Contraction algebra and singularity of three-dimensional flopping contraction

In [5], Donovan and Wemyss introduced the contraction algebra of flopping curves in 3-folds. They conjectured that the contraction algebra determines the formal neighborhood of the underlying singularity of the contraction. In this paper, we prove that the contraction algebra together with its natural $A_\infty$-structure constructed in [10], determine the formal neighborhood of the singularity.