Question

Has anyone defined what it means for a modular operad to be Koszul, or what the Koszul dual of a modular operad is? In particular, is it meaningful to say that a modular operad is quadratic? Merkulov, Markl and Shadrin (Wheeled PROPs, graph complexes and the master equation) give a definition for wheeled properads, which are basically modular operads with oriented edges, which are constrained to have at most one output.

Presumable the role of bar-cobar duality is then played by the Feynman transform and the dual Feynman transform.

Answer 1

With due thanks to Daniel for pumping my thesis, it does not really deal with operads, modular or otherwise. Judging by your question, you probably already have some basic references, like Loday and Vallette's monograph (available online) on Algebraic Operads, which deals in particular with Koszulness for general operads. As far as modular operads, I don't know what the status of Koszulness may be for them, but the original paper of Getzler and Kapranov on Modular Operads does deal with the use of Feynman transforms in a manner analogous to the use of the cobar construction, as indicated by the following quote:

"The cobar functor B [13] is an involution on the category of differential graded (dg) operads. We will construct an analogous functor F on the category of dg-modular operads, the Feynman transform. This functor generalizes Kontsevich’s graph complexes [24]."

Peter

Answer 2

Peter Lee's PhD Thesis is about quadraticity of the pure virtual braid group. Virtual tangles are an algebra over a modular operad (generated by crossings), so this is very much in the same world. The Koszul dual of the latter is the algebra of arrow diagrams... so it does make sense, and people are studying such things.