Topological quantization of self-dual Chern-Simons vortices on Riemann   Surfaces
  The self-duality equations of Chern-Simons Higgs theory in a background curved spacetime are studied by making use of the U(1) gauge potential decomposition theory and $\phi$-mapping method. The special form of the gauge potential decomposition is obtained directly from the first of the self-duality equations. Using this decomposition, a rigorous proof of magnetic flux quantization in background curved spacetime is given and the unit magnetic flux in curved spacetime is also found . Furthermore, the precise self-dual vortex equation with topological term is obtained, in which the topological term has always been ignored. 
Yongqiang Wang, Yuxiao Liu, Zhenhua Zhao, Yishi Duan