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 \begin{equation} S_{\text{3d CS}}=\int_M\, A\wedge dA, \end{equation} 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 \begin{equation} S_{\text{4d CS}}=\int_X\, dA\wedge dA. \end{equation}

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?


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