Abstract
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.
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Acknowledgements
We are grateful to the anonymous referee of the paper [1] who suggested the generalization discussed in the paper, demonstrated its potential importance and outlined a plan of proving it, although we didn’t follow that approach and have used our own method in this paper. We also thank Sasha Voronov for providing us the referee’s report of his paper.
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Communicated by Jim Stasheff.
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Yu, H. The equivalence between Feynman transform and Verdier duality. J. Homotopy Relat. Struct. 16, 427–449 (2021). https://doi.org/10.1007/s40062-021-00286-4
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DOI: https://doi.org/10.1007/s40062-021-00286-4