Formality of canonical symplectic complexes and Frobenius manifolds

TL;DR

This paper proves the formality of the de Rham complex for certain symplectic manifolds and constructs Frobenius manifolds from their cohomology using algebraic structures, advancing understanding in symplectic geometry and mathematical physics.

Contribution

It demonstrates the formality of the de Rham complex for symplectic manifolds satisfying the hard Lefschetz condition and constructs Frobenius manifolds from their cohomology.

Findings

De Rham complex is formal for these symplectic manifolds.

Frobenius manifold structures can be derived from the associated algebraic structures.

Provides a link between symplectic geometry and Frobenius manifolds.

Abstract

It is shown that the de Rham complex of a symplectic manifold $M$ satisfying the hard Lefschetz condition is formal. Moreover, it is shown that the differential Gerstenhaber-Batalin-Vilkoviski algebra associated to such a symplectic structure gives rise, along the lines explained in the papers of Barannikov and Kontsevich [alg-geom/9710032] and Manin [math/9801006], to the structure of a Frobenius manifold on the de Rham cohomology of $M$.