Higher-Dimensional Heegaard Floer

• Higher-Dimensional Heegaard Floer Homology is a framework that extends classical 3D Floer theory to exact and Liouville symplectic manifolds via pseudoholomorphic curve counts.
• It connects topological invariants, representation theory, and skein-type quantum algebras by using counts of holomorphic polygons with Lagrangian boundary conditions.
• The approach leverages exact Lagrangian submanifolds in settings like cotangent bundles and Milnor fibers to produce practical invariants for link, contact, and symplectic topology.

Higher-dimensional Heegaard Floer homology (HDHF) generalizes classical 3-dimensional Heegaard Floer homology to exact and Liouville symplectic manifolds of dimension greater than four, replacing the $n=1$ 3-manifold setup with higher-dimensional symplectic geometry. The theory harnesses counts of pseudoholomorphic curves with Lagrangian boundary conditions to encode algebraic data, and its distinguishing feature is its ability to connect topological invariants, representation theory, and skein-type quantum algebras in a unified symplectic package. HDHF operates naturally in cotangent bundles of surfaces and Milnor fibers of simple singularities, revealing direct symplectic realizations of Hecke-type and double affine Hecke algebras, and admits applications to link invariants, contact and symplectic topology, and spectral networks.

1. Geometric and Algebraic Foundations

The geometric input of HDHF consists of an exact symplectic manifold $M$, such as $T^*C$ for a closed or punctured surface $C$ endowed with its canonical Liouville form $\lambda = p\,dq$ and symplectic structure $\omega=dp\wedge dq$. Central objects are exact Lagrangian submanifolds—typically union of cotangent fibers $L = \bigsqcup_{i=1}^\kappa T^*_{q_i} C$ or real spectral curves $\Sigma \subset T^*C$ associated with branched coverings.

The wrapped HDHF chain complex $CW(L)$ is generated by time-one Hamiltonian chords from $L$ to itself, for a Hamiltonian growing quadratically at infinity. For generic data, all generators lie in degree zero, and the differential vanishes; hence the cohomology $HW^0$ is an ordinary associative algebra. The $A_\infty$-structure is determined by counts of holomorphic polygons with boundary on the Lagrangians, and all higher products vanish for degree reasons in the cotangent bundles of closed surfaces of $g>0$ (Honda et al., 2022).

HDHF moduli spaces parametrize holomorphic curves $(F,u)$, where $u:F\to X$ maps a bordered Riemann surface $F$ (punctured disk or polygon) into $X$, satisfying the perturbed Cauchy--Riemann equation

$(du - X_{H} \otimes d\tau)^{0,1} = 0$

with boundary lifted to the selected Lagrangians. The count is further refined by Euler characteristic and, in some cases, Maslov index.

2. Algebraic Structures: Hecke Algebras and Skein Relations

The algebraic targets of HDHF are Hecke-type or braid skein algebras. For cotangent fibers in $T^*\Sigma$, the wrapped Floer cohomology algebra $HW(L)$ is isomorphic to the (surface) Hecke algebra $H_k(\Sigma)$, realized as a quotient of the group algebra of the surface braid group by the HOMFLY skein relation $\sigma_{+} - \sigma_{-} = \hbar \sigma_0$. For cotangent fibers in $T^*D^2$, the resulting algebra is the finite Hecke algebra of $S_k$ (Tian et al., 2022).

For $T^*T^2$, the algebra is the double affine Hecke algebra (DAHA) of type A, and HDHF modules realize the polynomial representation of DAHA. Holomorphic curve counts match the Demazure–Lusztig operators' action, with the grading $\deg \hbar=2$ coinciding with the algebraic degree conventions (Gao et al., 9 Nov 2025).

Quantization parameters such as $\hbar$ appear as weights recording the Euler characteristic contributions from the moduli spaces.

3. HDHF, Skein Algebras, and Spectral Networks

A crucial development is the construction of an algebra homomorphism

$$\Phi \colon \mathrm{BSk}_\kappa(C) \longrightarrow \mathrm{Mat}\left(N^\kappa, \mathrm{BSk}_\kappa(\Sigma)\right)$$

where $\mathrm{BSk}_\kappa(C)$ is the Morton–Samuelson braid skein algebra of $C$ and $\Sigma$ is an $N$-fold real exact Lagrangian spectral curve in $T^*C$ (Honda et al., 22 Jan 2026). This map is defined via counts of HDHF-type holomorphic "open-string polygons" with boundary on cotangent fibers and on $\Sigma$, incorporating branch cover data and weighted by $\hbar$ per branch point. The outcome is a matrix-valued quantization of skein-theoretic nonabelianization.

Degenerations of the holomorphic moduli spaces enforce the skein relations and matrix algebra rules. Particular care is given to relations at branch points, introducing additional signs. Notably, when $\kappa=1$, $\Phi$ recovers the Gaiotto–Moore–Neitzke nonabelianization on algebras generated by fundamental group loops.

