Formal Frobenius Manifold Structure on Equivariant Cohomology

For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model. The notion of Frobenius manifolds was introduced by Dubrovin [11, 12]. It gives a coordinate free formulation of solutions to the WDVV equations. As surveyed in Manin [23], there are three major methods to construct solutions to WDVV equations. The first method involves the theory of quantum cohomology via GromovWitten invariants (or topological sigma model in physics literature), see e.g. RuanTian [25] and Kontsevich-Manin [19]. The second method is Saito’s theory of singularities (or Landau-Ginzburg model in physics literature). The third method exploits the so-called DGBV algebras, named after Gerstenhaber, Batalin and Vilkovisky. This last method first appeared in Barannikov-Kontsevich [2] in the context of extended moduli spaces of Calabi-Yau manifolds, based on the Kodaira-Spencer theory of gravity of Bershadsky-Cecotti-Ooguri-Vafa [5] which extends earlier works of Tian [26] and Todorov [27]. A detailed account of the construction for general DGBV algebras can be found in Manin [23]. GBV algebras have appeared in many places in Mathematics and Mathematical Physics, e.g. algebraic deformation theory and Hochschild cohomology (Gerstenhaber [13]), string theory (Lian-Zuckerman [20]), gauge theory (Batalin-Vilkovisky [3]), etc. However, examples of DGBV algebras in differential geometry were relatively rare. Earlier examples include Tian’s formula [26] in deformation theory of Calabi-Yau manifolds and Koszul’s operator in Poisson geometry [18]. But the recognizations of DGBV algebra structures in these theories seem to come later in e.g. Ran [24] and Xu [28] respectively. In a series of papers [6, 7, 8], the authors constructed many DGBV algebras from Kähler and hyperkähler manifolds, and showed that they satisfy the conditions to carry out the construction of formal Frobenius manifold structures on the cohomology. Also, it was shown that different DGBV algebra structures can yield the same solution to the WDVV equations. In particular, we get formal Frobenius manifold structures on the de Rham and Dolbeault cohomology of a closed Kähler manifold. In this paper, we carry over the same ideas to equivariant cohomology, in the case of closed Kähler manifolds with Hamiltonian actions of a Lie group by holomorphic isometries. The main result in this paper is related to the equivariant quantum cohomology. There are three models to define equivariant cohomology: the Borel model, the Cartan model and the Weil model. In Givental-Kim [15], a version of quantum cohomology based on Borel model was suggested and the rigorous formulation appeared in Lu [21]. Some discussions of WDVV equations and Frobenius manifold