Noncommutative crepant resolutions of $cA_n$ singularities via Fukaya categories

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.