Modular operads

TL;DR

This paper introduces the theory of modular operads, explores their properties through a duality called the Feynman transform, and calculates the Euler characteristic of this transform, generalizing Wick's theorem.

Contribution

It develops the concept of the Feynman transform for differential graded modular operads and computes its Euler characteristic using symmetric functions.

Findings

Feynman transform extends Kontsevich's graph complexes

Euler characteristic of the Feynman transform is computed

Generalizes Wick's theorem for Gaussian integrals

Abstract

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.