# Auslander orders over nodal stacky curves and partially wrapped Fukaya categories

## TL;DR

This paper establishes an equivalence between the derived categories of modules over Auslander orders on certain nodal stacky curves and partially wrapped Fukaya categories of punctured surfaces, linking algebraic and symplectic geometry.

## Contribution

It introduces a new equivalence connecting algebraic categories of nodal stacky curves with symplectic categories of punctured surfaces, extending known analogies.

## Key findings

- Derived categories of modules over Auslander orders are equivalent to partially wrapped Fukaya categories.
- Derived categories of coherent sheaves on nodal stacky curves correspond to wrapped Fukaya categories.
- Results apply to punctured surfaces of arbitrary genus with boundary stops.

## Abstract

It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409.8611 that for punctured surfaces $X$ with stops $\Lambda$ at their boundary, the partially wrapped Fukaya category $\mathcal{W}(X,\Lambda)$ provides a categorical (smooth and proper) resolution of the compact Fukaya category $\mathcal{F}(X)$. Inspired by this analogy, we establish an equivalence between the derived category of modules over the Auslander orders over certain nodal stacky curves and partially wrapped Fukaya categories associated to punctured surfaces of arbitrary genus equipped with stops at their boundary. As an application, we deduce equivalences between derived categories of coherent sheaves (resp. perfect complexes) on such nodal stacky curves and the wrapped (resp. compact) Fukaya categories of punctured surfaces of arbitrary genus.