## Elliptic Gromov-Witten invariants and the generalized mirror conjecture

Citations: | 47 - 5 self |

### BibTeX

@MISC{Givental_ellipticgromov-witten,

author = {Alexander Givental},

title = {Elliptic Gromov-Witten invariants and the generalized mirror conjecture},

year = {}

}

### OpenURL

### Abstract

A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov- Witten invariants of compact Kähler manifolds with isolated fixed points and for concave bundle spaces over such manifolds. Several results on genus 0 Gromov- Witten theory include: a non-linear Serre duality theorem, its application to the genus 0 mirror conjecture, a mirror theorem for concave bundle spaces over toric manifolds generalizing a recent result of B. Lian, K. Liu and S.-T. Yau. We also establish a correspondence (see the extensive footnote in section 4) between their new proof of the genus 0 mirror conjecture for quintic 3-folds and our proof of the same conjecture given two years ago. Research supported by NSF grants DMS-93-21915 and DMS-97-04774