nLab
modular operad

Idea

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.

Motivations

• 2d TQFT
• constructing graph complexes of Kontsevich as Feynman transforms of suitable twisted modular operad?s

References

• Ezra Getzler, Mikhail Kapranov, Modular operads, Compositio Math. 110 (1998), no. 1, 65–126, doi, arXiv:dg-ga/9408003, MR99f:18009

• Ezra Getzler, The semi-classical approximation for modular operads, arxiv:alg-geom/9612005

• Ezra Getzler, J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, hep-th/9403055

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

• Joseph Chuang, Andrey Lazarev, Dual Feynman transform for modular operads, arxiv/0704.2561

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