Operad structure on Kontsevich's admissible graphs

Question

In his celebrated 97' preprint q-alg/9709040, M. Kontsevich constructs a $L_\infty$-quasi-isomorphism $\mathcal U:\mathcal D_{\rm poly}\to\mathcal T_{\rm poly}$ between the differential graded algebra structure on the deformation complex of the associative algebra of functions on $\mathbb R^n$ (called $\mathcal D_{\rm poly}$) and its cohomology the differential graded algebra of polyvector fields (called $\mathcal T_{\rm poly}$). The explicit construction involves a sum $\mathcal U_n(X_1,\ldots,X_n)=\sum_{\Gamma}\omega_\Gamma B_\Gamma(X_1,\ldots,X_n)$ over a particular class of graphs $\Gamma$ (called admissible graphs) where $\omega$ is a map from these graphs to ℝ (the weight function) and $B$ maps a graph and a set of compatible polyvector fields to a multidifferential operator (i.e. an element of $\mathcal D_{\rm poly}$).

The graphs are characterised by two integers $n$ and $m$ standing for the number of aerial vertices (decorated by polyvector fields via the action of $B$) and terrestrial vertices (decorated by functions) respectively.

Imposing that the component $\mathcal U_n$ of the morphism $\mathcal U$ is of degree $1-n$ imposes the following constraint on the number of edges of admissible graphs: e=2n+m-2 which coincides with the dimension of configuration spaces over which the integral $\omega_\Gamma$ is integrated.

Questions:

1) Is there a natural operad structure on the set of admissible graphs ? (I know there is a large body of literature regarding operad structure on the Kontsevich graph complex introduced in [Formality conjecture] but which involves different (linear combinations of isomorphisms classes of) graphs).

2) More generally, is there a way to interpret the constraint e=2n+m-2 purely in terms of a certain algebraic structure on the set of admissible graphs (without referring to $\mathcal U$) ?

Answer 1

Yes, there is an operad structure. The answer essentially lies in Willwacher's paper Models for the $n$-Swiss-Cheese operad. It's actually a colored operad. The graphs you describe are bicolored: you have the aerial vertices and the terrestrial vertices. Inside a terrestrial vertex, you can insert a graph of the same kind (bicolored). Inside an aerial vertex, you can insert a unicolored graph, of the kind found in Kontsevich's paper Operads and motives in deformation quantization. Note that to do this, you have to consider graphs with $2 \times 2$ types of vertices: aerial/terrestrial, but also external/internal. In Kontsevich's $L_\infty$-morphism, only graphs with exclusively internal vertices are considered. The operad structure maps insert graphs into the external vertices.

(If you only want to consider graphs with exclusively internal vertices, as in Kontsevich's paper, the algebraic structure you get is a semi-direct product of Lie algebras $GC_n \rtimes SGC_n$, where $GC_n$ are unicolored graphs, and $SGC_n$ are bicolored graphs. The Lie bracket is induced by the operad structure.)

The question is, what differential do you put on this operad (or Lie algebra). The answer is tricky and contained in Willwacher's paper above. In the end, you obtain a model (in the sense of real homotopy theory) for Voronov's Swiss-Cheese operad, which appears in Kontsevich's paper as compactifications of configuration spaces of the upper half-plane.

As for interpreting the admissibility constraint in terms of algebraic structure, I don't know. As you say, the constraint is here to make sure the integral is nonzero. You can consider non-admissible graphs, but the weight you get for those is zero.