Homological mirror symmetry for K3 surfaces via moduli of $A_\infty$-structures

We show that several moduli spaces of lattice polarized K3 surfaces tied with Arnold's strange duality (and its generalization due to Berglund-H\"ubsch-Krawitz) arise as moduli spaces of $A_\infty$-structures on particular finite-dimensional graded algebras. The same algebras also appear in the Fukaya category of the mirror dual family. Based on these identifications, we discuss applications to homological mirror symmetry for K3 surfaces, and give a proof of homological mirror symmetry for the affine quartic surface. Along the way, we also give a proof of a conjecture of Seidel from math/0206155 which may be of independent interest.