When I read A.Zee's QFT in a Nutshell, he asked me to do the calculation of his exercise IV.4.6: (A.Zee called these the Pontryagin Index)(adsbygoogle = window.adsbygoogle || []).push({});

Let g(x) be the element of a group G. The 1-form v = gdg† is known as the Cartan-Maurer form. Then tr v^N is trivially closed on an N-dimensional manifold since it is already an N-form. Consider Q = SN ∫tr vN with SN the N-dimensional sphere. Discuss the topological meaning of Q. These con- siderations will become important later when we discuss topology in field theory in chapter V.7. [Hint: Study the case N = 3 and G = SU(2).]

I found that the result for G=SU(2) and N=3 is -24π^2, is it correct? I also calculated for the case N=1 and G=U(1), and i found the result is -2πi. What's on earth is the topological meaning of these results?

Thanks.

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# Calculation of the Pontryagin index

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