We show that the perfect derived categories of Iyama’sd-dimensional Auslander algebras of type${\mathbb {A}}$are equivalent to the partially wrapped Fukaya categories of thed-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 thed-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 by-product of our results, we deduce that the partially wrapped Fukaya categories associated to thed-fold symmetric product of the disk organise into a paracyclic object equivalent to thed-dimensional Waldhausen$\text {S}_{\bullet }$-construction, a simplicial space whose geometric realisation provides thed-fold delooping of the connective algebraicK-theory space of the ring of coefficients.

中文翻译：

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高等 Auslander 代数的辛几何：圆盘的对称积

我们证明了 Iyama 的完美派生类别d维 Auslander 类型的代数${\mathbb {A}}$相当于部分包裹的深谷类别d- 的折叠对称乘积$2$维单位圆盘，在其边界上具有有限多个停靠点。此外，我们观察到 Koszul 对偶性在与d- 盘的对称乘积和它的$(nd)$-折叠对称产品；这一观察导致了贝克特定理的辛证明，该定理涉及在相应的较高 Auslander 类型代数之间导出的 Morita 等价性${\mathbb {A}}$. 作为我们结果的副产品，我们推断部分包裹的 Fukaya 类别与d- 圆盘的折叠对称积组织成一个等效于d维瓦尔德豪森$\text {S}_{\bullet }$-构造，一个单纯的空间，其几何实现提供了d- 连接代数的折叠去环ķ- 系数环的理论空间。