Loop Hodge structure and harmonic bundles

TL;DR

This paper introduces loop Hodge structures as an infinite-dimensional extension of classical Hodge structures and demonstrates their equivalence to harmonic bundles, enabling the application of Hodge theory tools to study harmonic bundles.

Contribution

It defines loop Hodge structures and establishes their equivalence to harmonic bundles, providing a new framework for analysis using Hodge theory techniques.

Findings

Equivalence between loop Hodge structures and harmonic bundles

Existence of a period map for harmonic bundles

An integrality result for the Hitchin energy class

Abstract

We define the notion of a loop Hodge structure -- an infinite dimensional generalization of a Hodge structure -- and prove that a suitable variation of this object over a complex manifold is equivalent to the datum of a harmonic bundle. Hence one can study harmonic bundles using classical tools of Hodge theory, especially the existence of a period map (with values in an infinite dimensional period domain). Among other applications, we prove an integrality result for the Hitchin energy class of a harmonic bundle.