Higher-Dimensional Heegaard Floer Homology

• Higher-dimensional Heegaard Floer homology is a symplectic invariant defined via counts of pseudoholomorphic curves in cotangent bundles and Weinstein domains.
• It encodes explicit A∞ operations on chain complexes that correspond to polynomial representations of DAHA and Hecke algebras.
• The framework underpins applications in contact topology, link invariants, and symplectic cohomology, bridging geometry with representation theory.

Higher-dimensional Heegaard Floer homology (HDHF) is a symplectic topological invariant defined using counts of pseudoholomorphic curves in higher-dimensional symplectic manifolds, generalizing the classical Heegaard Floer homology constructed for three-manifolds. In recent years, HDHF has been systematically developed as an invariant of Lagrangian tuples in cotangent bundles of surfaces and higher-dimensional Weinstein domains, leading to new connections between symplectic topology, representation theory, Hecke algebras, and link invariants (Gao et al., 9 Nov 2025, Colin et al., 2020, Yuan, 2023, Honda et al., 2022, Tian et al., 2022, Krutowski et al., 2023). HDHF associates algebraic structures—often with explicit $A_\infty$ or graded algebra presentations—to configurations of cotangent fibers and conormal bundles, and exhibits deep relationships with polynomial representations of double affine Hecke algebras (DAHA) and their inner products.

1. Geometric and Symplectic Foundation

HDHF is formulated on symplectic manifolds $(X, \omega)$ constructed as cotangent bundles $T^*\Sigma$ of closed, orientable surfaces $\Sigma$ or higher-dimensional Weinstein domains $(W, \beta, \phi)$. Typical Lagrangian submanifolds of interest are cotangent fibers $T^*_{q_i}\Sigma$ at $n$ distinct basepoints $q = \{q_1, \ldots, q_n\}$, or the conormal bundle $N^*\alpha$ of a nontrivial simple closed curve $\alpha \subset \Sigma$. The standard choice of $\Sigma$ is the two-torus $T^2$ with points identified as $\Sigma = S^1_x \times S^1_y$, with $\alpha$ given by $x=0$ and $q \subset \alpha \setminus \{\star\}$.

Pseudoholomorphic curve theory is essential; holomorphic maps $u: (\Sigma, j) \to (X, J)$ are solved with boundary conditions on tuples of exact Lagrangians $L_i$ and $L_\infty$. The choice of quadratic-at-infinity Hamiltonians $H_V$ ensures non-degeneracy of Hamiltonian chords and rigid moduli spaces. Transversality is achieved via Sard-Smale perturbations and domain-dependent almost complex structures compatible under gluing (Gao et al., 9 Nov 2025, Honda et al., 2022).

2. Algebraic Structure: Chain Complexes and $A_\infty$ Operations

HDHF chain complexes $CF^*(L_1 \times \cdots \times L_n, L_\infty)$ are generated by $n$-tuples of time-$1$ Hamiltonian chords $x = (x_1, \ldots, x_n)$, each solving $\dot{x}_i = X_{H_V}(t, x_i(t))$ with $x_i(0) \in L_i$, $x_i(1) \in L_\infty$. In the generic setup, all chords are transverse and have zero Maslov grading.

The differential and higher products are encoded as an $A_\infty$ module over $$CF^*(L_1 \times \cdots \times L_n, L_1 \times \cdots \times L_n)$$, formed by counting rigid holomorphic branched covers in moduli spaces $D_m \times T^* \Sigma$. For each $m \geq 1$, disks with $m+1$ boundary punctures map to $T^* \Sigma$ with boundary conditions alternating between the $L_i$ and $L_\infty$, and projections to $D_m$ required to be degree-$n$ branched covers. The resulting operations $\mu^m$ are defined as

$$\mu^m: CF(L_1 \times \cdots \times L_n, L_\infty) \otimes CF(\ldots)^{\otimes (m-1)} \rightarrow CF(L_1 \times \cdots \times L_n, L_\infty),$$

with the differential corresponding to $m=1$. For HDHF with cotangent fibers and conormal bundles on $T^2$, only $\mu^2$ survives at the chain level, yielding a differential graded module with homology $HW^*(T^* \Sigma; L_1, \ldots, L_n; L_\infty)$ supported in degree $0$ (Gao et al., 9 Nov 2025, Honda et al., 2022, Tian et al., 2022).

3. Representation-Theoretic Identification: DAHA and Hecke Algebras

A central achievement of HDHF is its explicit equivalence with polynomial representations of double affine Hecke algebras (DAHA) of type $A$ (Gao et al., 9 Nov 2025). The DAHA ${DAHA}_{q,t}(GL_n)$ is generated over $\mathbb{C}[t^{\pm 1}, q^{\pm 1}]$ by elements $T_i$, $X_i^{\pm 1}$, $Y_i^{\pm 1}$ subject to braid and quadratic relations and specific commutation rules

$$(T_i - t)(T_i + t^{-1}) = 0, \quad X_i X_j = X_j X_i, \quad Y_i Y_j = Y_j Y_i, \quad T_i X_i T_i = X_{i+1}, \ldots$$

The module $P_n$ consists of Laurent polynomials $$\mathbb{C}[t^{\pm 1}, q^{\pm 1}][x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$$ with \begin{align*} X_i \cdot f(x) & = x_i f(x), \ T_i \cdot f(x) & = t^{1/2} s_i(f) + (t^{1/2} - t^{-1/2})\frac{f - s_i(f)}{x_i - x_{i+1}}, \ Y_i \cdot f(x) & = q^{x_i \partial_{x_i}} f(x), \end{align*} where $s_i$ exchanges $x_i \leftrightarrow x_{i+1}$.

