Hodge Correlators II

TL;DR

This paper introduces Hodge correlators for compact Kahler manifolds, establishing their role in defining a functorial mixed Hodge structure and connecting them to motivic correlators and periods, with implications for algebraic geometry.

Contribution

It defines Hodge correlators via Feynman integrals, linking them to mixed Hodge structures and motivic correlators, providing a new framework for understanding rational homotopy types.

Findings

Hodge correlators form a functorial real mixed Hodge structure.

They establish a canonical map from cyclic homology to complex numbers.

Motivic correlators relate to periods and motivic coalgebras.

Abstract

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.