Geometric quantization of the moduli space of the self-duality equations on a Riemann surface

Abstract

The self-duality equations on a Riemann surface arise as dimensional reduction of self-dual Yang-Mills equations. Hitchin showed that the moduli space M of solutions of the self-duality equations on a compact Riemann surface of genus g > 1 has a hyper-Kahler structure. In particular M is a symplectic manifold. In this paper we elaborate on one of the symplectic structures, the details of which are missing in Hitchin's paper. Next we apply Quillen's determinant line bundle construction to show that M admits a prequantum line bundle. The Quillen curvature is shown to be proportional to the symplectic form mentioned above.