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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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