Integral expression for a topological charge in the Faddeev–Niemi nonlinear sigma model

We have introduced Faddeev–Niemi type variables for static SU(3) Yang–Mills theory. The variables suggest that a nonlinear sigma model whose sigma fields take values in SU(3)/(U(1) × U(1)) and SU(3)/(SU(2) × U(1)) may be relevant to infrared limit of the theory. Shabanov showed that the energy functional of the nonlinear sigma model is bounded from below by certain functional. However, Shabanov's functional is not homotopy invariant, and its value can be an arbitrary real number—therefore it is not a topological charge. Since the third homotopy group of SU(3)/(U(1) × U(1)) is isomorphic to the group of integer numbers, there is a non-trivial topological charge (given by the isomorphism). We apply Novikov's procedure to obtain integral expression for this charge. The resulting formula is analogous to the Whitehead's realization of the Hopf invariant. (paper)