SO(4) Monopole As A New Topological Invariant And Its Topological Structure

By making use of the decomposition theory of gauge potential, the inner structure of SU(2) and SO(4) gauge theory is discussed in detail. We find the SO(4) monopole can be given via projecting the SO(4) gauge field onto an antisymmetric tensor. This projection fix the coset $% SU(2)/U(1)\bigotimes SU(2)/U(1)$ of SO(4) gauge group. The generalized Hopf map is given via a Dirac spinor. Further we prove that this monopole can be consider as a new topological invariant. Which is composed of two monopole structures. Local topological structure of the SO(4) monopole is discussed in detail, which is quantized by winding number. The Hopf indices and Brouwer degree labels the local property of the monopoles.