Fukaya categories in Koszul duality theory

TL;DR

This paper introduces a method to compute $A_{ abla}$-Koszul duals for directed $A_{ abla}$-categories using Fukaya categories and Dehn twists, revealing detailed algebraic structures.

Contribution

It provides a concrete formula for $A_{ abla}$-Koszul duals of path algebras with directed $A_n$-type quivers via Fukaya categories and Dehn twists, linking algebra and symplectic geometry.

Findings

Explicit formula for $A_{ abla}$-Koszul duals of path algebras

Unveiling of Ext groups and higher compositions

Construction of directed subcategories in Fukaya categories

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.