Hodge Correlators II

A. B. Goncharov

Published 2008-07-30, updated 2009-08-13Version 2

We define Hodge correlators for a compact Kahler 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.