3
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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 $F\wedge F$. $\endgroup$ – Meng Cheng Mar 18 '15 at 19:43
  • $\begingroup$ i saw it in witten article but i can not understand $\endgroup$ – Ali Mar 18 '15 at 20:02
5
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Isn't it rather that $A\wedge A = 0$ for $U(1)$? $\endgroup$ – Robin Ekman Mar 19 '15 at 11:44
  • $\begingroup$ @RobinEkman You are right, $A\wedge A=0$ is enough to rule out the other terms. $\endgroup$ – Meng Cheng Mar 19 '15 at 23:10
  • $\begingroup$ it was a very good answer for me.but i could not understand that why we must have trace in writing chern simons actin? $\endgroup$ – Ali Apr 3 '15 at 11:14
  • $\begingroup$ 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) $\endgroup$ – Ali Apr 3 '15 at 12:13
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. projecteuclid.org/euclid.cmp/1104180750 $\endgroup$ – Yingfei Gu Mar 29 '15 at 18:28
  • $\begingroup$ 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 $\endgroup$ – Ali Apr 3 '15 at 12:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.