# Lifting 3d Chern-Simons theory to 4d

For simplicity, let us only consider abelian Chern-Simons theory.

The usual way of lifting 3d Chern-Simons theory to 4d is achieved through the Stokes' theorem. Say, if the original Chern-Simons theory was defined on some 3-manifold $$M$$, the 3d action would be given by $$$$S_{\text{3d CS}}=\int_M\, A\wedge dA,$$$$ where $$A$$ is a 1-form potential. Then by Stokes' theorem, the 4d analogue is obtained by taking $$M$$ to be the boundary of some 4-manifold $$X$$ (i.e. take $$M=\partial X$$), so that the action becomes $$$$S_{\text{4d CS}}=\int_X\, dA\wedge dA.$$$$

What happens to the 3d Chern-Simons term if we want to lift $$M$$ to $$M\times S^1$$ ($$S^1$$ is the circle) instead?

Looking at this post, perhaps a better way to ask the question would be this: What 4d theories give rise to 3d Chern-Simons/ BF terms upon Kaluza-Klein reduction on a circle?