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

TL;DR

This paper establishes a connection between higher-dimensional Heegaard Floer homology and the polynomial representation of double affine Hecke algebras, providing new topological insights into Cherednik's inner product.

Contribution

It demonstrates that certain Floer homologies recover the polynomial representation of double affine Hecke algebras and offers a topological interpretation of Cherednik's inner product.

Findings

Floer homology recovers polynomial representations of DAHA

Topological interpretation of Cherednik's inner product

Connection between Floer theory and algebraic structures

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.