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Morse theory of loop spaces and Hecke algebras

Ko Honda, Roman Krutowski, Yin Tian, Tianyu Yuan

TL;DR

This work constructs the based multiloop $A___$-algebra $CM_{-*}(\,oldsymbol{q} ight)$ from Morse theory on based loop configurations, viewing it as a graded deformation of the braid skein algebra. The authors establish an $A___$-equivalence with the wrapped high-dimensional Heegaard Floer algebra of $oldsymbol{q}$-based cotangent fibers after a base-change by a deformation parameter $$ (or its $$-adic completion when $n>2$). They prove that, for closed surfaces not equal to $S^2$, the based multiloop algebra recovers the braid skein algebra, while for $M=S^2$ the algebra is quasi-equivalent to a derived Hecke-type dga $H_$. A key part of the work is the construction of a linear map $oldsymbol{}$ from wrapped HDHF to $CM_{-*}(\,oldsymbol{q} ight)$ via mixed moduli spaces, together with a direct-limit argument showing the $S^2$ case reduces to explicit presentations and higher operations vanish under Katok-type metrics. The results illuminate a deformation- and derived-topology perspective on Hecke algebras and suggest a path toward a derived HOMFLYPT skein framework via factorization homology. The approach blends Morse theory, pseudo-gradients, and sophisticated holomorphic/Morse moduli spaces to build and compare $A_$-structures across geometric contexts.

Abstract

Given a smooth closed $n$-manifold $M$ and a $κ$-tuple of basepoints $\boldsymbol{q}\subset M$, we define a Morse-type $A_\infty$-algebra $CM_{-*}(Ω(M,\boldsymbol{q}))$, called the based multiloop $A_\infty$-algebra, as a graded generalization of the braid skein algebra due to Morton and Samuelson. For example, when $M=T^2$ the braid skein algebra is the Type A double affine Hecke algebra (DAHA). The $A_\infty$-operations couple Morse gradient trees on a based loop space with Chas-Sullivan type string operations. We show that, after a certain "base change", $CM_{-*}(Ω(M,\boldsymbol{q}))$ is $A_\infty$-equivalent to the wrapped higher-dimensional Heegaard Floer $A_\infty$-algebra of $κ$ disjoint cotangent fibers which was studied in the work of Honda, Colin, and Tian. We also compute the based multiloop $A_\infty$-algebra for $M=S^2$, which we can regard as a derived Hecke algebra of the $2$-sphere.

Morse theory of loop spaces and Hecke algebras

TL;DR

This work constructs the based multiloop -algebra from Morse theory on based loop configurations, viewing it as a graded deformation of the braid skein algebra. The authors establish an -equivalence with the wrapped high-dimensional Heegaard Floer algebra of -based cotangent fibers after a base-change by a deformation parameter (or its -adic completion when ). They prove that, for closed surfaces not equal to , the based multiloop algebra recovers the braid skein algebra, while for the algebra is quasi-equivalent to a derived Hecke-type dga . A key part of the work is the construction of a linear map from wrapped HDHF to via mixed moduli spaces, together with a direct-limit argument showing the case reduces to explicit presentations and higher operations vanish under Katok-type metrics. The results illuminate a deformation- and derived-topology perspective on Hecke algebras and suggest a path toward a derived HOMFLYPT skein framework via factorization homology. The approach blends Morse theory, pseudo-gradients, and sophisticated holomorphic/Morse moduli spaces to build and compare -structures across geometric contexts.

Abstract

Given a smooth closed -manifold and a -tuple of basepoints , we define a Morse-type -algebra , called the based multiloop -algebra, as a graded generalization of the braid skein algebra due to Morton and Samuelson. For example, when the braid skein algebra is the Type A double affine Hecke algebra (DAHA). The -operations couple Morse gradient trees on a based loop space with Chas-Sullivan type string operations. We show that, after a certain "base change", is -equivalent to the wrapped higher-dimensional Heegaard Floer -algebra of disjoint cotangent fibers which was studied in the work of Honda, Colin, and Tian. We also compute the based multiloop -algebra for , which we can regard as a derived Hecke algebra of the -sphere.

Paper Structure

This paper contains 34 sections, 43 theorems, 273 equations, 23 figures.

Table of Contents

  1. Introduction
  2. The based multiloop $A_\infty$-algebra
  3. The Lagrangian action functional and pseudogradient vector fields
  4. The switching map
  5. Multidiagonals
  6. Construction of reparametrizations
  7. Switching
  8. The based multiloop complex
  9. The $A_\infty$-structure
  10. The isomorphism with HDHF
  11. Review of HDHF
  12. Half-disk Hurwitz spaces
  13. The linear map $\mathcal{F}^1$
  14. Mixed moduli spaces $\mathcal{P}_r(\bm y,\bm \gamma)$
  15. Ghost bubbles and Kuranishi replacements
  16. ...and 19 more sections

Key Result

Theorem 1.3

$\mathcal{F}$ is an $A_\infty$-equivalence when $n=2$. When $n>2$ it induces an $A_\infty$-equivalence of $\hbar$-adic completions.

Figures (23)

  • Figure 2.1: Here we schematically illustrate the effect of the switching map $sw^{ij}_{I_1}$ on the left tuple of paths, where we have locally drawn $3$ out of $\kappa$ paths. All paths except $\gamma_i$ and $\gamma_j$ are unaffected, and the portion of $\gamma_i$ after the intersection point at $\theta$ gets replaced with the reparametrized portion of $\gamma_j$ and vice versa. Dashed lines depict the portions of $\gamma_i$ and $\gamma_j$ affected by the smoothing reparametrization.
  • Figure 2.2: Here we depict the perturbation scheme for an MFLS with $3$ switches, marked by red crosses. Away from the location of switching markers the trajectories are the flow lines of the fixed pseudogradient vector field $X$. Near the switches they are flow lines of the perturbed vector fields $X+Y_i$. Note that $Y_i$ for $i=1, \dots, \ell-1$ is nonzero near the $i$th and $(i+1)$st switching markers as depicted.
  • Figure 2.3: A schematic depiction of an MFLS with $2$ switchings. As before the red crosses denote switches and the thick blue segments denote the perturbed parts of the trajectories.
  • Figure 2.4:
  • Figure 2.5: Schematic depiction of an MFTS with $2$ switchings contributing to $[\bm \gamma_0]\hbar^2$ in $\mu^5_M([\bm \gamma_5],[\bm \gamma_4],[\bm \gamma_3], [\bm \gamma_2], [\bm \gamma_1])$. As before we depict in blue the areas where additional perturbations are turned on.
  • ...and 18 more figures

Theorems & Definitions (129)

  • Remark 1.1: Remark on ground rings
  • Remark 1.2: Deformation parameter $\hbar$ and base change
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • proof
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8
  • ...and 119 more