  By the generalization of Chern--Simons topological current and
Gauss--Bonnet-Chern theorem, the purpose of this paper is to make a non-Abelian
gauge field theory foundation of the topological current of $\tilde{p}$-branes
formulated in our previous work. Using $\phi $--mapping topological current
theory proposed by Professor Duan, we find that the topological $\tilde
p$-branes are created at every isolated zero of vector field $\vec \phi (x)$.
It is shown that the topological charges carried by $\tilde p$-branes are
topologically quantized and labeled by Hopf index and Brouwer degree, i.e., the
winding number of the $\phi $--mapping. The action of topological $\tilde
p$--branes is obtained and is just Nambu action for multistrings when $D-
\tilde d=2$.
