The O(N) nonlinear sigma model has topological solitons only when N=3 in the(adsbygoogle = window.adsbygoogle || []).push({});

planar geometry. There exists a generalization of the O(3) sigma model so that the

new model possess topological solitons for arbitrary N in the planar geometry. It is

the CP^{N-1} sigma model,whose group manifold is

[tex]CP^{N-1}= U(N)/[U(1)\bigotimes U(N-1)] =SU(N)/[U(1)\bigotimes SU(N)\bigotimes SU(N-1)][/tex]

The homotopy theorem tells

[tex] \pi_2(CP^{N-1})=Z [/tex]

since [tex]\pi_2(G/H) =\pi_1(H) [/tex] (when G is simply connected) and [tex]\pi_n(G\bigotimes G')=\pi_n (G)\bigoplus \pi_n(G') [/tex] . It is also called the SU(N) sigma model.

Would anyone gives a more detailed hints to the following sentences:

[tex]CP^{N-1}= U(N)/[U(1)\bigotimes U(N-1)] =SU(N)/[U(1)\bigotimes SU(N)\bigotimes SU(N-1)][/tex]

The homotopy theorem tells

[tex] \pi_2(CP^{N-1})=Z [/tex]

since [tex]\pi_2(G/H) =\pi_1(H) [/tex] (when G is simply connected) and [tex]\pi_n(G\bigotimes G')=\pi_n (G)\bigoplus \pi_n(G') [/tex]

U(N) seems to be not simply connected.

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# Please give some hints on the Complex projective group

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