Hidden Hodge symmetries and Hodge correlators

TL;DR

This paper explores the symmetries of Hodge structures, defining a natural action of the twistor Galois group on derived categories of sheaves, with Hodge correlators interpreted as Feynman integrals.

Contribution

It introduces a new action of the twistor Galois group on derived categories using Hodge correlators, linking Hodge theory with Feynman integral interpretations.

Findings

The twistor Galois group acts via A-infinity autoequivalences.

Hodge correlators are interpreted as Feynman integrals.

A potential extension to holonomic D-modules is proposed.

Abstract

The Hodge Galois group is the Tannakian Galois group of the category of real mixed Hodge structures. It has a subgroup, called the twistor Galois group, which is the Galois group of the category of mixed twistor structures, defined by C. Simpson. It is isomorphic to a semidirect product of C* and the unipotent radical of the Hodge Galois group. We define a natural action of the twistor Galois group by A-infinity autoequivalences of the derived category of complexes of sheaves with smooth cohomology on a compact smooth Kahler manifold X. The action of C* is provided by Simpson's action on irreducible local systems, and the action of the unipotent radical is given explicitly by the Hodge correlators for irreducible local systems on X. The Hodge correlators can be interpreted as correlators of cetrtain Feynman integrals. One should have a similar construction for the whole derived category of all holonomic D-modules.