A modular operad is an operad that has the kind of extra structure which is exhibited by the operad whose $n$-ary operations are Riemann surfaces with $(n+1)$-punctures, and whose composition is sewing of such surfaces. Every modular operad has a structure of a cyclic operad and also some additional structure.

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André Joyal, Joachim Kock, Feynman graphs, and nerve theorem for compact symmetric multicategories, proceedings “Quantum Physics and Logic VI”, arxiv/0908.2675

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Serguei Barannikov, Modular operads and Batalin-Vilkovisky geometry, Int. Math. Res. Not. IMRN 2007, no. 19, Art. ID rnm075, 31 pp. arxiv/0912.5484

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