  We define a higher-dimensional analogue of symplectic Khovanov homology.
Consider the standard Lefschetz fibration $p\colon W\to D\subset\mathbb{C}$ of
a $2n$-dimensional Milnor fiber of the $A_{2\kappa-1}$ singularity. We
represent a link by a $\kappa$-strand braid, which is expressed 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. This work constitutes part of the author's PhD thesis.
