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.

49pages, 28 figures

References in corpus (1)

• On the Fukaya-Seidel categories of surface Lefschetz fibrations