Non-abelian Hodge correspondence and moduli spaces of flat bundles on Sasakian manifolds with fixed basic structures

TL;DR

This paper explores the structure of moduli spaces of flat bundles on compact Sasakian manifolds, establishing a non-abelian Hodge correspondence and extending Hitchin's properness results to this setting.

Contribution

It provides a detailed description of the non-abelian Hodge correspondence for Sasakian manifolds and introduces an analogue of Hitchin's properness theorem.

Findings

Moduli space of flat bundles decomposes into fixed basic structure components.

Established a non-abelian Hodge correspondence at the moduli space level.

Proved an analogue of Hitchin's properness for Sasakian manifolds.

Abstract

We show that the moduli space of simple flat bundles over a compact Sasakian manifold is a finite disjoint union of moduli spaces of simple flat bundles with fixed basic structures. This gives a detailed description of the non-abelian Hodge correspondence on a compact Sasakian manifold at the level of moduli spaces. As an application, we give an analogue of Hitchin's properness of maps defined by the coefficients of the characteristic polynomial of Higgs fields.