  We consider coupling an ordinary quantum field theory with an infinite number
of degrees of freedom to a topological field theory. On R^d the new theory
differs from the original one by the spectrum of operators. Sometimes the local
operators are the same but there are different line operators, surface
operators, etc. The effects of the added topological degrees of freedom are
more dramatic when we compactify R^d, and they are crucial in the context of
electric-magnetic duality. We explore several examples including
Dijkgraaf-Witten theories and their generalizations both in the continuum and
on the lattice. When we couple them to ordinary quantum field theories the
topological degrees of freedom allow us to express certain characteristic
classes of gauge fields as integrals of local densities, thus simplifying the
analysis of their physical consequences.
