Quantum K -theory of Grassmannians

We show that (equivariant) K -theoretic 3 -point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) K -theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we s...

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Bibliographic Details
Published in:Duke mathematical journal 2011-02, Vol.156 (3), p.501-538
Main Authors: Buch, Anders S., Mihalcea, Leonardo C.
Format: Article
Language:English
Subjects:
14E08
14M15
14N15
14N35
19E08
51M35
Classical problems
Donaldson-Thomas invariants [See also 53D45]
flag manifolds [See also 32M10
Gopakumar-Vafa invariants
Grassmannians
Gromov-Witten invariants
K$-theory of schemes [See also 14C35]
quantum cohomology
Rationality questions [See also 14M20]
Schubert calculus
Schubert varieties
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Summary:We show that (equivariant) K -theoretic 3 -point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) K -theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through three general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K -theory ring of a Grassmannian, which determine the multiplication in this ring. We also compute the dual Schubert basis for this ring and show that its structure constants satisfy S 3 -symmetry. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.
ISSN:0012-7094
1547-7398
DOI:10.1215/00127094-2010-218