Modular operads as modules over the Brauer properad

TL;DR

This paper establishes an equivalence between modular operads and modules over the Brauer properad, extending bar and cobar constructions and proposing a Koszul duality framework for modular operads.

Contribution

It introduces the Brauer properad as a new algebraic structure for modular operads and generalizes classical bar and cobar constructions to modules over properads.

Findings

Modular operads are equivalent to modules over the Brauer properad.

The Feynman transform corresponds to the cobar construction in this setting.

A framework for Koszul duality for modular operads is sketched.

Abstract

We show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore, we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. This generalizes the classical case of algebras over an operad and might be of independent interest. As an application, we sketch a Koszul duality theory for modular operads.