Feynman diagrams and minimal models for operadic algebras

TL;DR

This paper constructs explicit minimal models for operadic algebras using decorated trees and Feynman graphs, generalizing known results for A-infinity algebras and proving gauge-independence of certain graph cohomology classes.

Contribution

It introduces a general method for minimal models of operadic algebras, extending to modular operads and Feynman graphs, with applications to graph cohomology.

Findings

Minimal models expressed via sums over decorated trees and Feynman graphs

Homotopy equivalence of minimal models to original algebras established

Gauge-independence of Kontsevich's dual construction proven

Abstract

We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A-infinity algebras. Further, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's `dual construction' producing graph cohomology classes from contractible differential graded Frobenius algebras.