The symplectic geometry of higher Auslander algebras: Symmetric products of disks

TL;DR

This paper establishes a deep connection between higher Auslander algebras of type A and symplectic geometry by showing their derived categories are equivalent to partially wrapped Fukaya categories of symmetric products of disks, revealing new dualities and structures.

Contribution

It demonstrates an equivalence between derived categories of higher Auslander algebras and symplectic Fukaya categories of symmetric products, and uncovers Koszul duality and categorical structures related to algebraic K-theory.

Findings

Derived categories of higher Auslander algebras are equivalent to Fukaya categories of symmetric products.

Koszul duality induces an equivalence between Fukaya categories of symmetric products of different dimensions.

Partially wrapped Fukaya categories form a paracyclic object related to the Waldhausen S-construction.

Abstract

We show that the perfect derived categories of Iyama's $d$-dimensional Auslander algebras of type $\mathbb{A}$ are equivalent to the partially wrapped Fukaya categories of the $d$-fold symmetric product of the $2$-dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the $d$-fold symmetric product of the disk and those of its $(n-d)$-fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type $\mathbb{A}$. As a byproduct of our results, we deduce that the partially wrapped Fukaya categories associated to the $d$-fold symmetric product of the disk organise into a paracyclic object equivalent to the $d$-dimensional Waldhausen $\operatorname{S}$-construction, a simplicial space whose geometric realisation provides the $d$-fold delooping of the connective algebraic $K$-theory space of the ring of coefficients.