Equivariant Cohomological Crepant Resolution Conjecture for ADE orbifolds

Let $G\subset \mathrm{SL}_2(\mathbb{C})$ be a nontrivial finite subgroup, let $p:X\to \mathbb{C}^2/G$ be the minimal resolution, and equip the spaces with the relevant $\mathbb{C}^*$ -action. After specializing the quantum parameters of $X$ at suitable roots of unity and applying the Bryan–Gholampour change of variables, there is an isomorphism of $\mathbb{C}^*$ -equivariant algebras $QH^*_{\mathbb{C}^*}(X)\cong H^*_{\mathrm{CR},\mathbb{C}^*}([\mathbb{C}^2/G])$ .

References

Primary source

arXiv

Additional references

Equivariant Cohomological Crepant Resolution Conjecture for ADE-orbifolds — arXiv — Qike Li, Fabio Perroni

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September preprint claims to settle the correspondence for all ADE types, but the available evidence does not yet verify the full quantum-theoretic identification.

The conjecture asserts that equivariant quantum cohomology of ADE resolutions matches equivariant Chen–Ruan cohomology of the corresponding orbifolds after root-of-unity specialization and suitable changes of variables. The reported claim covers all ADE types, but its only direct support is an unrefereed preprint.

September 2026 evidence and claimed proof

On September 15, 2026, Qike Li and Fabio Perroni reported that the Bryan–Gholampour transformation establishes the proposed correspondence. A separate 2026 preprint studies cyclic foldings of ADE resolutions, proves specialization and integral-structure compatibility results, and formulates the full identification after analytic continuation; it explicitly says these results do not prove the full quantum-product correspondence.

Current status (as of September 2026): A preprint claims a complete resolution, but the full equivariant quantum correspondence remains unverified and is not established by the independently described folding results.

Sources

No solutions have been posted yet.