Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4)

We prove Zagier's conjecture on the value at s=4 of the Dedekind zeta-function of a number field F. For any field F, we define a map from the appropriate pieces of algebraic K-theory of F to the cohomology of the weight 4 polylogarithmic motivic complex. When F is the function field of a complex variety, composing this map with the regulator map on the polylogarithmic complex to the Deligne cohomology, we get a rational multiple of Beilinson's regulator. This plus Borel's theorem implies Zagier's conjecture. Another application is a formula expressing the value at s=4 of the L-function of an elliptic curve E over Q via generalized Eisenstein-Kronecker series. We get a strong evidence for the part of Freeness Conjecture describing the weight four part of the motivic Lie coalgebra of F via higher Bloch groups. Our main tools are motivic correlators and a new link of cluster varieties to polylogarithms.