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

TL;DR

This paper proves Wemyss's conjecture that contraction algebras can be deformed to a semisimple algebra, providing a new intrinsic method for calculating Gopakumar-Vafa invariants of 3D flops.

Contribution

It establishes the conjecture for types A and D and introduces new techniques for constructing and comparing algebra deformations.

Findings

Proof of Wemyss' conjecture for types A and D

Development of new methods for flat algebra deformations

Proposal of two new conjectures for deeper theory

Abstract

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.