Semi-infinite A-variations of Hodge structure over extended Kähler cone

In this paper we propose a symplectic analogue of Barannikov's construction of semi-infinite Hodge structure variations over extended moduli spaces of complex structures. It is shown that the duality transformation for mirror torus fibrations over the same Monge-Ampere manifold isomorphically exchanges symplectic semi-infinite A-variations of Hodge structure introduced in this paper with Barannikov's semi-infinite B-variations of Hodge structure.