[Paper Review] Hodge correlators

A. B. Goncharov|arXiv (Cornell University)|Mar 3, 2008

Advanced Algebra and Geometry18 references4 citations

TL;DR

This paper introduces Hodge correlators as periods of a Feynman integral and defines motivic correlators in the motivic Lie algebra whose periods recover these Hodge correlators. It establishes that Hodge correlators arise as coefficients of a twistor connection describing variations of real mixed Hodge structures on curves, unifying classical polylogarithms and Rankin-Selberg integrals within a motivic framework.

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.

Motivation & Objective

• To define Hodge correlators as complex numbers arising from integrals on smooth complex curves.
• To establish a connection between Hodge correlators and Feynman integrals, linking mathematical physics with Hodge theory.
• To introduce motivic correlators in the motivic Lie algebra whose periods are the Hodge correlators, thereby describing the motivic fundamental group.
• To generalize the description of variations of real mixed Hodge structures using twistor connections on the product of a variety with an affine line.
• To propose a DG enhancement of Saito's Hodge complexes with smooth cohomology, extending the framework for motivic and Hodge-theoretic constructions.

Proposed method

• Define Hodge correlators as specific integrals over smooth complex curves, linking them to Feynman amplitude structures.
• Construct the pronilpotent completion of the fundamental group of the curve and describe its real mixed Hodge structure.
• Introduce motivic correlators as elements in the motivic Lie algebra, with periods equal to the Hodge correlators.
• Define twistor connections as connections on the product of a variety with an affine line, generalizing variations of real mixed Hodge structures.
• Develop a DG enhancement of the subcategory of Saito's Hodge complexes with smooth cohomology to capture higher homotopical and motivic data.
• Demonstrate that when the curve varies, the Hodge correlators emerge as coefficients of the twistor connection encoding the variation of real MHS.

Experimental results

Research questions

• RQ1 How can Hodge correlators on smooth complex curves be interpreted as coefficients of a Feynman integral?
• RQ2 What is the role of motivic correlators in describing the motivic fundamental group of a curve?
• RQ3 How do twistor connections encode variations of real mixed Hodge structures on algebraic varieties?
• RQ4 In what way do Hodge correlators generalize classical and elliptic polylogarithms?
• RQ5 How do the motivic correlators relate to known elements in motivic cohomology, such as Beilinson's elements and the Beilinson-Kato Euler system?

Key findings

• Hodge correlators are shown to be coefficients of a twistor connection that describes the variation of real mixed Hodge structures on a curve.
• Motivic correlators are defined as elements in the motivic Lie algebra whose periods are precisely the Hodge correlators, providing a motivic lift of the fundamental group.
• The construction unifies classical polylogarithms and elliptic polylogarithms as special cases of Hodge correlators.
• The simplest Hodge correlators on modular curves are identified as Rankin-Selberg integrals, linking automorphic forms to Hodge theory.
• Beilinson's elements in motivic cohomology, including those forming the Beilinson-Kato Euler system on modular curves, are realized as motivic correlators.
• The paper provides a DG enhancement of Saito's Hodge complexes with smooth cohomology, enabling a derived enhancement of the category for motivic and Hodge-theoretic analysis.