nLab Feynman transform

Idea

The Feynman transform is an operation on the category of twisted modular operads. It gives a way to parametrize various versions of Kontsevich’s graph complex, by various modular operads. Every modular operad is in particular cyclic (some say “symplectic”). The Feynman transform, up to a shift, reduces to the cobar operad? of the underlying cyclic operad.

The name “Feynman transform” is due to Getzler and Kapranov.

Properties

Relation to “Feynman categories” (colored operads)

R. Kaufmann and B. C. Ward have introduced Feynman categories, a concept that turns out to be biequivalent to coloured operads (see here).

Utilizing the pushforward functors defined using left Kan extensions, they define the bar and cobar constructions in the setup of Ab-enriched Feynman categories with some additional structure; assuming certain duality for categories of chain complexes over the coefficient additive category they define Feynman transform as composition of a (co)bar and duality. See Definition 7.11 in their paper Feynman categories arXiv:1312.1269.

References

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

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

• 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

• Martin Markl, Steven Shnider, Jim Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, Amer. Math. Soc. 2002. x+349 pp. MR2003f:18011(for Feynman transform see page 251)

• Michael Slawinski, The quantum $A_\infty$-relations on the elliptic curve, arxiv/1711.07940

We define and prove the existence of the Quantum $A_\infty$-relations on the Fukaya category of the elliptic curve, using the notion of the Feynman transform of a modular operad, as defined by Getzler and Kapranov. Following Barannikov, these relations may be viewed as defining a solution to the quantum master equation of Batalin-Vilkovisky geometry.