nLab modular operad

Idea

A modular operad (Getzler & Kapranov 1998) 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 further additional structure.

Motivations

• 2d TQFT

• constructing graph complexes of Kontsevich as Feynman transforms of suitable twisted modular operads

References

The original article:

• Ezra Getzler, Mikhail Kapranov: Modular operads, Compositio Math. 110 1 (1998) 65–126 [doi:10.1023/A:1000245600345, arXiv:dg-ga/9408003, MR99f:18009]

Further discussion:

• 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

• Martin Doubek, Branislav Jurčo, Lada Peksová, Ján Pulmann, Connected sum for modular operads and Beilinson-Drinfeld algebras, arXiv:2210.06517

• Jeffrey Giansiracusa, Moduli spaces and modular operads, Morfismos 17:2, 101–125 journal author pdf

On an (∞,1)-operad version of modular operads:

• Philip Hackney, Marcy Robertson, Donald Yau: A graphical category for higher modular operads, Advances in Mathematics 365 (2020) 107044 [doi:10.1016/j.aim.2020.107044, arXiv:1906.01143]