Topological tensor current of $\tilde{p}$-branes in the $ϕ$-mapping theory

We present a new general topological tensor current of $\tilde{p}$-branes by making use of the $ϕ$-mapping theory. It is shown that the current is identically conserved and behave as $δ(\vecϕ),$ and every isolated zero of the vector field $\vecϕ(x)$ corresponds to a `magnetic' $\tilde{p}$-brane. Using this topological current, the generalized Nambu action for multi $\tilde{p}$-branes is given, and the field strength $F$ corresponding to this topological tensor current is obtained. It is also shown that the `magnetic' charges carried by $\tilde{p}$-branes are topologically quantized and labeled by Hopf index and Brouwer degree, the winding number of the $ϕ$-mapping.

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