## Equivariant Gromov-Witten invariants (1996)

Venue: | Internat. Math. Res. Notices |

Citations: | 92 - 10 self |

### BibTeX

@ARTICLE{Givental96equivariantgromov-witten,

author = {Alexander B. Givental and Alexander B. Givental},

title = {Equivariant Gromov-Witten invariants},

journal = {Internat. Math. Res. Notices},

year = {1996},

volume = {13},

pages = {613--663}

}

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### Abstract

The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the Gromov-Witten (GW) theory, i.e., intersection theory on spaces of (pseudo-) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a compact Kähler manifold X, the equivariant GW-theory provides, as we will show in Section 3, the equivariant cohomology space H ∗ G (X) with a Frobenius structure (see [11]). We will discuss applications of the equivariant theory to the computation ([15], [18]) of quantum cohomology algebras of flag manifolds (Section 5), to the simultaneous diagonalization of the quantum cupproduct operators (Sections 7, 8), to the S1-equivariant Floer homology theory on the loop space LX (see Section 6 and [14], [13]), and to a “quantum ” version of the Serre duality theorem (Section 12). In Sections 9–11 we combine the general theory developed in Sections 1–6 with the fixed-point localization technique [21], in order to prove the mirror conjecture (in the form suggested in [14]) for projective complete intersections. By the mirror conjecture, one usually means some intriguing relations (discovered by physicists) between symplectic and complex geometry on a compact Kähler Calabi-Yau n-fold and, respectively, complex and symplectic geometry on another Calabi-Yau n-fold, called the mirror partner of the former one. The remarkable application [8]ofthe mirror conjecture to the enumeration of rational curves on Calabi-Yau 3-folds (1991, see the theorem below) raised a number of new mathematical problems—challenging tests of maturity for modern methods of symplectic topology. On the other hand, in 1993 I suggested that the relation between symplectic and complex geometry predicted by the mirror conjecture can be extended from the class of Calabi-Yau manifolds to more general compact symplectic manifolds if one admits non-