I have a simple question: What is a Topological Action?
Certain actions depend only on the topology of the underlying space on which the action is defined (as an integral on a smooth space or a sum on a lattice). This may happen even though the way the action is presented may look like it depends on more than topology. For example, the Einstein-Hilbert action looks like it depends on a metric on the underlying spacetime, and it usually does depend on a metric. But in two spacetime dimensions it happens to be proportional to the Euler characteristic of the spacetime, which is a topological invariant.
Consider a (two)-sphere and then squash it a bit. Relative to the un-squashed case, certain points on the sphere are now further apart while others are closer together. This is an active transformation. It is equivalent to a passive transformation where you don't squash the sphere, you simply change the metric on it. Evaluate the E-H action on both spaces and you will get the same answer. I've been purposely vague about what it means for two spaces to be topologically equivalent because the intuitive notion of a continuous deformation, like squishing and stretching, usually suffices.
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