We define a higher-dimensional analogue of symplectic Khovanov homology.
Consider the standard Lefschetz fibration p:W→D⊂C of
a 2n-dimensional Milnor fiber of the A2κ−1 singularity. We
represent a link by a κ-strand braid, which is expressed as an element
h of the symplectic mapping class group Symp(W,∂W). We
then apply the higher-dimensional Heegaard Floer homology machinery to the pair
(a,h(a)), where a is a collection of
κ 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