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In Topological Solitons by Nicholas Manton where he considers "compact Lie groups" to be the gauge groups for generalizing gauge theoretic concepts. But, he does not mention why that condition is required

The killing is non-degenerate as per the requirement of the kinetic terms.Kinetic part of the Yang Mills action $$ \int Tr({\bf{F^2}}) dV$$ to be positive definite. I do think there is relevancy of the compactness of the lie group as apposed to the views in Why is the Yang-Mills gauge group assumed compact and semi-simple?

I've found this result. http://www.math.columbia.edu/~woit/notes9.pdf

For compact groups the Killing form will be negative definite

I'm not able to find any such result for a Lie group. Can anyone tell me how this and the above result are result and its implementation in gauge theory?

Thanks, Sai


marked as duplicate by ACuriousMind, Gert, John Rennie, Qmechanic Nov 10 '15 at 17:44

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  • $\begingroup$ compact groups have finite reps? $\endgroup$ – innisfree Nov 10 '15 at 5:42
  • $\begingroup$ @innisfree and why finite representations are required? $\endgroup$ – diffeomorphism Nov 10 '15 at 6:43
  • 3
    $\begingroup$ I don't see how this question is different from the one you link, might you explain? $\endgroup$ – ACuriousMind Nov 10 '15 at 14:28