Noncommutative crepant resolutions of $cA_n$ singularities via Fukaya categories

TL;DR

This paper establishes a mirror symmetry correspondence between wrapped Fukaya categories of a cylinder with marked points and categories of perfect complexes on crepant resolutions of $cA_n$ singularities, revealing new geometric and algebraic structures.

Contribution

It computes the wrapped Fukaya category of a punctured cylinder and proves a mirror equivalence with derived categories of crepant resolutions of $cA_n$ singularities, introducing braid group actions and geometric models.

Findings

Mirror equivalence between Fukaya categories and perfect complexes

Braid group actions on derived categories

Geometric model of the contraction algebra

Abstract

We compute the wrapped Fukaya category $\mathcal{W}(T^*S^1, D)$ of a cylinder relative to a divisor $D= \{p_1,\ldots, p_n\}$ of $n$ points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over $k[t_0,\ldots, t_n]$) of the singularity $uv=t_0t_1\ldots t_n$. Upon making the base-change $t_i= f_i(x,y)$, we obtain the derived category of any crepant resolution of the $cA_{n}$ singularity given by the equation $uv= f_0\ldots f_n$. These categories inherit braid group actions via the action on $\mathcal{W}(T^*S^1,D)$ of the mapping class group of $T^*S^1$ fixing $D$. We also give a geometric model of the derived contraction algebra of a $cA_n$ singularity in terms of the relative Fukaya category of the disc.