We study the deformation complex of the dg wheeled properad of $mathbb{Z}$-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmu
ller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of $mathbb{Z}$-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps.
This paper attempts to provide a more or less self-contained introduction into theory of the Grothendieck-Teichmueller group and Drinfeld associators using the theory of operads and graph complexes.
For any integer $d$ we introduce a prop $RHra_d$ of oriented ribbon hypergraphs (in which edges can connect more than two vertices) and prove that it admits a canonical morphism of props, $$ Holieb_d^diamond longrightarrow RHra_d, $$ $Holieb_d^diamon
d$ being the (degree shifted) minimal resolution of prop of involutive Lie bialgebras, which is non-trivial on every generator of $Holieb_d^diamond$. We obtain two applications of this general construction. As a first application we show that for any graded vector space $W$ equipped with a family of cyclically (skew)symmetric higher products the associated vector space of cyclic words in elements of $W$ has a combinatorial $Holieb_d^diamond$-structure. As an illustration we construct for each natural number $Ngeq 1$ an explicit combinatorial strongly homotopy involutive Lie bialgebra structure on the vector space of cyclic words in $N$ graded letters which extends the well-known Schedlers necklace Lie bialgebra structure from the formality theory of the Goldman-Turaev Lie bialgebra in genus zero. Second, we introduced new (in general, non-trivial) operations in string topology. Given any closed connected and simply connected manifold $M$ of dimension $geq 4$. We show that the reduced equivariant homology $bar{H}_bullet^{S^1}(LM)$ of the space $LM$ of free loops in $M$ carries a canonical representation of the dg prop $Holieb_{2-n}^diamond$ on $bar{H}_bullet^{S^1}(LM)$ controlled by four ribbon hypergraphs explicitly shown in this paper.
We introduce a new category of differential graded multi-oriented props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of $k$ linear subspaces in that space, $k$ being the number
of extra directions (if $k=0$ this structure recovers an ordinary prop); symplectic vector spaces equipped with $k$ Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context of this paper Manin triples are precisely symplectic Lagrangian representations of the {em 2-oriented} generalization of the classical operad of Lie algebras. In a sense, the theory of multi-oriented props provides us with a far reaching strong homotopy generalization of Manin triples type constructions. The homotopy theory of multi-oriented props can be quite non-trivial (and different from that of ordinary props). The famous Grothendieck-Teichmuller group acts faithfully as homotopy non-trivial automorphisms on infinitely many multi-oriented props, a fact which motivated much the present work as it gives us a hint to a non-trivial deformation quantization theory in every geometric dimension $dgeq 4$ generalizing to higher dimensions Drinfeld-Etingof-Kazhdans quantizations of Lie bialgebras (the case $d=3$) and Kontsevichs quantizations of Poisson structures (the case $d=2$).
