Floer Theory: Foundations & Applications

• Floer theory is an advanced framework that adapts Morse theory to infinite-dimensional settings by counting solutions to elliptic PDEs.
• It employs chain complexes and A∞-structures to derive invariants in symplectic geometry, bridging analytic techniques with topological applications.
• Variants like wrapped and quilted Floer theory extend its scope to open manifolds and categorical field theories, offering new insights into geometric structures.

Floer theory is a rich suite of techniques in symplectic topology and low-dimensional topology centered around the infinite-dimensional Morse theory of action functionals on loop or path spaces, and counts of pseudo-holomorphic curves in symplectic manifolds. Developed initially to address the Arnold conjectures regarding fixed points of Hamiltonian diffeomorphisms and intersections of Lagrangian submanifolds, Floer theory now underpins major advancements in symplectic geometry, low-dimensional topology, mirror symmetry, and mathematical physics. Its incarnations—spanning Hamiltonian and Lagrangian Floer cohomology, wrapped and quilted theories, symplectic and instanton variants, and Floer-theoretic invariants for 3- and 4-manifolds—share a common methodology: defining chain complexes whose differentials count solutions to elliptic PDEs, yielding invariants sensitive to subtle symplectic and topological information.

1. Foundations and Key Analytical Principles

The core methodological innovation in Floer theory is the transposition of Morse-theoretic techniques to infinite-dimensional manifolds, specifically loop spaces or spaces of paths or connections. For Hamiltonian Floer theory, the central object is the action functional

$$\mathcal{A}_H([\gamma, A]) = -\int_A \omega - \int_0^1 H(t, \gamma(t))\,dt,$$

with $[\gamma, A]$ denoting a loop $\gamma$ in $M$ capped by a disc $A$. Critical points are $1$-periodic orbits of the Hamiltonian vector field $X_{H_t}$, and the Floer differential counts finite-energy solutions $u: \mathbb{R} \times S^1 \rightarrow M$ to

$\partial_s u + J_t(\partial_t u - X_{H_t}(u)) = 0,$

interpolating between $1$-periodic orbits. In Lagrangian Floer theory, intersection points (or Hamiltonian chords) between Lagrangian submanifolds serve as generators, with the differential counting $J$-holomorphic strips.

The construction requires delicate analytic foundations, including the control of bubbling phenomena and the achievement of transversality in relevant moduli spaces. Monotonicity or aspherical conditions are often imposed to preclude disk or sphere bubbling, ensuring that only the combinatorial boundary strata appear in degenerations of 1-dimensional moduli spaces. Fredholm theory is used to analyze the relevant linearized operators and compactness results (e.g., Gromov compactness, energy bounds) are foundational for the convergence analysis of sequences of curves.

Variants such as wrapped Floer theory, reduced and local Floer theory, or low-area Floer theory on non-monotone Lagrangians introduce additional technical modifications, such as non-Archimedean completions for open manifolds, localized bounding of energy, or restriction to minimal-area holomorphic disks (Groman, 2015, Tonkonog et al., 2015).

2. Algebraic and Categorical Structures

Beyond the abelian group structure of Floer (co)homology, the full power of Floer theory is realized in its $A_\infty$-algebraic and categorical frameworks, most notably the Fukaya category. For an (exact or monotone) symplectic manifold $M$, the Fukaya category $F(M)$ has as objects suitably decorated Lagrangian submanifolds (often together with gradings, relative spin structures, or bounding chains), and $\operatorname{hom}(L_0, L_1) = CF^*(L_0, L_1)$ is the Floer complex. The $A_\infty$-structure maps

$$\mu^d : CF^*(L_{d-1}, L_d) \otimes \cdots \otimes CF^*(L_0, L_1) \rightarrow CF^*(L_0, L_d)$$

are defined by counts of rigid pseudo-holomorphic polygons, subject to higher associativity relations encoding the combinatorics of the moduli spaces' boundaries.

Bounding cochains, bulk deformations, and other $A_\infty$-algebraic tools are used to address situations where the curvature $\mu^0$ does not vanish, allowing extension to wider classes of Lagrangians in monotone or non-exact settings.

The $A_\infty$-categorical viewpoint is also essential for the formulation and proof of local-to-global principles for the wrapped Fukaya category of Liouville sectors, Mayer-Vietoris gluing formulas, and the functoriality of invariants in TQFT and field-theoretic settings (Auroux, 26 Oct 2025).

3. Geometric Constructions, Quilted and Wrapped Floer Theory

Advances in Floer theory have expanded the scope of moduli spaces well beyond the original context. Quilted Floer theory extends Floer complexes to sequences of Lagrangian correspondences between symplectic manifolds, defining “quilted” moduli spaces of pseudo-holomorphic curves mapping patches of a domain (the quilt) to different target manifolds, with seam and boundary conditions given by the Lagrangian correspondences (Lekili et al., 2010). The main technical achievement in such contexts is the construction of isomorphisms between Floer groups under geometric composition of correspondences, with precise control over bubbling at seams (e.g., via monotonicity and minimal Maslov index bounds) and detailed index calculations. For instance, the isomorphism

$$HF(L_0, L_{01}, L_{12}, L_2) \cong HF(L_0, L_{02}, L_2),\quad L_{02} = L_{01}\circ L_{12}$$

is established under embedded composition and suitable transversality (Lekili et al., 2010).

