Chern-Simons action in 4 dimensions

Question

I can not understand why we do not have a Chern-Simons action for four or even forms?

And why is it not a good theory for (3+1) dim?

Answer 1

The rule of the game is to use $A$ and $F=dA$ to write a topological action, and in $d+1$-space time dimension you need to come up with a gauge-invariant $d+1$-form which can then be integrated over the manifold to give you the action. Such an action does not depend on metric at all. Take $U(1)$ gauge field as an example. In $2+1$, the only thing you can write down is $AF$($AAA$ vanishes identically), which is the Chern-Simons. Then in $3+1$, you can guess $FF, AAF, AAAA$. $AAF$ and $AAAA$ vanishes due to antisymmetrization of the wedge product. So you are left with $FF$. This can be generalized to other Lie groups.

Answer 2

1. By definition, the Lagrangian form $\mathbb{L}$ of Chern-Simons (CS) theory (wrt. a Lie algebra valued one-form gauge field $A$) is a CS form, i.e. the CS action reads $$S[A]~=~\int_M\mathbb{L}.$$ The exterior derivative $\mathrm{d}\mathbb{L}$ of a CS form is (also by definition) the Lie algebra trace of a polynomial of the 2-form field strength $F$. In the other words, $\mathrm{d}\mathbb{L}$ must have even form-degree, or equivalently, $\mathbb{L}$ must have odd form-degree, and hence the dimension of spacetime $M$ must be odd.

2. Of course, one could introduce a new definition of generalized CS theory. More generally, there is e.g. the notion of TQFT. TQFTs can exist in any dimensions.

3. In particular, we should mention that there exists a generalized 4D CS theory by Costello, Witten & Yamazaki defined on $\mathbb{R}^2\times\mathbb{C}$, cf. e.g. this Phys.SE post.

References:

1. M. Nakahara, Geometry, Topology and Physics, 2003; Section 11.5.

2. Higher CS theories on nLab.