Chern Simons action in 4 dimensions

Question

I can not understand why we do not have a Chern-Cimons action for 4 or even forms?

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

This post imported from StackExchange Physics at 2015-03-21 18:34 (UTC), posted by SE-user Ali rezaie

Comments

You can try to write down an action using 1-form gauge field and wedge product, and the only thing you get in 3+1 is the theta term $F\wedge F$.

This post imported from StackExchange Physics at 2015-03-21 18:34 (UTC), posted by SE-user Meng Cheng

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i saw it in witten article but i can not understand

This post imported from StackExchange Physics at 2015-03-21 18:34 (UTC), posted by SE-user Ali rezaie

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.

This post imported from StackExchange Physics at 2015-03-21 18:34 (UTC), posted by SE-user Meng Cheng

Comments

Isn't it rather that $A\wedge A = 0$ for $U(1)$?

This post imported from StackExchange Physics at 2015-03-21 18:34 (UTC), posted by SE-user Robin Ekman

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@RobinEkman You are right, $A\wedge A=0$ is enough to rule out the other terms.

This post imported from StackExchange Physics at 2015-03-21 18:34 (UTC), posted by SE-user Meng Cheng

Answer 2

Comments to the question (v4):

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. This definition is what Witten refers to.

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.

References:

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

2. Higher CS theories on nLab.

This post imported from StackExchange Physics at 2015-03-21 18:34 (UTC), posted by SE-user Qmechanic

Comments

thank you from your answers.

This post imported from StackExchange Physics at 2015-03-21 18:34 (UTC), posted by SE-user Ali rezaie