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Hodge correlators

Alexander B. Goncharov

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 twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing 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.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-0400449

Award Identifier / Grant number: DMS-0653721

Award Identifier / Grant number: DMS-1059129

Award Identifier / Grant number: DMS-1301776

Funding statement: The author was supported by the NSF grants DMS-0400449, DMS-0653721, DMS-1059129 and DMS-1301776.

Acknowledgements

I am very grateful to Alexander Beilinson for many fruitful discussions, and especially for encouragement over the years to work on Hodge correlators. Many ideas of this work were worked out and several sections written during my stays at the MPI (Bonn) and IHES. A part of this work was written during my stay at the Okayama University (Japan) in May 2005. I am indebted to Hiraoki Nakamura for the hospitality there. I am grateful to these institutions for the hospitality and support. I am very grateful to referees for the extraordinary job. I am grateful to Claus Hertling for pointing out some errors, and sending me notes of his seminar talks in 2008–2009 on the paper.

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Received: 2012-04-26
Revised: 2016-01-03
Published Online: 2016-07-28
Published in Print: 2019-03-01

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