Hodge correlators

TL;DR

Hodge correlators are complex integrals associated with smooth complex curves, linked to Feynman integrals, and relate to the motivic fundamental group and mixed Hodge structures, with applications to polylogarithms and Euler systems.

Contribution

The paper introduces motivic correlators and twistor connections, providing a new framework for understanding variations of mixed Hodge structures and their motivic counterparts.

Findings

Hodge correlators are periods of Feynman integrals.

Motivic correlators relate to motivic Lie algebras and fundamental groups.

Examples include classical polylogarithms and Beilinson's elements.

Abstract

Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by an afine line. We call them twistor connections. Generalising this, we suggest a DG enhancement of the subcategory of Saito's Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin-Selberg integrals. Examples of the motivic correlators include Beilinson's elements in the motivic cohomology, e.g. the ones delivering the Beilinson - Kato Euler system on modular curves.