Centres and semisimple deformations of contraction algebras

Abstract

Let $A_{\rm{con}}$ be the contraction algebra of a $3$-fold flop of an irreducible curve, with genus-zero Gopakumar-Vafa invariants $n_1, \dots, n_l$. We show that $\prod_j M_j(\mathbb{C})^{n_j}$ is the only semisimple algebra to which $A_{\rm{con}}$ deforms. Hence $A_{\rm{con}}$ encodes its Gopakumar-Vafa invariants intrinsically, answering a question of Wemyss. Hua and Toda showed that this deformation exists, and Jelisiejew and Smoktunowicz proved its uniqueness in types A and D. The key step in our proof is that the centre of $A_{\rm{con}}$ has dimension $n_1 + \dots + n_l$. To prove this, we lift the contraction algebra of the universal flop to the Cartan subalgebra, where it becomes a corner of a deformed preprojective algebra. We show that the universal contraction algebra and its zeroth Hochschild homology are free over a polynomial ring, with presentations whose determinants are products of roots. Pulling these presentations back along the arc that classifies the flop gives the count, and recovers Toda's formula for $\dim A_{\rm{con}}$. We also prove Hua and Toda's flatness conjecture, and determine which Gopakumar-Vafa invariants occur in lengths 3 to 6.