Lagrangian Floer Theory

• Lagrangian Floer theory is a framework that uses pseudo-holomorphic curves to define invariants for intersecting Lagrangian submanifolds in symplectic spaces.
• It employs advanced techniques such as RGW compactification and spectral sequences to ensure well-defined cohomology and robust analytical control.
• The theory underlies the construction of Fukaya categories and plays a pivotal role in homological mirror symmetry and symplectic rigidity results.

Lagrangian Floer theory is a foundational component of modern symplectic topology, encoding intersection-theoretic data of Lagrangian submanifolds within symplectic manifolds through analytical, algebraic, and topological structures. It provides invariants for pairs or collections of Lagrangians, undergirds the construction of Fukaya categories, and interfaces with subjects ranging from homological mirror symmetry to enumeration of holomorphic curves. The theory encompasses a broad arsenal of techniques: transversality analysis for moduli spaces of pseudo-holomorphic curves, sophisticated compactification schemes, spectral sequences, and homotopical enhancements such as Floer homotopy types. The following sections synthesize the contemporary research territory of Lagrangian Floer theory, emphasizing technical definitions, constructions, and structural results.

1. Foundational Structures: Floer Complexes, Grading, and Differential

The classical Lagrangian Floer chain complex is constructed from two compact Lagrangians $L_0,L_1\subset (M^{2n},\omega)$ intersecting transversely in a symplectic manifold, usually assumed to be closed or convex at infinity. The chain group is

$$CF^*(L_0,L_1) = \bigoplus_{p\in L_0\cap L_1} \Lambda\cdot p,$$

graded by the Maslov–Viterbo index when $2c_1(TM) = 0$ and Maslov classes of $L_i$ vanish; otherwise, one obtains a $\mathbb{Z}/N$-graded theory if the minimal Maslov number $N$ is defined. The Floer differential $\partial$ counts rigid $J$-holomorphic strips $u:\mathbb{R}\times[0,1]\to M$, mapping edges of the strip to $L_0$ and $L_1$, respectively, and asymptotic to intersection points as $s\to\pm\infty$:

$$\partial x_+ = \sum_{x_-}\#\mathcal{M}(x_+, x_-) \, x_-.$$

With generic domain-dependent almost complex structure $J$ and Hamiltonian perturbation $H_t$, one proves compactness and transversality for the moduli spaces of expected dimension 0 and 1, ensuring $\partial^2=0$ and a well-defined cohomology $HF^*(L_0,L_1)$ (Auroux, 26 Oct 2025).

Novikov coefficients are required in general to account for nontrivial symplectic areas of pseudo-holomorphic strips and bubbling phenomena. When $L_0$ and $L_1$ have "clean intersection"—i.e., their intersection is a submanifold $C$ with $T_pC = T_pL_0 \cap T_pL_1$—one uses the Morse–Bott or "pearl" complex, with generators given by critical points of Morse functions on $C$ and a boundary operator built from cascades alternating Morse flows and strips. This setup is foundational for the application of spectral sequences and for naturality of the grading conventions (Schmäschke, 2016).

Abstractions to scenarios such as immersed, orbifold, or equivariant Lagrangians introduce further ingredients: grading shifts, local systems, or group actions, which permeate throughout the differential and higher $A_\infty$ operations. Exactness or monotonicity often serve to preclude disc bubbling in dimension 1 or lower.

2. Moduli Spaces, Compactification, and RGW Theory

The analytical heart of Lagrangian Floer theory is the study of moduli spaces of (perturbed) $J$-holomorphic curves. For open symplectic manifolds $X\setminus D$—complements of smooth divisors in closed symplectic manifolds—standard stable map compactification fails due to boundary strata retaining holomorphic curves in the divisor $D$. This deficiency leads to violation of $\partial^2=0$ unless additional structure is imposed.

To address this, the RGW (Relative Gromov–Witten) compactification is employed (Daemi et al., 2018). The boundary strata of the RGW compactified moduli are indexed by decorated ribbon trees whose vertices represent discs, spheres, or relative components (maps to $D$) at different "levels", with combinatorial data encoding intersection multiplicities and a hierarchical approach to $D$. Gluing parameters and balancing conditions imposed at $D$-vertices ensure that positive-level spheres in $D$ do not occur in codimension-1 boundary components, which is essential for restoring the chain-level relation $\partial^2=0$.

Analytically, the RGW-topology refines the Gromov topology to accommodate weighted convergence in normal bundle directions, leveraging exponential decay on necks and meromorphic sections. This yields a moduli space $\mathcal{M}^{\mathrm{RGW}}$ that is second countable, Hausdorff, and metrizable, with a natural stratification by decorated trees (Daemi et al., 2018). Kuranishi structures on RGW-compactified moduli, constructed level-by-level and compatible with group actions and obstruction bundles, are essential for oriented virtual fundamental chains and for algebraic structures in Floer theory (Daemi et al., 2018).

