Root systems and the quantum cohomology of ADE resolutions

Bryan, Jim; Gholampour, Amin

doi:10.2140/ant.2008.2.369

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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

DOI: 10.2140/ant.2008.2.369

Abstract

We compute the $ℂ^{*}$ -equivariant quantum cohomology ring of $Y$ , the minimal resolution of the DuVal singularity $ℂ^{2} ∕ G$ where $G$ is a finite subgroup of $S U (2)$ . 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 $[ℂ^{2} ∕ G]$ .

Keywords

quantum cohomology, root system, ADE

Mathematical Subject Classification 2000

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/

Vol. 2, No. 4, 2008