Hodge correlators II

We define Hodge correlators for a compact Kähler manifold $X$. They are complex numbers which can be obtained by perturbative series expansion of a certain Feynman integral which we assign to $X$. We show that they define a functorial real mixed Hodge structure on the rational homotopy type of $X$.
The Hodge correlators provide a canonical linear map from the cyclic homomogy of the cohomology algebra of $X$ to the complex numbers.
If $X$ is a regular projective algebraic variety over a field $k$, we define, assuming the motivic formalism, motivic correlators of $X$. Given an embedding of $k$ into complex numbers, their periods are the Hodge correlators of the obtained complex manifold.
Motivic correlators lie in the motivic coalgebra of the field $k$. They come togerther with an explicit formula for their coproduct in the motivic Lie coalgebra.