3. Algebraic Structures: $A_\infty$-Operations and Fukaya Categories

Lagrangian Floer theory naturally generalizes into an $A_\infty$-framework via higher-order operations $\mu^k$. These are defined by counting solutions to Floer's equation on domains modeled on $(k+1)$-punctured discs, with cyclically ordered boundary components labeled by possibly different Lagrangians. The $A_\infty$-relations,

$$\sum_{i+j\le k} (-1)^{\star} \mu^{k-j+1}(a_k,\dots,\mu^{j}(a_{i+j},\dots),\dots,a_1) = 0,$$

are verified by analyzing the 1-dimensional moduli's boundary strata, which correspond to different degenerations (e.g., strip breaking) (Auroux, 26 Oct 2025). In the presence of obstructions from Maslov index 2 discs, bounding cochains $b$ are employed to deform the $A_\infty$-structure, recasting the theory uncurved at $b$ and preserving $\mu^1_b\circ\mu^1_b=0$.

The Fukaya category is then the $A_\infty$-category with (possibly local-system-equipped and bounding-cochain-equipped) Lagrangians as objects, Floer complexes as morphism spaces, and these higher operations as structure maps. This categorical structure is pivotal to applications in homological mirror symmetry, which identifies—under diverse correspondences—Fukaya categories of symplectic manifolds with derived categories of coherent sheaves or matrix factorizations on mirror Landau–Ginzburg models (Fukaya et al., 2010, Auroux, 26 Oct 2025).

4. Filtration, Spectral Sequences, and Computations

Floer complexes admit natural filtrations, either by action (Novikov exponents), Maslov index, or both. Action or Maslov filtrations enable the construction of spectral sequences, as in the Oh–Pozniak and Leray–Serre frameworks (Schultz, 2017, Schmäschke, 2016). In the clean intersection setting, the Morse–Bott theory facilitates spectral sequences beginning on pages

$$E_{1}^{p,q} \cong \bigoplus_{A(C_\ell)=a_p} H_{p+q-\mu(C_\ell)}(C_\ell),$$

with $d_r$ differentials governed by cascades of length $r$, and convergence to $HF^*(L_0,L_1)$ is ensured under monotonicity or asphericity (Schmäschke, 2016). In fibered settings, base-energy filtrations provide computational tools for reducing Floer-theoretic questions on total spaces to those on fibers and bases (Schultz, 2017).

Vanishing results can be proved by employing such spectral sequences. If a fiber Lagrangian $L_{F_p}$ in a fibration has vanishing Floer cohomology, the total space Lagrangian’s Floer cohomology also vanishes, providing obstruction-theoretic consequences and computational streamlining for complex fibrations (Schultz, 2017).

5. Gluing Theorems, Hardy Spaces, and Analytic Techniques

The analytic machinery underpinning algebraic structures in Lagrangian Floer theory relies on gluing results for moduli spaces of pseudo-holomorphic strips and polygons. The Hardy-space approach (Simcevic, 2014) provides a robust functional-analytic context: moduli spaces of finite and infinite strips are realized as embedded submanifolds in Hilbert manifolds of paths with $E^{3/2}$-regularity, and $C^1$-convergence of gluing maps is established via elliptic and interpolation estimates. Intersection-theoretic gluing facilitates rigorous proofs of

$\partial^2 = 0$

even in settings with monotonicity or minimal Maslov number $\ge 3$. This analytic apparatus also extends to invariance under Hamiltonian isotopy and to the composition of morphisms and ring isomorphisms in Floer–Donaldson and Seidel representations (Simcevic, 2014).

6. Structural and Rigidity Results, Applications

Lagrangian Floer invariants possess significant rigidity implications. In open symplectic manifolds $X\setminus D$ with monotone Lagrangians, Daemi–Fukaya’s construction (Daemi et al., 2018, Daemi et al., 2018) establishes that the Floer homology is well-defined and Hamiltonian isotopy invariant, and provides Arnold–type lower bounds:

$$\operatorname{dim}_{\Lambda} HF(L_0,L_1; X\setminus D) \leq \#(L_0\cap L_1).$$

Displaceability by Hamiltonian isotopy implies Floer cohomology vanishes. Pair-of-pants and module structures over quantum cohomology, as well as spectral sequences from the ordinary homology of $L$ converging to its Floer homology in the open complement, are established. The RGW compactification ensures that the only codimension 1 boundaries relevant for $\partial^2=0$ are those from broken strips—not from disc/sphere bubbling in $D$, as would obstruct the theory in the standard stable-map framework (Daemi et al., 2018, Daemi et al., 2018).

These foundational results permeate the structure of the monotone Fukaya category of $X\setminus D$, extend Oh's Floer theory without convexity at infinity or positivity/negativity assumptions, and enable further development of invariants for pairs, module categories, and spectral sequences allied to Hamiltonian group actions and quantum invariants.

7. Advances, Generalizations, and Homotopical Enhancements

Recent developments---including Floer homotopy theory, equivariant and local system enhancements, and generalizations to immersed and singular Lagrangians---expand the landscapes of both theoretical and computational symplectic topology. Floer homotopy types (Bonciocat, 20 Jun 2025) and equivariant theories (Xiao, 2023) introduce new generalized cohomological structures and operations (e.g., Steenrod operations on Floer complexes), further constraining the topology of intersection loci and amplifying the reach of rigidity theorems.

Utility in mirror symmetry, classification of objects in Fukaya categories, and applications to cobordism, toric and orbifold Lagrangians, and singular configurations (including trivalent graphs and non-commutative correspondences) testifies to the pervasiveness of Lagrangian Floer theoretic methods as both a computational and structural backbone in symplectic geometry, topology, and beyond.