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 ${\mathbb {A}}$ 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 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 $\text {S}_{\bullet }$ -construction, a simplicial space whose geometric realisation provides the d-fold delooping of the connective algebraic K-theory space of the ring of coefficients.

Homological mirror symmetry for higher-dimensional pairs of pants

Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.

Computation of Quantum Cohomology From Fukaya Categories

Assume the existence of a Fukaya category $\textrm{Fuk}(X)$ of a compact symplectic manifold $X$ with some expected properties. In this paper, we show $\mathscr{A} \subset \textrm{Fuk}(X)$ split generates a summand $\textrm{Fuk}(X)_e \subset \textrm{Fuk}(X)$ corresponding to an idempotent $e \in QH^{\bullet }(X)$ if the Mukai pairing of $\mathscr{A}$ is perfect. Moreover, we show $HH^{\bullet }(\mathscr{A}) \cong QH^{\bullet }(X) e$. As an application, we compute the quantum cohomology and the Fukaya category of a blow-up of $\mathbb{C} P^2$ at four points with a monotone symplectic structure.

Relative Calabi–Yau structures

We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.