Morse theory of loop spaces and Hecke algebras

Given a smooth closed $n$-manifold $M$ and a $\kappa$-tuple of basepoints $\boldsymbol{q}\subset M$, we define a Morse-type $A_\infty$-algebra $CM_{-*}(\Omega(M,\boldsymbol{q}))$, 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", $CM_{-*}(\Omega(M,\boldsymbol{q}))$ is $A_\infty$-equivalent to the wrapped higher-dimensional Heegaard Floer $A_\infty$-algebra of $\kappa$ 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.