Higher-dimensional Heegaard Floer homology and the polynomial representation of double affine Hecke algebras

TL;DR

The paper constructs a wrapped higher-dimensional Heegaard Floer framework between the conormal bundle of a single loop $T^*_{\alpha}\Sigma$ and a $\kappa$-tuple of cotangent fibers $T^*_{\mathbf{q}}\Sigma$ to realize the polynomial representation of the type A double affine Hecke algebra $\ddot{H}_{\kappa}$, focusing on $\Sigma=T^2$. It develops a robust link to braid skein modules by proving an isomorphism $HW(T^*\Sigma,\mathbf{q},\alpha,\mathbf{p}_0)^{\varphi} \cong \mathrm{BSk}_{\kappa}(\Sigma,\mathbf{q},\alpha,\mathbf{p}_0) \otimes_{\mathbb{Z}[\hbar]}\mathbb{Z}\llbracket\hbar\rrbracket$ and identifies $\mathrm{BSk}_{\kappa}(T^2,\mathbf{q})$ with $\ddot{H}_{\kappa}$, yielding an explicit isomorphism to the polynomial representation $P_{\kappa}$. A topological interpretation of Cherednik's inner product is obtained via a bilinear form on the skein module induced by a combination of braids, reproducing the defining properties of Cherednik's inner product. The results provide a geometric realization of DAHA representations and suggest avenues toward realizing Cherednik's inner product within Floer theory. Overall, the work connects topological skein theory, Floer homology, and algebraic structures from Macdonald theory in a coherent, computable framework.

Abstract

We show that the higher-dimensional Heegaard Floer homology between tuples of cotangent fibers and the conormal bundle of a homotopically nontrivial simple closed curve on $T^2$ recovers the polynomial representation of double affine Hecke algebra of type A. We also give a topological interpretation of Cherednik's inner product on the polynomial representation.

Figures (11)

• Figure 1: The $A_\infty$ base direction $D_{m}$.
• Figure 2: After resolving at $s_i=-\infty$, the strip like ends $e^1_\lambda\sqcup e^2_\lambda$ is replaced by $e^n_\lambda$.
• Figure 3: $u_\lambda$ restricted to $e^1_\lambda\sqcup e^2_\lambda$. In the upper case, after resolving $u_0$ we get a new family of nodal curves, where $e^1_\lambda\sqcup e^2_\lambda$ is replaced by $e^n_\lambda$.
• Figure 4: As the image of $e^1_\lambda$ and $e^2_\lambda$ are getting closer on the $z_3$-direction, a new family of curves $u^n_\lambda$, $\lambda\geq0$ occurs, where $u^n_0$ is the nodal curve resolved from $u_0$. The projection of $u^n_\lambda$ to $z_i$-direction, $i=1,2$, has a double branched point on the red dashed arc, which moves away from the origin as $\lambda$ increases.
• Figure 5: As the image of $e^1_\lambda$ and $e^2_\lambda$ are getting closer on the $z_3$-direction, there cannot be a new family of resolved curves.

Theorems & Definitions (41)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.1: Morton-Samuelson
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Lemma 3.1
• proof
• Remark 3.2