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

Joachim Jelisiejew, Agata Smoktunowicz

TL;DR

The paper addresses how contraction algebras arising from 3‑dimensional flops encode GV invariants, proving that contraction algebras of type $A$ and $D$ deform to a single semisimple algebra, thereby enabling GV invariants to be read off intrinsically from the algebra. It develops and compares multiple deformation frameworks—including formal, convergent, polynomially split, and Beauville–Laszlo descent—to establish a unified theory (Theorem J1) and then applies it to obstruction and existence results for semisimple degenerations of type $A$ and $D$ contraction algebras. The authors provide explicit presentations for these algebras, prove semicontinuity constraints on possible semisimple degenerations, and construct explicit deformations (via induction and transitivity) to semisimple targets, culminating in the conclusion that both $A$ and $D$ contraction algebras admit flat deformations to a single semisimple algebra. This advances an intrinsic, deformation-based route to GV invariants and hints at a deeper, unobstructed noncommutative deformation theory for contraction algebras with potential broader implications in noncommutative algebraic geometry. The work also proposes conjectures about unobstructedness and regularity of centers under deformations, supported by computational checks in small cases.

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.

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

TL;DR

The paper addresses how contraction algebras arising from 3‑dimensional flops encode GV invariants, proving that contraction algebras of type and deform to a single semisimple algebra, thereby enabling GV invariants to be read off intrinsically from the algebra. It develops and compares multiple deformation frameworks—including formal, convergent, polynomially split, and Beauville–Laszlo descent—to establish a unified theory (Theorem J1) and then applies it to obstruction and existence results for semisimple degenerations of type and contraction algebras. The authors provide explicit presentations for these algebras, prove semicontinuity constraints on possible semisimple degenerations, and construct explicit deformations (via induction and transitivity) to semisimple targets, culminating in the conclusion that both and contraction algebras admit flat deformations to a single semisimple algebra. This advances an intrinsic, deformation-based route to GV invariants and hints at a deeper, unobstructed noncommutative deformation theory for contraction algebras with potential broader implications in noncommutative algebraic geometry. The work also proposes conjectures about unobstructedness and regularity of centers under deformations, supported by computational checks in small cases.

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.
Paper Structure (13 sections, 20 theorems, 58 equations)

This paper contains 13 sections, 20 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Deformations of finite-dimensional algebras
  3. Algebraic geometry context
  4. Outline of the paper
  5. Deformations of finite-dimensional associative algebras
  6. Moduli space of finite-dimensional algebras
  7. Zariski topology on the moduli space
  8. Comparing various notions of deformations
  9. From $\Bbbk[\![t]\!]$ to $\Bbbk[t]$ by descent
  10. Contraction algebras
  11. Description of contraction algebras of type A and D
  12. Obstructions to deformations to semisimple algebras
  13. Existence of deformations of $D_{m,n}, D_{n, \infty}$ to semisimple algebras

Key Result

Theorem 1.3

Contraction algebras of type $A$ and $D$ deform to a single semisimple algebra, as listed in Theorem ref:obstructionsMain:thm.

Theorems & Definitions (51)

  • Theorem 1.3: GV invariants are determined by contraction algebras, Theorem \ref{['ref:obstructionsMain:thm']}, Theorem \ref{['ref:existence:thm']}
  • Conjecture 1.4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5: Adding roots of $t$
  • proof
  • Example 2.6
  • Proposition 2.7
  • ...and 41 more