Quadratic invariant in Chern-Simons Th. w/ ISO(2,1) group.

[zwicky]

Hi dudes.

I'm studying the paper of Witten: 2+1 Dimensional gravity as an exactly soluble system.

Before eq (2.8) the author justifies as a way to find the inner product the fact that in this theory we have the casimir $$\epsilon_{abc}P^aJ^b$$. Then he introduce the invariant quadratic form
$$<J_a,P_b>=\delta_{ab}, <J_a,J_b>=0, <P_a,P_b>=0.$$
I', trying to realize about this step. Any comment would be very helpful. I looked at many books in group theory. I was able to construct the Killing metric but only with a cosmological constant, in the case of the Poincare group, it seems like the Killing metric is degenerate.

Thank you very much in advance.

 

[LightHero]

The inner product that Witten is using in the paper is related to the Killing metric. It is a general property of Lie groups, that they have a Killing metric associated with them, which is an inner product invariant under the group action. In the case of the Poincare group, the Killing metric is degenerate, meaning that there are directions in which the inner product is zero. In this particular case, Witten is using the Casimir element of the Poincare algebra, $\epsilon_{abc}P^aJ^b$, to define the inner product for the group. This inner product is different from the usual one used for the Poincare group, which makes use of the Casimir element $\epsilon_{abc}P^aP^bP^c$. By using the Casimir element $\epsilon_{abc}P^aJ^b$ instead, Witten is able to construct an inner product that is invariant under the group action.