Moduli Spaces of the Basic Hitchin Equation on Sasakian Threefolds

TL;DR

This paper introduces the basic Hitchin equation on Sasakian threefolds, constructs its moduli space, and demonstrates that it admits a hyperKähler metric, extending known structures to a three-dimensional setting.

Contribution

It constructs the moduli space of the basic Hitchin equation on Sasakian threefolds and proves it admits a hyperKähler metric, a novel extension of Hitchin theory.

Findings

The moduli space admits a hyperKähler metric.

The dimension of the moduli space is explicitly calculated.

The moduli space of flat bundles over Sasakian threefolds also admits a hyperKähler metric.

Abstract

In this paper, we study an equation which we call the basic Hitchin equation. This is an equation defined on Sasakian threefolds and is a three-dimensional analog of the Hitchin equation, which is defined on Riemann surfaces. We construct the moduli space of the basic Hitchin equation and show that such a space admits a hyperK\"ahler metric. This also shows that the moduli space of flat bundles over Sasakian threefolds admits a hyperK\"ahler metric. We also calculate the dimension of the moduli space.