Topological Couplings and Contact Terms in 2d Field Theory

In string theory and in topological quantum field theory one encounters operators whose effect in correlation functions is simply to measure the topology of 2d spacetime. In particular these 'dilation'-type operators count the number of other operators via contact terms with the latter. While contact terms in general have a reputation for being convention-dependent, the ones considered here are perfectly well-defined by virtue of their simple geometrical meaning: They reflect the geometry of the stable-curve compactification. We give an unambiguous prescription for their evaluation which involves no analytic continuation in momenta.