分野別セミナー

【講演要旨】Given a smooth manifold $M$ and a k-tuple q of basepoints in $M$, we define a Morse-type
$A_\infty$-algebra $CM(\Omega(M,q))$, called the based multiloop $A_\infty$-algebra, as a graded generalization of the braid skein algebra due to Morton-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, $CM(\Omega(M,q))$ is $A_\infty$-equivalent to the wrapped higher-dimensional
Heegaard Floer $A_\infty$-algebra of k disjoint cotangent fibers.