Morse theory of loop spaces and Hecke algebras

2025/03/10 by Honda, Ko, Krutowski, Roman, Tian, Yin +1
#53D40 (Primary) 55P50 #57K31 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2503.07543

Abstract

Given a smooth closed n-manifold M and a κ-tuple of basepoints \boldsymbolq⊂ M, we define a Morse-type A_∞-algebra CM_-*(Ω(M,\boldsymbolq)), called the based multiloop A_∞-algebra, as a graded generalization of the braid skein algebra due to Morton and Samuelson. For example, when M=T² the braid skein algebra is the Type A double affine Hecke algebra (DAHA). The A_∞-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,\boldsymbolq)) is A_∞-equivalent to the wrapped higher-dimensional Heegaard Floer A_∞-algebra of κ disjoint cotangent fibers which was studied in the work of Honda, Colin, and Tian. We also compute the based multiloop A_∞-algebra for M=S², which we can regard as a derived Hecke algebra of the 2-sphere.

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