## Abstract

We study an analog for higher-dimensional Calabi-Yau manifolds of the standard predictions of Mirror Symmetry. We introduce periods associated with "non-commutative" deformations of Calabi-Yau manifolds. These periods define a map M → ⊕_{k}H^{k}(X^{n}, ℂ) [n - k] on the moduli space of such deformations which is a local isomorphism. Using these non-commutative periods we introduce invariants of variations of semi-infinite generalized Hodge structures living over the moduli space M. It is shown that the generating function of such invariants satisfies the system of WDVV-equations exactly as in the case of Gromov-Witten invariants. We prove that the total collection of rational Gromov-Witten invariants of complete intersection Calabi-Yau manifold can be identified with the collection of invariants of variations of generalized (semi-infinite) Hodge structures attached to the mirror variety. The basic technical tool utilized is the deformation theory.

Original language | English |
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Pages (from-to) | 281-325 |

Number of pages | 45 |

Journal | Communications in Mathematical Physics |

Volume | 228 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2002 |

Externally published | Yes |