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.

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

Dismiss Notice

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!

# Please give some hints on the Complex projective group

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