Hi dudes.(adsbygoogle = window.adsbygoogle || []).push({});

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 [tex]\epsilon_{abc}P^aJ^b[/tex]. Then he introduce the invariant quadratic form

[tex]<J_a,P_b>=\delta_{ab}, <J_a,J_b>=0, <P_a,P_b>=0.[/tex]

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**