UCLA

Given a closed oriented Riemann surface $\Sigma$ of genus greater than zero, we construct a map $\mathcal{F}$ from the higher-dimensional Heegaard Floer homology of cotangent fibers of $T^*\Sigma$ to the Hecke algebra associated to $\Sigma$. We show that $\mathcal{F}$ is an isomorphism of algebras. We also define a higher-dimensional analog of symplectic Khovanov homology. Consider the standard Lefschetz fibration $p:W\to D\subset\mathbb{C}$ of a $2n$-dimensional Milnor fiber of the $A_{2\kappa-1}$ singularity. We represent a link by a $\kappa$-strand braid, which in turn is represented as an element $h$ of the symplectic mapping class group $\mathrm{Symp}(W,\partial W)$. We then apply the higher-dimensional Heegaard Floer homology machinery to the pair $(\boldsymbol{a},h(\boldsymbol{a}))$, where $\boldsymbol{a}$ is a collection of $\kappa$ unstable manifolds of $W$ which are Lagrangian spheres. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant.