#### Vol. 2, No. 4, 2008

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Root systems and the quantum cohomology of ADE resolutions

### Jim Bryan and Amin Gholampour

Vol. 2 (2008), No. 4, 369–390
##### Abstract

We compute the ${ℂ}^{\ast }$-equivariant quantum cohomology ring of $Y$, the minimal resolution of the DuVal singularity ${ℂ}^{2}∕G$ where $G$ is a finite subgroup of $SU\left(2\right)$. The quantum product is expressed in terms of an ADE root system canonically associated to $G$. We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an $n$ parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov–Witten potential of $\left[{ℂ}^{2}∕G\right]$.

##### Keywords
quantum cohomology, root system, ADE
Primary: 14N35
##### Milestones
Received: 10 August 2007
Revised: 9 May 2008
Accepted: 9 May 2008
Published: 15 June 2008
##### Authors
 Jim Bryan 1984 Mathematics Road Vancouver, BC V6T 1Z2 Canada http://www.math.ubc.ca/~jbryan/ Amin Gholampour Mathematics 253-37 Caltech Pasadena, CA 91125 United States http://www.its.caltech.edu/~agholamp/