U(1) CS theory vs SL(2) CS formulation: Boundary theory and Wilson line
We first derive the boundary theory from the U(1) Chern-Simons theory.The boundary action on an n-sheet manifold appears from its back-reaction of the Wilson line.The reason is that the U(1) Chern-Simons theory can provide an exact effective action MINTS SPEARMINT when introducing the Wilson line.The Wilson line in the pure AdS3 Einstein gravity is