Algebra & Number Theory

Root systems and the quantum cohomology of ADE resolutions

Jim Bryan and Amin Gholampour

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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].

Article information

Algebra Number Theory, Volume 2, Number 4 (2008), 369-390.

Received: 10 August 2007
Revised: 9 May 2008
Accepted: 9 May 2008
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

quantum cohomology root system ADE


Bryan, Jim; Gholampour, Amin. Root systems and the quantum cohomology of ADE resolutions. Algebra Number Theory 2 (2008), no. 4, 369--390. doi:10.2140/ant.2008.2.369.

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