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

We show that the perfect derived categories of Iyama’s d-dimensional Auslander algebras of type 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 A. As a by-product 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-dot-construction, a simplicial space whose geometric realisation provides the d-fold delooping of the connective algebraic K-theory space of the ring of coefficients.