Equivariant formality of transversely symplectic foliations and Frobenius manifolds

TL;DR

This paper proves an equivariant formality theorem and a symplectic $d\,\delta$-lemma for transversely symplectic foliations with group actions, leading to a Frobenius manifold structure on their equivariant basic cohomology.

Contribution

It introduces new formality results and a $d\,\delta$-lemma for such foliations, and constructs a Frobenius manifold structure in the Riemannian case.

Findings

Establishment of equivariant formality theorem.

Proof of equivariant symplectic $d\delta$-lemma.

Existence of Frobenius manifold structure on equivariant basic cohomology.

Abstract

Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation which satisfies the transverse hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic $d\delta$-lemma in this setting. As an application, we show that if the foliation is also Riemannian, then there exists a natural formal Frobenius manifold structure on the equivariant basic cohomology of the foliation.