Why are boundary terms important in Chern-Simons theory? I am learning about Chern-Simons theory. I work in Euclidean space. The action is given by
$$
I=\int_{\mathbb{R}^4}d\omega=\int_{\partial\mathbb{R}^4}\omega
$$
where $\omega$ is the usual Chern-Simons form, and I have used stokes' theorem. My first question is, what is the boundary of for dimensional Euclidean space? and why are boundary terms important in this theory? I mean, up till now I have shamelessly ignored boundary terms. Why are they important now?
 A: I can't comment on the string theory applications, but Chern-Simons theories are often used in condensed matter to model topologically ordered systems and symmetry protected topological (SPT) systems (not the same thing!).  Such systems are in some senses trivial in the bulk - all bulk excitations are gapped, so nothing happens in the bulk at low temperatures.  However, there are interesting gapless excitations on the physical boundary of the system, if it's a finite-volume chunk of an actual material.  These excitations do survive down to arbitrarily low temperatures.
For example, topological insulators do not conduct electricity through their bulk, but they do conduct on their surfaces (and in fact, these conducting modes are very robust against perturbations).  So the fact that the usual bulk integral reduces entirely to a surface integral tells you that all the interesting "action" (heh heh) happens at the physical surface of your system.  Since nothing happens in the bulk (that can be captured by this theory), it's not a very interesting theory to study on systems with no boundary.
