The Lagrangian for a quantum field theory typically depends on the metric, that is, on the geometry of spacetime. This is not surprising -- it is natural that the propagation of a field on a curved manifold should depend on the curvature. There are certain special theories, however, whose actions are independent of the metric. Quantities such as partition functions computed from such theories cannot "see" the geometry of a manifold, but only its topology.
Typical field theories have an infinite number of degrees of freedom -- while they involve only a finite number of fields, each field has one or more degrees of freedom per point. In certain cases, though, symmetries are strong enough to reduce the physical degrees of freedom to a finite number. One example of such a reduction occurs in (2+1)-dimensional gravity.
Usage varies, but a "topological field theory" is usually defined as one with both of these properties, metric independence and a finite number of physical degrees of freedom. The archetypical topological field theory is Chern-Simons theory, a type of gauge theory in three dimensions. In the past few years, especially because of the work of Ed Witten, such theories have become increasingly important, both in physics and in mathematics.
- Steve Carlip (Physics, UC Davis)