Title: Fukaya categories in Koszul duality theory,
Abstract: In this paper, we define $A_{\infty}$-Koszul duals for directed
$A_{\infty}$-categories in terms of twists in their $A_{\infty}$-derived
categories. Then, we compute a concrete formula of $A_{\infty}$-Koszul duals
for path algebras with directed $A_n$-type Gabriel quivers. To compute an
$A_\infty$-Koszul dual of such an algebra $A$, we construct a directed
subcategory of a Fukaya category which are $A_\infty$-derived equivalent to the
category of $A$-modules and compute Dehn twists as twists. The formula unveils
all the ext groups of simple modules of the parh algebras and their higher
composition structures.