Gromov-Witten invariants on Grassmannians

We prove that any three-point genus zero Gromov-Witten invariant on a type AA Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step flag variety is replaced by a sub-maximal is...

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Bibliographic Details
Published in:Journal of the American Mathematical Society 2003-10, Vol.16 (4), p.901-915
Main Authors: Buch, Anders Skovsted, Kresch, Andrew, Tamvakis, Harry
Format: Article
Language:English
Subjects:
Algebra
Geometry
Integers
Lagrangian function
Mathematical manifolds
Mathematical rings
Mathematics
Morphisms
Polynomials
Research article
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Summary:We prove that any three-point genus zero Gromov-Witten invariant on a type AA Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step flag variety is replaced by a sub-maximal isotropic Grassmannian. Our theorems are applied, in type AA, to formulate a conjectural quantum Littlewood-Richardson rule, and in the other classical Lie types, to obtain new proofs of the main structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians.
ISSN:0894-0347
1088-6834
DOI:10.1090/S0894-0347-03-00429-6