ar X iv : m at h / 99 04 16 8 v 1 [ m at h . D G ] 2 9 A pr 1 99 9 ON QUASI - ISOMORPHIC DGBV ALGEBRAS

One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functorial property of a construction of deformations of the multiplicative structures of their cohomology. In particular, we show that quasi-isomorphic DGBV algebras yield identifiable Frobenius manifold structures. String theorists are interested in two kinds of conformal field theories defined on a Calabi-Yau manifold X : an A-type theory which depends only on the Kähler structure but not the complex structure X , and an B-type theory which depends only on the complex structure but not the Kähler structure. Conceivably, an A-type theory should then be related to the deformations of the Kähler structure, while an B-type theory should be related to the deformations of the complex structure. The mysterious mirror symmetry [22] can be formulated as the identification of an A-type theory on X with a B-type theory on its mirror manifold X̂. Physicists also provide us some examples of such theories: the topological sigma model [21] and the Kähler theory of gravity [3] are A type theories, while the Kodaira-Spencer theory of gravity [2] is a B type theory. Through the efforts of many mathematicians, the topological sigma model now has rigorous mathematical formulation in terms of suitably defined Gromov-Witten invariants and has led to vast progress in symplectic geometry and algebraic geometry. On the other hand, based on the work of Tian [19] and Todorov [20], the Kodaira-Spencer theory of gravity was analyzed in details by Bershadsky-Cecotti-Ooguri-Vafa [2], and the theory of Kähler gravity by Bershadsky and Sadov [3]. Barannikov-Kontsevich [1] reformulated the results in [2] in terms of Frobenius manifolds introduced by Dubrovin [8, 9], and made the important observation that there is an algebraic structure called DGBV algebra hidden in the theory, and the method to obtain formal Frobenius manifold structure by the Kodaira-Spencer Lagrangian can be generalized to any DGBV algebra satisfying certain conditions. See the detailed account in Manin [17]. In two earlier papers [4, 5], we pointed out two DGBV algebra structures in the theory of Kähler gravity, one on Dolbeault cohomology and the other on de Rham cohomology, and showed that they satisfy the conditions in [1, 17] for constructing Frobenius manifold structures. Furthermore, we were able to identify the Frobenius manifold structures from these two different DGBV algebras. Subsequently, we also generalized these results to hyperkähler manifolds [6] and equivariant cohomology [7]. We conjectured that for a Calabi-Yau manifold X with a mirror manifold X̂, one should be able to identify the Frobenius manifold structure constructed in [1] for X with that constructed in [4, 5] for X̂ (maybe after some coordinate change).