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Quantum cohomology ring for Hermitian symmetric spaces of type DIII. (English) Zbl 1046.53056

Kluev, V. V. (ed.) et al., Topics in applied and theoretical mathematics and computer science. Athens: WSEAS Press (ISBN 960-8052-47-6/hbk). Mathematics and Computers in Science and Engineering. A Series of Reference Books and Textbooks, 232-237 (2001).
Summary: We determine the quantum cohomology ring for Hermitian symmetric spaces of type DIII. Among the manifolds whose quantum cohomology ring has been rigorously computed, two Hermitian symmetric spaces have been reported so far. The one we show in this paper, following a result of Sievert and Tian, is the third example. For this purpose encounting of the number of rational curves over the spaces satisfying a certain dimension condition is needed and it is accomplished by cell decomposition of Hermitian symmetric spaces which are an analogue of the Schubert cell for complex Grassmann manifolds.
For the entire collection see [Zbl 1028.00013].

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C35 Differential geometry of symmetric spaces
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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