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 -equivariant quantum cohomology ring of Y , the minimal resolution of the DuVal singularity 2G where G is a finite subgroup of SU(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 [2G].

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/