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Equivariant Gromov-Witten invariants. (English) Zbl 0881.55006
The construction and some applications of the equivariant counterpart of the Gromov-Witten (GW) theory are described. The GW theory is considered as the intersection theory on spaces of (pseudo-)holomorphic curves in (almost) Kähler manifolds. It is shown that GW theory provides the equivariant cohomology space \(H_G^*(X)\) with a Frobenius structure, where \(G\) is a compact group acting on the compact Kähler manifold \(X\). Some applications of the equivariant theory to the computation of quantum cohomology algebras of flag manifolds, to the \(S^1\)-equivariant Floer homology theory, and to a “quantum” version of the Serre duality theorem are discussed. Then, the general theory is combined with the fixed-point localization technique, in order to prove the mirror conjecture for projective complete intersections. The relation with the Calabi-Yau manifolds is also analyzed.

MSC:
55N91 Equivariant homology and cohomology in algebraic topology
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53D40 Symplectic aspects of Floer homology and cohomology
53D42 Symplectic field theory; contact homology
81T70 Quantization in field theory; cohomological methods
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