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Unobstructed Immersed Lagrangian Correspondence and Filtered $A_{\infty}$ Functor

Kenji Fukaya

TL;DR

This work establishes a comprehensive 2-categorical framework connecting unobstructed immersed Lagrangian Floer theory with filtered $A_{ obreak ext{∞}}$ categories. It constructs a 2-functor from the unobstructed immersed Weinstein category to the 2-category of all strict, unital filtered $A_{ obreak ext{∞}}$ categories, using immersed Lagrangian correspondences with bounding cochains and Yoneda-type representability. A key innovation is the algebraic–geometric bridge via Künneth bi-functors and tri-modules, realized through quilted moduli spaces, and the use of Lekili–Lipyanskiy’s Y-diagram to prove compatibility and associativity results without relying on strip shrinking or Figure-8 bubbles. The results generalize prior work (Wehrheim–Woodward, Mau–Wehrheim–Woodward) to complete compact settings, providing a robust foundation for composing and representing Lagrangian correspondences in the immersed setting, with potential applications to gauge theory and beyond. The paper also develops the obstruction theory via bounding cochains, guaranteeing unobstructedness under geometric transformations and establishing a coherent functorial calculus for immersed Lagrangian Floer theory.

Abstract

In this paper, we 'construct' a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered $A_{\infty}$ categories. We consider arbitrary (compact) symplectic manifolds and its arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered $A_{\infty}$ category associated to $(X,ω)$ is defined by using Lagrangian Floer theory in such generality, see Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009). The morphism of unobstructed immersed Weinstein category (from $(X_1,ω_1)$ to $(X_2,ω_2)$) is by definition a pair of an immersed Lagrangian submanifold of the direct product and its bounding cochain (in the sense of Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009)). 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 unobstructedness 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-Lipyanskiy'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.

Unobstructed Immersed Lagrangian Correspondence and Filtered $A_{\infty}$ Functor

TL;DR

This work establishes a comprehensive 2-categorical framework connecting unobstructed immersed Lagrangian Floer theory with filtered categories. It constructs a 2-functor from the unobstructed immersed Weinstein category to the 2-category of all strict, unital filtered categories, using immersed Lagrangian correspondences with bounding cochains and Yoneda-type representability. A key innovation is the algebraic–geometric bridge via Künneth bi-functors and tri-modules, realized through quilted moduli spaces, and the use of Lekili–Lipyanskiy’s Y-diagram to prove compatibility and associativity results without relying on strip shrinking or Figure-8 bubbles. The results generalize prior work (Wehrheim–Woodward, Mau–Wehrheim–Woodward) to complete compact settings, providing a robust foundation for composing and representing Lagrangian correspondences in the immersed setting, with potential applications to gauge theory and beyond. The paper also develops the obstruction theory via bounding cochains, guaranteeing unobstructedness under geometric transformations and establishing a coherent functorial calculus for immersed Lagrangian Floer theory.

Abstract

In this paper, we 'construct' a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered categories. We consider arbitrary (compact) symplectic manifolds and its arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered category associated to is defined by using Lagrangian Floer theory in such generality, see Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009). The morphism of unobstructed immersed Weinstein category (from to ) is by definition a pair of an immersed Lagrangian submanifold of the direct product and its bounding cochain (in the sense of Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009)). Such a morphism transforms an (immersed) Lagrangian submanifold of to one of . The key new result proved in this paper shows that this geometric transformation preserves unobstructedness 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-Lipyanskiy'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.

Paper Structure

This paper contains 39 sections, 62 theorems, 181 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Filtered $\boldsymbol{A_{\infty}}$ category: Review
  3. $\boldsymbol{A_{\infty}}$ category
  4. $\boldsymbol{A_{\infty}}$ functor
  5. Functor category
  6. $\boldsymbol{A_{\infty}}$-Whitehead theorem
  7. $\boldsymbol{A_{\infty}}$-Yoneda embedding
  8. Floer theory of immersed Lagrangian submanifolds: Review
  9. Immersed Lagrangian submanifold
  10. Moduli space of pseudo-holomorphic polygons
  11. The filtered $\boldsymbol{A_{\infty}}$ algebra associated to an immersed Lagrangian submanifold
  12. Filtered $\boldsymbol{A_{\infty}}$ categories of immersed Lagrangian Floer theory
  13. Opposite $\boldsymbol{A_{\infty}}$ category and $\boldsymbol{\omega \mapsto -\omega}$
  14. Preliminary on Lagrangian correspondence
  15. The Künneth bi-functor in Lagrangian Floer theory
  16. ...and 24 more sections

Key Result

Theorem 1.3

Let $(X,\omega)$ be a compact symplectic manifold and $\mathbb L$ a finite set of its spin immersed Lagrangian submanifolds.In the introduction, we assume spinness of Lagrangian submanifolds rather than relatively-spinness, for simplicity. The statement in the relatively spin case will be given in t

Figures (11)

  • Figure 1: Domains $Z_+$, $Z_-$.
  • Figure 5: $\lim_{z \uparrow z_i}\gamma(z)$.
  • Figure 6: Tree $\Gamma$.
  • Figure 7: $e_i({\rm v})$.
  • Figure 8: $v_s({\rm e})$, $v_t({\rm e})$.
  • ...and 6 more figures

Theorems & Definitions (173)

  • Definition 1.1: informal definition
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • Remark 1.11
  • ...and 163 more