Dual Feynman transform for modular operads

Abstract

We introduce and study the notion of a dual Feynman transform of a smodular operad. This generalizes and gives a conceptual explanation of Kontsevich's dual construction producing graph cohomology classes from a contractible differential graded Frobenius algebra. The dual Feynman transform of a modular operad is indeed linear dual to the Feynman transform introduced by Getzler and Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman transform, the dual notion admits an extremely simple presentation via generators and relations; this leads to an explicit and easy description of its algebras. We discuss a further generalization of the dual Feynman transform whose algebras are not necessarily contractible. This naturally gives rise to a two-colored graph complex analogous to the Boardman-Vogt topological tree complex.

Noncommutative geometry and compactifications of the moduli space of curves

“…One should also be able to associate a differential graded Lie algebra to this modular operad which recovers the homology of this graph complex. For a treatment of graph complexes from the perspective of modular operads, the reader may consult [CL07].…”

Modular envelopes, OSFT and nonsymmetric (non-$\Sigma$) modular operads

“…Since the symmetrization (18) clearly commutes with the non-Σ modular envelope functor, Theorem 37 implies the isomorphisms (27) Mod(Ass) ∼ = Sym Span( * M ) proved in [6,9] though not expressed in this form there. Let us start proving the isomorphisms (26a) and (26b) of Theorem 35.…”

“…Guided by the example of Mod( * C ), one would expect Mod( * C ) to be the (symmetrization of) the terminal Set-operad in the conjectural category of non-Σ modular operads. The description of Mod( * C ) given in [9,Theorem 3.1] and its relation to the moduli space of Riemann surfaces with boundaries [6,Theorem 3.7] therefore gives some feeling what non-Σ modular operads should be.…”

“…The modular envelope Mod( * C ) of the terminal non-Σ cyclic Set-operad, described much later in [6,9], turned out to be surprisingly complex. (The authors of [6,9] worked with the linearized version, i.e.…”

“…(The authors of [6,9] worked with the linearized version, i.e. with the cyclic operad Ass for associative algebras, but the linear structure is irrelevant here as everything important happens inside Set.)…”