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

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 $S$-construction, a simplicial space whose geometric realisation provides the $d$-fold delooping of the connective algebraic $K$-theory space of the ring of coefficients.

arXiv comment43 pages, 6 figures

References in corpus (22)

• Flat surfaces and stability structures
• An open string analogue of Viterbo functoriality
• Covariantly functorial wrapped Floer theory on Liouville sectors
• A universal characterization of higher algebraic K-theory
• Bimodules in bordered Heegaard Floer homology
• $n$-abelian and $n$-exact categories
• Triangulated surfaces in triangulated categories
• n-representation-finite algebras and twisted fractionally Calabi-Yau algebras
• On partially wrapped Fukaya categories
• Higher Nakayama algebras I: Construction
• On the representation dimension of artin algebras
• Differential Graded Categories are k-linear Stable Infinity Categories
• Auslander orders over nodal stacky curves and partially wrapped Fukaya categories
• Smoothness of equivariant derived categories
• Simplicial structures in higher Auslander-Reiten theory
• Higher Auslander correspondence for dualizing $R$-varieties
• Homological mirror symmetry for higher dimensional pairs of pants
• Higher Segal structures in algebraic $K$-theory
• Cyclic structures and broken cycles
• On auto-equivalences and complete derived invariants of gentle algebras
• Higher preprojective algebras, Koszul algebras, and superpotentials
• A1-homotopy invariants of topological Fukaya categories of surfaces

Cited by in corpus (4)

• The category of extensions and a characterisation of $n$-exangulated functors
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• On higher torsion classes
• A characterisation of higher torsion classes