  Let M be a U(1) bundle over a smooth Riemann surface. I show that for
Chern-Simons theory on M, with structure group G, the path integral is an
integral over the space of G-connections on the Riemann surface involving
characteristic classes as well as a certain 4-dimensional class that comes from
a universal bundle. When M is the product of a Riemann surface with a circle
the 4-dimensional class does not enter and the path integral takes the form of
a Riemann-Roch formula albeit in infinite dimensions. The discussion is
generalised to include Wilson lines along the fibre direction in M.
