Hypertrees and embedding of the $\mathrm{FMan}$ operad

Abstract

The operad $\mathrm{FMan}$ encodes the algebraic structure on vector fields of Frobenius manifolds, in the same way as the operad $\mathrm{Lie}$ encodes the algebraic structure on vector fields of a smooth manifold. It is well known that the operad $\mathrm{Lie}$ admits an embedding in the operad $\mathrm{PreLie}$ encoding pre-Lie algebras. We prove a conjecture of Dotsenko stating that the operad $\mathrm{FMan}$ admits an embedding in the operad $\mathrm{ComPreLie}$. The operad $\mathrm{ComPreLie}$ is the operad encoding pre-Lie algebras with an additional commutative product such that right pre-Lie multiplications act as derivations. To prove this result, we first remark a link between the Greg trees and the so-called operadic twisting of $\mathrm{PreLie}$. We then give a combinatorial description of the operad $\mathrm{ComPreLie}$ \emph{\`a la} Chapoton-Livernet with forests of rooted hypertrees. We generalize this construction to forests of rooted Greg hypertrees, and then use operadic twisting techniques to prove the conjecture.