Noncommutative crepant resolutions of $c{A}_n$ singularities via Fukaya categories

We compute the wrapped Fukaya category $\mathcal{W}(T^{\ast }{S}^1,D)$ of a cylinder relative to a divisor $D=\{p_1,\dots ,p_n\}$ of $n$ points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over $k[t_0,\dots ,t_n]$) of the singularity $uv=t_0{t}_1\dots t_n$. Upon making the base-change $t_i=f_i(x,y)$, we obtain the derived category of any crepant resolution of the $c{A}_n$ singularity given by the equation $uv=f_0\dots f_n$. These categories inherit braid group actions via the action on $\mathcal{W}(T^{\ast }{S}^1,D)$ of the mapping class group of $T^{\ast }{S}^1$ fixing $D$. We also give a geometric model of the derived contraction algebra of a $c{A}_n$ singularity in terms of the relative Fukaya category of the disc.