Instantons, Monopoles and the Flux Quantization in the Faddeev-Niemi Decomposition

We study how instantons arise in the low energy effective theory of the SU(2) Yang-Mills theory in the context of the non-linear sigma model recently propose by Faddeev and Niemi. We find a simple relation between the instanton number $\nu$ and the charge m of the monopole that appears in the effective theory. It is given by $\nu =m\mathrm{\Phi }\mathrm{/}(2\pi )$, where $\mathrm{\Phi }$ is the quantized flux associated with a U(1) gauge field passing through the loop formed by the singularity of the monopole.