Integral expression for a topological charge in the Faddeev-Niemi non-linear sigma model

Understand

We have introduced Faddeev-Niemi type variables for static SU(3) Yang-Mills theory.

• The variables suggest that a non-linear sigma model whose sigma fields take values in SU(3)/(U(1)xU(1)) and SU(3)/(SU(2)xU(1)) may be relevant to infrared limit of the theory.
• Shabanov showed that the energy functional of the non-linear sigma model is bounded from below by certain functional.
• However, the Shabanov's functional is not homotopy invariant, and its value can be an arbitrary real number -- therefore it is not a topological charge.