Quantum periods - I. Semi-infinite variations of Hodge structures

Abstract

We introduce a generalization of variations of Hodge structures living over the moduli spaces of non-commutative deformations of complex manifolds. The Hodge structure associated with a point of such moduli space is an element of Sato type grassmanian of semi-innite subspaces in H (X; C )[[~ \\Gamma1 ; ~]]. Periods associated with such semi-innite Hodge structures serve in order to extend mirror symmetry relations in dimensions greater then three.