I study an analog for higher-dimensional Calabi–Yau manifolds of the standard predictions of Mirror Symmetry. I introduce periods associated with “non-commutative” deformations of Calabi–Yau manifolds. These periods define a map 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. I 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.
@article{hal-00486064,
author = {Barannikov, Sergey},
title = {Non-Commutative Periods and Mirror Symmetry in Higher Dimensions},
journal = {HAL},
volume = {2002},
number = {0},
year = {2002},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00486064}
}
Barannikov, Sergey. Non-Commutative Periods and Mirror Symmetry in Higher Dimensions. HAL, Tome 2002 (2002) no. 0, . http://gdmltest.u-ga.fr/item/hal-00486064/