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Noncommutative crepant resolutions of $cA_n$ singularities via Fukaya categories. (arXiv:2307.06592v1 [math.SG])

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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 $k[t_1,\ldots, t_n]$) of the singularity $uv=t_1t_2\ldots t_n$. Upon
making the base-change $t_i= f_i(x,y)$, we obtain the derived category of any
crepant resolution of the $cA_{n-1}$ singularity given by the equation $uv=
f_1\ldots f_n$. 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 $D$.

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