Identification of Two Frobenius Manifolds in Mirror Symmetry

We identify two Frobenius manifolds obtained from two different differential Gerstenhaber-Batalin-Vilkovisky (dGBV) algebras on a compact Kähler manifold by Barannikov-Kontsevich-Manin (BKM) construction [1, 13]. One is constructed on the Dolbeault cohomology in Cao-Zhou [5], and the other on the de Rham cohomology in the present paper. This can be considered as a generalization of the identification of the Dolbeault cohomology ring with the complexified de Rham cohomology ring. The mysterious Mirror Conjecture [23] in string theory has enabled the physicists to write down a formula [3] on the number of rational curves of any degree on a quintic in CP4. Recently, Lian-Liu-Yau [12] have given a rigorous proof of this important formula. The rapidly progressing theory of quantum cohomology, also suggested by physicists, has lead to a better mathematical formulation of the Mirror Conjecture. Now a version of the Mirror Conjecture can be formulated as the identification of Frobenius manifold structures obtained by different constructions. (For an exposition of this point of view, the reader is referred to a recent paper by Manin [13].) More precisely, on a Calabi-Yau manifold X , there are two natural algebras A(X) = ⊕p,qH(X,Ω), B(X) = ⊕p,qH(X,Ω), where Ω is the sheaf of holomorphic sections to ΛTX . By Bogomolov-TianTodorov theorem, the moduli space of complex structures on X is an open subset inH(X,Ω). Witten [20] suggested the construction of an extended moduli space of complex structures. A construction in Barannikov-Kontsevich [1], together with a remark in Cao-Zhou [5] (see also §2.3), has realized it as a supermanifold with Bosonic part an open set in B(X) = ⊕p+q=evenH(X,Ω). Furthermore, there is a structure of Frobenius supermanifold on this extended moduli space by Barannikov-Kontsevich [1]. This construction of Frobenius manifold structure has been generalized by Manin [13] to general differential GerstenhaberBatalin-Vilkovisky (dGBV) algebras with some mild conditions. We call this the BKM construction. On the other hand, by Hodge theory, A(X) is isomorphic to the de Rham cohomology with complex coefficients H dR(X,C). Now the the complexified deformations of Kähler form is parameterized by H(X) ⊂ H dR(X,C). Also surveyed in Manin [13] is the construction of a formal Frobenius manifold structure modelled on H dR(X,C) (or H (X,R)) by quantum cohomology. (For another survey of quantum cohomology, see Tian [16]. For some remarks on recent development, see the introduction of Li-Tian [11].) In Cao-Zhou [5], the authors Authors’ research was supported in part by NSF 1 2 HUAI-DONG CAO & JIAN ZHOU gave a construction of Frobenius supermanifold modelled on A(X), and conjectured it can be identified with the construction via quantum cohomology. Notice that in general there is a problem of convergence in the construction via quantum cohomology. This problem has been solved only in the case of complete intersection Calabi-Yau manifolds. See Tian [16]. In our construction, a standard argument in Kodaira-Spencer-Kuranishi deformation theory [15] guarantees the convergence . In this paper, we give a construction of a Frobenius supermanifold modelled on H dR(X,C). We then identify it with the Frobenius supermanifold modelled on A(X) constructed in Cao-Zhou [5]. As to the knowledge of the authors, this gives the first nontrivial result on the identification of two Frobenius supermanifolds. Acknowledgements. The work in this paper is carried out while the second author is visiting Texas A&M University. He likes to express his appreciation for the hospitality and financial support of the Mathematics Department and the GeometryAnalysis-Topology group. 1. Barannikov-Kontsevich-Manin Construction The construction of Barannikov-Kontsevich [1] has been algebraically formulated by Manin [13], we call the general construction the BKM (Barannikov-KontsevichManin) construction. We review the related definitions and constructions here. For details, the reader should consult the original papers by Barannikov-Kontsevich [1] and Manin [13]. 1.1. Gerstenhaber algebras. A Gerstenhaber algebra consists of a triple (A = ⊕i∈Z2A,∧, [· • ·]), such that (A,∧) is a Z2-graded commutative associative algebra over a filed k, (A[−1], [· • ·]) is a graded Poisson algebra with respect to the multiplication ∧. Here A[−1] stands for the vector space A with a new grading: A[−1]i = Ai+1. An operator ∆ of odd degree is said to generate the Gerstenhaber bracket if for all homogeneous a, b ∈ A, [a • b] = (−1)(∆(a ∧ b)−∆a ∧ b− (−1)a ∧∆b). 1.2. Gerstenhaber-Batalin-Vilkovisky algebras. If there is an operator ∆ on a Gerstenhaber algebra A which generates the Gerstenhaber bracket, such that ∆ = 0, then A is called a Gerstenhaber-Batalin-Vilkovisky algebra (GBV algebra). 1.3. Differential Gerstenhaber-Batalin-Vilkovisky algebras. A differential Gerstenhaber-Batalin-Vilkovisky algebra (dGBV algebra) is a GBV algebra with a k-linear derivation δ of odd degree with respect to ∧, such that δ = δ∆+∆δ = 0. We will be interested in the cohomology group H(A, δ). 1.4. Integral on dGBV algebras. A k-linear functional ∫ : A → k on a dGBValgebra is called an integral if for all a, b ∈ A,