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