| My talk consists of two parts. First part("genus zero"): the negative cyclic homology subspace moving inside periodic cyclic homology defines for the noncommutative varieties the analogue of the variations of Hodge structure, which I've described more than 10 years ago. For the deformations of the derived categories of coherent sheaves of Calabi-Yau hypersurfaces the periods of these noncommutative Hodge structures give the generating function of the totality of the genus zero Gromov-Witten invariants of the mirror manifolds. Second part ("arbitrary genus"): I'll describe the higher dimensional analogs of the matrix Airy integral, which I've introduced in my 2006 paper on noncommutative Batalin-Vilkovisky formalism (hal-00102085). My matrix integrals are constructed from Calabi-Yau A-infinity algebras. The asympthotic expansions of my matrix integrals are related with cohomology of the compactified moduli spaces. |
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