This construction aligns HDHF directly with constructions in quantum topology (HOMFLY skein modules, DAHA) and spectral networks (quantum trace, nonabelianization) (Honda et al., 22 Jan 2026).

4. Analytic and Morse-Theoretic Techniques: Hybrid Floer–Morse Theory

A salient analytical ingredient is the compatibility of the HDHF holomorphic-curve formalism with Morse-theoretic models. In the adiabatic/degeneration regime, index-zero holomorphic curve counts (core of HDHF) are shown to correspond to combinatorial data of "folded Morse trees": gradient trees in the base manifold built from differences of spectral curve branches [(Honda et al., 22 Jan 2026), Fukaya–Oh].

Squid-like configurations (three-punctured disk with attached trees) model the gradient flow on the configuration space, and a hybrid moduli of pairs ((HDHF disk), (folded Morse tree)) computes a Morse-theoretic realization of the $\Phi$ map. An adiabatic limit theorem, leveraging techniques of Fukaya–Oh and Ekholm, rigorously matches holomorphic counts of polygons to Morse flow tree counts, ensuring the equivalence between holomorphic and combinatorial models for skein quantization.

5. Invariants, Applications, and Examples

HDHF gives rise to a suite of topological invariants:

• Link invariants: In the Milnor fiber or cotangent bundle setup, HDHF produces well-defined link invariants, generalizing symplectic Khovanov homology. The machinery is insensitive to arc slides (Reidemeister II, III) and Markov stabilizations, satisfying all moves required for link invariance (Yuan, 2023).
• Contact class and fillability obstructions: The HDHF contact class generalizes the 3-dimensional contact invariant and provides obstructions to Liouville fillability and sufficient conditions for the Weinstein conjecture (Colin et al., 2020).
• Recovery of quantum algebras: For $T^*D^2$ and $\kappa$ cotangent fibers, HDHF recovers the finite Hecke algebra. For $T^*T^2$, the DAHA representation is realized entirely within HDHF (Gao et al., 9 Nov 2025, Tian et al., 2022).

Table: Representative Correspondence between HDHF and Algebraic Targets

Geometric Setting	HDHF Algebra	Algebraic Target
$T^*D^2$, $\kappa$ fibers	$HW_\hbar$	Finite Hecke algebra $H_\kappa$
$T^*(S^1\times \mathbb{R})$	$HW_\hbar$ for fibers	Affine Hecke algebra
$T^*T^2$	$HW_\hbar$ for fibers	Double affine Hecke algebra (DAHA)
$T^*\Sigma$ (genus $g>0$)	$HW_\hbar$ for fibers	Surface Hecke algebra $H_\kappa(\Sigma)$

Explicit calculations show that the HDHF product $m_2$ reproduces the quadratic and braid relations of the respective Hecke algebras, and in the DAHA setting, the Floer-theoretic module realizes the polynomial representation, including Cherednik's pairing (Gao et al., 9 Nov 2025).

6. Extensions, Open Questions, and Future Directions

HDHF exhibits flexibility for extension to broader contexts:

• q-Deformation and quantum traces: Incorporating additional homological variables $c$ in the curve counts and considering intersection with branch cuts aims to lift HDHF to a $q$-deformed version matching quantum trace (Bonahon–Wong) constructions and categorical quantum group structures (Honda et al., 22 Jan 2026).
• Partially wrapped Fukaya categories: For punctured surfaces, HDHF extends to partially wrapped settings, modifying the analytic setup to ensure compactness and proper algebraic structure (Honda et al., 2022).
• Betti–Langlands and link to categorical DT: Comparison with the Betti–Langlands correspondence, categorical Donaldson–Thomas theory for quivers, and physical connections via knot/contact homology and the theory of M2/M5-branes are highlighted as promising research avenues (Honda et al., 22 Jan 2026).

Several technical open questions remain, such as the full scope of invariance under symplectic handleslides, potential plus/infinity versions of HDHF, and the detection of more refined symplectic and contact geometric structures (e.g., plastikstufe, $PS$-structures) (Colin et al., 2020).

7. Significance and Interdisciplinary Connections

HDHF bridges symplectic topology, low-dimensional and high-dimensional topological invariants, and the representation theory of quantum groups and Hecke-type algebras. The framework provides a symplectic/Fukaya-categorical realization of quantum algebras arising in topology and physics and connects to mirror symmetry and coherent Springer theory (Honda et al., 2022).

The quantized nonabelianization maps constructed via HDHF mark a substantial technical advance over previous braid or skein-theoretic approaches, leveraging higher-dimensional holomorphic curve counting on real spectral Lagrangians—within reach only through recent progress in symplectic geometry and adiabatic analysis (Honda et al., 22 Jan 2026).

HDHF establishes a robust dictionary from holomorphic polygon counts in symplectic geometry to algebraic structures controlling braid and link invariants, laying foundational tools for further exploration in geometry, topology, and mathematical physics.