Wrapped Floer theory adapts the framework to open (Liouville) manifolds, using Hamiltonians with controlled growth at infinity and analyzing the corresponding wrapped Fukaya category $\mathcal{W}(M)$ and symplectic cohomology $SH^*(M)$ (Groman, 2015). The analytic challenge is controlling the energy and diameter of solutions in noncompact settings (“energy confinement”) via i-bounded and loop-wise dissipative data.

4. Floer Theory in Low-Dimensional Topology

Several versions of Floer theory provide deep invariants for 3- and 4-manifolds:

• Instanton Floer Homology: Adapts Morse theory to the Chern-Simons functional, with the differential counting anti-self-dual connections on $Y\times \mathbb{R}$ (Manolescu, 2015).
• Monopole (Seiberg-Witten) Floer Homology: Utilizes the Seiberg-Witten equations, yielding invariants $\widehat{HM}$, $\overline{HM}$, etc., and provides a bridge to Heegaard Floer theory (Lee, 2017).
• Heegaard Floer Homology: Employs combinatorial data from Heegaard diagrams, with the chain complex generated by intersection points of tori in symmetric products (Manolescu, 2015).
• Embedded Contact Homology: Counts embedded $J$-holomorphic curves in the symplectization of a contact 3-manifold. Bridges with other theories are established via equivalence theorems (Lee, 2017).

Applications include the paper of homology cobordism, surgery exact triangles in knot theory, and the disproof of the topological triangulation conjecture in high dimensions via refined Pin(2)-equivariant Floer homotopy types (Manolescu, 2015, Cohen, 2019). Bordered Floer theory and TQFT formulations extend the framework to manifolds with boundary, using dg algebras associated to surfaces and modules for 3-manifolds with boundary, leading to powerful pairing and gluing theorems (Lipshitz et al., 2011).

5. Homotopical Enhancements and Bordism Interpretations

Floer theory admits further enrichment to stable homotopy types, particularly through flow categories whose objects are critical points (or generators) and whose morphisms are (compactified) moduli spaces of solutions, typically (derived) manifolds with corners equipped with extra tangential structure (Cohen, 2019, Abouzaid et al., 4 Apr 2024). The mapping spectra (in stable $\infty$-categories of flow categories) are constructed from these data, refining Floer (co)homology to stable homotopy invariants whose homotopy groups may be interpreted as Floer bordism groups.

In the framed case, the endomorphism spectrum recovers the sphere spectrum. In the complex-oriented setting, one expects module structures over periodic complex bordism, with the tangential data derived from the natural complex orientation on the moduli spaces. The construction of mapping spectra is sufficiently explicit to identify the Floer homotopy groups directly with Floer bordism groups (Abouzaid et al., 4 Apr 2024).

Such homotopical enhancements have led to (a) the detection of phenomena invisible to abelian group-valued invariants (e.g., Steenrod operations on Khovanov homology), (b) applications to the paper of Lagrangian immersions, and (c) new tools for addressing transversality and formal enumeration in settings with higher-dimensional moduli spaces (Cohen, 2019).

6. Field-Theoretic and Categorical Extensions

Floer theory is fundamentally linked to categorified TQFT and field theory constructions, where invariants for manifolds (and cobordisms) are constructed via functors to (quilted) bicategories/2-categories of symplectic manifolds, Lagrangian correspondences, and Floer-theoretic data (Wehrheim, 2016, Wehrheim et al., 2016). The extension principles encode invariance under changes of decomposition (Cerf moves), and the quilted framework underpins the interplay of composition, functoriality, and adjunctions at the categorical level.

For example, in tangle field theory and related topological gauge-theoretic contexts, objects are moduli spaces of flat bundles (equipped with symplectic forms), bordism-induced Lagrangian correspondences define the morphisms, and functor-valued invariants arise from quilted Floer theory (Wehrheim et al., 2015). Key results such as invariance of cohomology under geometric composition: $$HF(L_0, L_{01}, L_{12}, L_2) \cong HF(L_0, L_{01}\circ L_{12}, L_2)$$ are instrumental for realizing TQFT-like gluing properties.

Floer field theory provides a categorical lens for understanding highly structured invariants, compatibility with gluing, and interaction with derived geometry and mirror symmetry (Wehrheim, 2016).

7. Connections and Applications: Mirror Symmetry, Rigidity, and Dynamics

Floer theory is a central player in homological mirror symmetry, where it furnishes the $A$-model category (Fukaya category) that is conjecturally equivalent to the derived category of coherent sheaves on the mirror ($B$-model). The internal algebraic structures—Hochschild (co)homology, deformation theory, and open-closed string maps—translate symplectic geometric data to the mirror's algebraic geometry (Auroux, 26 Oct 2025). Family Floer cohomology, local-to-global structures for Fukaya categories, and the computation of superpotentials via Maslov index 2 disc counts expose further connections to mirror symmetry's predictions.

In dynamical applications, Floer-theoretic methods yield obstructions to symplectic embeddings, non-displaceability results (including low-area criteria for non-monotone Lagrangians (Tonkonog et al., 2015)), and spectral invariants that relate action selectors to quantum and classical Hamiltonian dynamics. For three-dimensional Anosov flows, the associated Anosov Liouville domains admit computation of symplectic and wrapped Fukaya categories, with orbit categories reflecting flow dynamics and open-closed maps capturing subtle generational properties (Cieliebak et al., 2022).

The interplay with low-dimensional topology, categorical field theory, and higher homotopical invariants positions Floer theory as an organizing principle in modern geometry, part of a unifying framework for symplectic topology, gauge theory, and mathematical aspects of string theory.