# 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?

• To start off, any theory stated in terms of differential equations (respectively, action principle with Stokes theorem) strongly depends on the boundary conditions. – gented Jul 4 '16 at 14:30
• @GennaroTedesco I am sorry to admit it but I don't even know what you mean with boundary conditions on this context – Yossarian Jul 4 '16 at 14:40
• The boundary conditions in Chern-Simons theory aren’t more important than in any other theory, they are just as important as. If the question is then why are boundary conditions important, in general, it is because the behaviour of the solutions of the equations of motion (as well as the other type of equations involved) usually depends on what boundary conditions the system is subject to. – gented Jul 4 '16 at 14:50