Chen-Ruan cohomology of ADE singularities

TL;DR

This paper investigates Ruan's cohomological crepant resolution conjecture for orbifolds with ADE singularities, computing Chen-Ruan cohomology and quantum corrected cohomology rings, and verifying the conjecture in specific cases.

Contribution

It explicitly computes Chen-Ruan cohomology rings for ADE singularities and proves a modified form of Ruan's conjecture in the A2 case.

Findings

Verified Ruan's conjecture for A1 singularities.

Constructed explicit isomorphism between cohomology rings in A1 case.

Proposed and proved a modified conjecture for A2 case.

Abstract

We study Ruan's \textit{cohomological crepant resolution conjecture} for orbifolds with transversal ADE singularities. In the $A_n$-case we compute both the Chen-Ruan cohomology ring $H^*_{\rm CR}([Y])$ and the quantum corrected cohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between $H^*_{\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case, verifying Ruan's conjecture. In the $A_n$-case, the family $H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that the conjecture should be slightly modified. We propose a new conjecture in the $A_n$-case which we prove in the $A_2$-case by constructing an explicit isomorphism.