Moduli Spaces of the Basic Hitchin Equation on Sasakian Threefolds

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 hyperKähler metric. This also shows that the moduli space of flat bundles over Sasakian threefolds admits a hyperKähler metric. We also calculate the dimension of the moduli space.