# Higgs bundles and flat connections over compact Sasakian manifolds

## TL;DR

This paper extends the known correspondence between flat bundles and Higgs bundles from Kähler to compact Sasakian manifolds, establishing an equivalence of categories and existence of special metrics.

## Contribution

It generalizes the flat bundle-Higgs bundle correspondence to Sasakian manifolds and proves existence of Yang--Mills--Higgs metrics for stable basic Higgs bundles.

## Key findings

- Equivalence between semi-simple flat bundles and polystable basic Higgs bundles on Sasakian manifolds
- Existence of Yang--Mills--Higgs metrics on stable basic Higgs bundles
- Extension of Kähler results to Sasakian geometry

## Abstract

Given a compact K\"ahler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, \theta)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that any stable basic Higgs bundle over a compact Sasakian manifold admits a basic Hermitian metric that satisfies the Yang--Mills--Higgs equation.