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Quantum Cohomology of Minuscule Homogeneous Spaces III. Semi-Simplicity and Consequences

  Published:2010-06-18
 Printed: Dec 2010
  • P. E. Chaput,
    Laboratoire de Mathématiques Jean Leray, UFR Sciences et Techniques, Nantes, France
  • L. Manivel,
    Institut Fourier, Université de Grenoble I, Saint-Martin d'Héres, France
  • N. Perrin,
    Institut de Mathématiques, Université Pierre et Marie Curie, PARIS, France
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Abstract

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q=1$, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa--Intriligator type formulas for the Gromov--Witten invariants.
Keywords: quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler class quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler class
MSC Classifications: 14M15, 14N35 show english descriptions Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]
Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
14M15 - Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]
14N35 - Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
 

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