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Quantum \(K\)-theory. I: Foundations. (English) Zbl 1051.14064

The author studies the foundations of quantum \(K\)-theory, a \(K\)-theoretic version of quantum cohomology theory, giving a deformation of the ordinary \(K\)-ring \(K(X)\) of a smooth projective variety \(X\), analogous to the relation between quantum cohomology and ordinary cohomology. As a result, he develops a new class of Frobenius manifolds, and answers an open question of A. Bayer and Y. I. Manin [http://arxiv.org/abs/math.AG/0103164].

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
19E08 \(K\)-theory of schemes
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
55N15 Topological \(K\)-theory
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References:

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