Morse theory of loop spaces and Hecke algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913728241336320 |
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| author | Honda, Ko Krutowski, Roman Tian, Yin Yuan, Tianyu |
| author_facet | Honda, Ko Krutowski, Roman Tian, Yin Yuan, Tianyu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07543 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Morse theory of loop spaces and Hecke algebras Honda, Ko Krutowski, Roman Tian, Yin Yuan, Tianyu Symplectic Geometry Algebraic Topology Geometric Topology Quantum Algebra 53D40 (Primary) 55P50, 57K31 (Secondary) 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. |
| title | Morse theory of loop spaces and Hecke algebras |
| topic | Symplectic Geometry Algebraic Topology Geometric Topology Quantum Algebra 53D40 (Primary) 55P50, 57K31 (Secondary) |
| url | https://arxiv.org/abs/2503.07543 |