The equivalence between Feynman transform and Verdier duality

The equivalence between dg duality and Verdier has been established for cyclic operads earlier. We propose a generalization of this correspondence from to twisted modular the Feynman transform. Specifically, each operad $$\mathcal {P}$$ (taking values in dg-vector spaces over field k characteristic 0), there is certain sheaf {F}$$ associated with it on moduli space stable metric graphs such that dual $$D\mathcal transform $$F\mathcal . In course proof, we also prove relation originally proposed pioneering work Getzler Kapranov; however, best knowledge author, no proof appeared. This geometric interpretation theory fundamental importance. believe result will illuminate many aspects find applications future. illustrate an application result, giving another homotopy properties transform, which quite intuitive simpler than original proof.