Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields

  • Sergey Barannikov, Maxim Kontsevich
  • Published 1998

Abstract

We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus = 0 GromovWitten invariants. Introduction Probably the best mathematically understood part of the Mirror Symmetry program is the theory of Gromov Witten invariants (see [KM]). In this paper we will construct a Mirror partner for the genus = 0 sector of this theory. It may be considered as a generalization of the theory of variations of Hodge structures on Calabi-Yau manifolds. One of the puzzles in Mirror symmetry was to find an interpretation of the mysterious objects involved in the famous predictions of the numbers of rational curves. Such an interpretation should, in particular, give the meaning to the “extended” moduli space H(M,ΛTM )[2], thought as generalized deformations of complex structure. This moduli space should be equipped with the analog of the 3-tensor Cijk(t) (“Yukawa coupling”) arising from a generalization of the variation of Hodge structure on H(M). To find such structure is essential for the extension of the predictions of Mirror Symmetry in the dimensions n > 3. 0.1 Background philosophy. The Mirror Symmetry conjecture, as it was formulated in [K1], states that the derived category of coherent sheaves on a Calabi-Yau manifold M is equivalent to the derived category constructed from (conjectured) Fukaya category associated with the dual Calabi-Yau manifold M̃ . The conjecture implies existence of the structure of Frobenius manifold on the moduli space of A∞-deformations of the derived category of coherent sheaves on M . This structure coincides conjecturally with the Frobenius structure on formal neighborhood of zero in H∗(M̃,C) constructed via Gromov-Witten classes of the dual Calaby-Yau manifold M̃ . S. B. was partially supported by the Fellowship for Graduate Study of the University of California at Berkeley Typeset by AMS-TEX 1 2 SERGEY BARANNIKOV, MAXIM KONTSEVICH 0.2 Formulation of the results. Consider the differential graded Lie algebra

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@inproceedings{Barannikov1998FrobeniusMA, title={Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields}, author={Sergey Barannikov and Maxim Kontsevich}, year={1998} }