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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?

Qmechanic
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Ali
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    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 $F\wedge F$. – Meng Cheng Mar 18 '15 at 19:43
  • i saw it in witten article but i can not understand – Ali Mar 18 '15 at 20:02

2 Answers2

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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.

Meng Cheng
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  • it was a very good answer for me.but i could not understand that why we must have trace in writing chern simons actin? – Ali Apr 03 '15 at 11:14
  • i am a student in MS in physics and recently i attached to study Chern-Simons action and i started with reading arXiv:1307.3200 i read it until The gauge algebra and i really want to continue it a i attached after it.i know about relation between Chern-simons and knots theory.please help me step by step(i can not understand very well Chern-Simons from only this article and Witten work) – Ali Apr 03 '15 at 12:13
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  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.

Qmechanic
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  • In case you are interested in generalized CS theory, I'd like to mention an example: I guess the most natural generalization of CS theory is the Dijkgraaf-Witten theory, which works for both compact Lie group case(give back to the usual CS for 3d case, for example G=SU(2)), and finite groups. https://projecteuclid.org/euclid.cmp/1104180750 – Yingfei Gu Mar 29 '15 at 18:28
  • i am a student in MS in physics and recently i attached to study Chern-Simons action and i started with reading arXiv:1307.3200 i read it until The gauge algebra and i really want to continue it a i attached after it.i know about relation between Chern-simons and knots theory.please help me step by step – Ali Apr 03 '15 at 12:01