or local Hamiltonian flows, with implications for the structure of the Fukaya category and potential mirror-symmetry applications.

Abstract

The goal of this paper is to prove an equivalence relation between the immersed Lagrangian Floer theory, defined using pearly tree discs, and local Hamiltonian flows, i.e., Hamiltonian flows performed in the Weinstein tubular neighborhood. This is a generalization of Alston-Bao's work.

Equivalence of the pearly tree immersed Lagrangian Floer theory and the Hamiltonian immersed Lagrangian Floer theory

TL;DR

This work resolves an equivalence between immersed Lagrangian Floer theory formulated via pearly-tree discs and the Hamiltonian-perturbed immersed Floer theory in Weinstein neighborhoods. By extending the Alston–Bao framework to obstructed chain complexes, the authors show a canonical identification between the pearly-tree chain complex defined by a Morse function on an immersion and the Hamiltonian chain complex induced by a small local Hamiltonian flow, in both directions. The proof leverages a detailed analysis of -holomorphic discs, a four-type classification of pearly-tree discs, and a key intermediate Hamiltonian to transport generators and boundary data across the two models, supported by a transversality/regularity framework and a Sard–Smale argument for families of Lagrangian immersions. The results unify two perspectives on immersed Lagrangian Floer theory, enabling computations via either pearly-trees or local Hamiltonian flows, with implications for the structure of the Fukaya category and potential mirror-symmetry applications.

Abstract

The goal of this paper is to prove an equivalence relation between the immersed Lagrangian Floer theory, defined using pearly tree discs, and local Hamiltonian flows, i.e., Hamiltonian flows performed in the Weinstein tubular neighborhood. This is a generalization of Alston-Bao's work.
Paper Structure (6 sections, 11 theorems, 41 equations, 7 figures)

This paper contains 6 sections, 11 theorems, 41 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Pearly tree immersed Lagarangian Floer theory
  4. Hamiltonian immersed Lagrangian Floer theory
  5. Proof of the main theorem
  6. Calculation of examples on surfaces

Key Result

Theorem 1.1

Let $(M^n,\omega)$ be a closed symplectic manifold. Suppose $g:L\looparrowright M$ is a Lagrangian immersion. Assume $f$ is a morse function defined on $L$ whose critical points are distinct from the self-intersections of $L$. Given a regular almost complex structure on $M$, there is a local Hamilto

Figures (7)

  • Figure 1: The curves are gradient flows and the discs are holomorphic discs. The top dots are inputs and the bottom dot is the output.
  • Figure 2: The four types of pearly tree discs. The left dots are inputs and the right dots are outputs.
  • Figure 3: The left picture is the holomorphic disc for Hamiltonian immersed Lagrangian Floer theory. The right picture is the disc part of the pearly tree from $a$ to $(g,f)$ (the morse trajectory part is constant).
  • Figure 4: Hamiltonian version
  • Figure 5: Pearly tree version
  • ...and 2 more figures

Theorems & Definitions (36)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Corollary 2.7
  • Theorem 2.8
  • proof
  • ...and 26 more