Modular operads

Details Modular operads Modular operads

1994-08-17

arxiv.org/abs/dg-ga/9408003v2

Physics

High Energy Physics

High Energy Physics - Theory

46 pages, latex2e. Substantially revised version.

Scientific paper

Modular operads are a special type of operad: in fact, they bear the same relationship to operads that graphs do to trees (i.e. simply connected graphs). One of the basic examples of a modular operad is the collection of Deligne-Mumford-Knudsen moduli spaces $\bar{M}_{g,n}$ of stable pointed algebraic curves; hence the word ``modular.'' In this paper, we introduce various constructions on differential graded modular operads, notably a duality which we call the Feynman transform, which extends Kontsevich's graph complexes. Our main result is the calculation of the Euler characteristic of the Feynman transform of a modular operad, using the theory of symmetric functions: the result is a generalization of Wick's theorem for Gaussian integrals.

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