Decomposition of SU(2) connection and knot structure in Skyrme-Faddeev model

In this paper, spinor and vector decomposition of SU(2) gauge potential are presented and their equivalence is constructed using a simple proposal. We also obtain the action of Skyrme-Faddeev model from the SU(2) massive gauge field theory which is proposed according to the gauge invariant principle. Then, the knot structure in Skyrme-Faddeev model is discussed in terms of the so-called phi-mapping topological current theory. The topological charge of the knot is characterized by the Hopf indices and the Brouwer degrees of phi-mapping, naturally. At last, we briefly discussed the topological invariant Hopf invariant which describes the topology of these knots. It is shown that Hopf invariant is the total number of all the linking numbers and self-linking numbers of these knots.