Geometric realisations of type $\tilde{A}_n$ preprojective algebras in homological mirror symmetry

TL;DR

This paper explores the geometric realization of type A preprojective algebras within homological mirror symmetry, establishing equivalences between symplectic and algebraic categories through hyper-Ke4hler geometry and mirror symmetry techniques.

Contribution

It introduces a geometric framework linking A preprojective algebras to hyper-Ke4hler manifolds and demonstrates homological mirror symmetry for these structures, including computations of Fukaya and derived categories.

Findings

Wrapped Fukaya categories match algebraic categories of preprojective algebras.

Homological mirror symmetry holds after adding divisors and stops.

Categories are triangulated equivalent to modules over A preprojective algebras.

Abstract

The type $A_n$-singularity $\mathbb{C}^2/\mathbb{Z}_{n+1}$ can be resolved by hyper-K\"ahler manifolds $X_{\zeta}$ with underlying smooth manifolds diffeomorphic to the resolution of singularities $X_{\text{res}}$, whose hyper-K\"ahler structure depends on a parameter $\zeta\in H_2(X_{\text{res}};\mathbb{R})$. The structure as a complex manifold of each such hyper-K\"ahler manifold is equivalent to the resolution of singularities at the poles and the structure of a Milnor fibre with roots determined by $\zeta$ elsewhere; the symplectic structure is exact along the equator and is deformed by areas depending on $\zeta$ on the exceptional $(-2)$-spheres away from the equator. We show that removing suitable divisors $D_u$ from a fixed $X_{\zeta}$ varying with $u$ in the underlying upper hemisphere of the $S^2$-family of K\"ahler-structures yields a log Calabi--Yau hyper-K\"ahler family (in particular a family of log Calabi--Yau submanifolds), and that mirror symmetry is satisfied (partly conjectural in one direction) for this family by hyper-K\"ahler rotation, in particular by interchanging the structures over the equator and the pole. We furthermore show homological mirror symmetry after adding the missing divisors, which is related to attaching stops and computing singularity categories of certain Landau--Ginzburg potentials on the $A$-side and $B$-side, respectively. More concretely: we compute wrapped Fukaya categories and compare them with (previous and new) computations of derived categories of coherent sheaves and derived categories of singularities in algebraic geometry. We show that the relevant categories (with two exceptions) are triangulated equivalent to module categories over the additive and the multiplicative preprojective algebras of type $\tilde{A}_n$, or to deformations of these algebras depending on the parameters $\zeta$.