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

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

2006-05-08

arxiv.org/abs/math-ph/0605026v2

Rept.Math.Phys.57:179-188,2006

Physics

Mathematical Physics

12 pages

Scientific paper

10.1016/S0034-4877(06)80016-9

The self-duality equations on a Riemann surface arise as dimensional reduction of self-dual Yang-Mills equations. Hitchin had showed that the moduli space ${\mathcal M}$ of solutions of the self-duality equations on a compact Riemann surface of genus $g >1$ has a hyperK\"{a}hler structure. In particular ${\mathcal M}$ is a symplectic manifold. In this paper we elaborate on one of the symplectic structures, the details of which is missing in Hitchin's paper. Next we apply Quillen's determinant line bundle construction to show that ${\mathcal M}$ admits a prequantum line bundle. The Quillen curvature is shown to be proportional to the symplectic form mentioned above. We do it in two ways, one of them is a bit unnatural (published in R.O.M.P.) and a second way which is more natural.

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