On a conjecture by Michael Wemyss regarding the calculation of GV invariants

Contraction algebras are noncommutative algebras introduced by Donovan and Wemyss to classify of 3-dimensional flops. Wemyss conjectures that contraction algebras can be deformed to a single semisimple algebra. This gives an intrinsic method of calculating Gopakumar-Vafa invariants of the flop. The main result is a proof of Wemyss’ conjecture for types A and D. In the course of the proof, we recall and introduce new techniques for constructing flat deformations of associative algebras and compare various notions of deformations. We also put forward two conjectures which hint towards a deeper theory.

💡 Research Summary

The paper addresses a conjecture of Michael Wemyss concerning the calculation of Gopakumar‑Vafa (GV) invariants via contraction algebras, a class of finite‑dimensional non‑commutative algebras introduced by Donovan and Wemyss to classify three‑dimensional flops. Wemyss proposed that every contraction algebra can be deformed to a single semisimple algebra of the form (k \times M_{1}(k)^{n_{1}} \times M_{2}(k)^{n_{2}} \times \dots), where the integers (n_{i}) are precisely the GV invariants associated to the flopped curve. If true, this would give an intrinsic algebraic method for extracting GV invariants directly from the abstract algebra (A), without reference to the underlying geometry.

The authors prove this conjecture for the two families of contraction algebras arising from type A and type D flops. Their approach is two‑fold. First, they develop a systematic treatment of deformations of finite‑dimensional associative algebras, comparing several notions that appear in the literature: formal deformations over (k