Non-commutative crepant resolutions of $cA_{n}$ singularities via Fukaya categories

Evans, Jonny and Lekili, Yankı (2025) Non-commutative crepant resolutions of $cA_{n}$ singularities via Fukaya categories. Documenta Mathematica. ISSN 1431-0635

Abstract

We compute the wrapped Fukaya category \mathcal{W}(T^{*}S^{1}, D) of a cylinder relative to a divisor D= \{p_{0},\dots,p_{n}\} of n+1 points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over k\llbracket t_{0},\dots,t_{n}\rrbracket ) of the singularity uv=t_{0}t_{1}\cdots 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}\cdots 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 geometric models for the derived contraction algebras associated to a cA_{n} singularity in terms of the relative Fukaya category of the disc.