On the HDHF side, $HW^*(T^* \Sigma; L_1, \ldots, L_n; L_\infty)$ is shown (Honda–Tian–Yuan; Morton–Samuelson) to be isomorphic to the polynomial representation $P_n$, mapped via "braid skein" arguments to the braid-skein algebra of the surface. For $\Sigma = T^2$, this algebra equals $\mathrm{DAHA}_{q,t}$ by algebraic skein-theoretic calculation. The chain-level $A_\infty$-module matches induced module relations in $P_n$, with geometric operations (e.g., chord slides) producing DAHA generator relations (Gao et al., 9 Nov 2025, Honda et al., 2022, Tian et al., 2022).

4. Geometric Realization of Cherednik Inner Product

Cherednik's inner product on $P_n$,

$\langle f, g \rangle$

is characterized by Macdonald polynomial orthogonality and the involutions $T_i^* = T_i^{-1}$, $X_i^* = X_i^{-1}$, $Y_i^* = Y_i^{-1}$. The form satisfies $$\langle x^\mu, x^\lambda \rangle = \delta_{\mu, \lambda} c_\mu(q,t)$$ for normalized constants $c_\mu(q, t)$.

HDHF geometrically incarnates this Hermitian product via the symplectic pairing of HDHF modules, realized as cap-cup compositions in the Floer category: modules for $(q, \alpha)$ are composed with those for $(\alpha, q)$, both ends capped along $\alpha$, and evaluation performed in the skein module for $\alpha \to \alpha$. Rigid holomorphic curves in this topology correspond precisely to the codimension-one degenerations enforcing adjointness relations $\langle H f, g \rangle = \langle f, H^*g \rangle$ for any DAHA generator, with $\langle 1, 1 \rangle = 1$ from the trivial strip (Gao et al., 9 Nov 2025).

5. Main Theorems and Generalizations

Two principal results characterize the HDHF-DAHA relationship:

Theorem A (Gao–Reisin-Tzur–Tian–Yuan):

Let $\Sigma = T^2$, $\alpha \subset \Sigma$ a nontrivial loop, $q = \{q_i\} \subset \alpha$, $L_\infty = N^* \alpha$, $L_i = T^*_{q_i}\Sigma$, then

$$HF^*(L_1 \times \cdots \times L_n, L_\infty) \cong P_n$$

as modules over $\mathrm{DAHA}_{q,t}(GL_n)$.

Theorem B:

Under this isomorphism, the Floer-geometric cap-cup pairing matches Cherednik's inner product.

The construction requires genus$(\Sigma) \geq 1$ to ensure transversality and well-defined gradings. Further, it generalizes to higher genus surfaces (yielding modules over surface-DAHA) and to other Lie types via conormals of webs or branes in cotangent bundles. Extension to Morse-theoretic or topological Fukaya models is proposed, paralleling Nadler–Zaslow's approach for Riemann surfaces (Gao et al., 9 Nov 2025, Honda et al., 2022).

6. Related Constructions and Applications

Contact Topology: HDHF forms the basis for constructing contact invariants, such as the contact class $c(\xi)$ in the framework of open-book decompositions, providing obstructions to Liouville fillability and verifying the Weinstein conjecture (Colin et al., 2020).

Link Invariants: Higher-dimensional analogues of symplectic Khovanov homology have been defined using HDHF machinery on Milnor fibers of $A_{2\kappa-1}$ singularities. Link invariance is established by arc-slide and Markov stabilization invariance, and explicit computations recover classical link homologies in graded form (Yuan, 2023).

Hecke Algebra Realizations: HDHF of cotangent fibers for disks and general Riemann surfaces yields isomorphisms with finite, affine, and double affine Hecke algebras of type $A$ or $GL_n$, via explicit identification of products and generators in Floer homology (Tian et al., 2022, Honda et al., 2022).

Symplectic Cohomology and Viterbo Theorem: The closed-string analogue of HDHF (Heegaard Floer symplectic cohomology) generalizes Viterbo's theorem to multiloop complexes and extends ordinary symplectic cohomology to the $\kappa$-particle context (Krutowski et al., 2023).

7. Outlook and Open Directions

Active directions include formulating HDHF for higher genus surfaces and corresponding surface DAHAs, extending to other Lie algebra types via generalized conormals, constructing Morse-theoretic models, and categorification via Fukaya categories of Hilbert schemes. Integral algebraic structures and potential relationships with quantum group categorifications are suggested. The explicit topological realization of representation-theoretic entities marks HDHF as a central structure at the interface of symplectic geometry, algebraic topology, and modern representation theory (Gao et al., 9 Nov 2025, Honda et al., 2022).

Context	HDHF Output	Algebraic Identification
$T^2$, cotangent fibers + conormal	$HF^*(L_1 \times \cdots \times L_n, L_\infty)$	Polynomial DAHA module $P_n$
Disk, disjoint fibers	$HF^*(\sqcup_i T^*_{q_i} D^2)$	Hecke algebra $H_\kappa$
Closed surface genus $g>0$	$HW^*(\sqcup_i T^*_{q_i} \Sigma)$	Surface Hecke algebra $\mathcal{H}_\kappa(\Sigma)$

This framework establishes higher-dimensional Heegaard Floer homology as a powerful invariant unifying Floer-theoretic, representation-theoretic, and quantum algebraic structures.