Deformation theory of the wheeled properad of strongly homotopy Lie bialgebras and graph complexes

It is well-known that the Lie algebra of homotopy non-trivial degree zero derivations of the properad of strongly homotopy Lie bialgebras $\mathcal{H}olieb$ can be identified with the Grothendieck-Teichmuller Lie algebra $\mathfrak{g}\mathfrak{r}\mathfrak{t}$. We study in this paper the derivation complex of the wheeled closure $\mathcal{H}olie{b}^{\circlearrowleft }$ (and of its degree shifted version $\mathcal{H}olie{b}_{p,q}^{\circlearrowleft },\mathrm{\forall }p,q\in \mathbb{Z}$) and establishing a quasi-isomorphism to a version of the Kontsevich graph complex. This result leads us to a surprising conclusion that the Lie algebra of homotopy non-trivial derivations of the wheeled properad $\mathcal{H}olie{b}^{\circlearrowleft }$ can be identified with the direct sum of $\mathit{\text{two}}$ copies of $\mathfrak{g}\mathfrak{r}\mathfrak{t}$. As an illustrative example, we describe explicitly how the famous tetrahedron class in $\mathfrak{g}\mathfrak{r}\mathfrak{t}$ acts as a derivation of $\mathcal{H}olie{b}^{\circlearrowleft }$ in two homotopy inequivalent ways.