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 the 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 appeared. 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 the Feynman transform, which is quite intuitive and simpler than the original proof.

dg 对偶性和 Verdier 对偶性之间的等价性早先已为循环操作数建立。我们建议将这种对应关系从循环操作数和 dg 对偶性推广到扭曲模操作数和费曼变换。具体来说，对于每个扭曲的模操作数\(\mathcal {P}\)（在特征为 0的字段k上取 dg 向量空间中的值），有一个特定的层\(\mathcal {F}\)与之关联上稳定的度量图的模空间，使得迭尔双札\（d \ mathcal {F} \）与飞漫转换相关\（F \ mathcal {P} \）的\（\ mathcal {P} \）. 在证明过程中，我们还证明了最初在 Getzler 和 Kapranov 的开创性工作中提出的循环操作数和模操作数之间的关系；然而，据作者所知，没有出现任何证据。操作数理论中的这种几何解释具有根本的重要性。我们相信这个结果将阐明模操作数理论的许多方面，并在未来找到许多应用。我们说明了这个结果的应用，给出了费曼变换的同伦性质的另一个证明，这比原始证明非常直观和简单。