3 Conclusion

We conclude this letter with a remark on a general property of interacting dual theories in regards to external currents. As it can be seen from equation (23) the current $j_e$ which is coupled to the gauge potential $A$ can be expressed in terms of the bulk current as $j_e=\partial J$. On the other hand, in the absence of a magnetic current, one can see from (26) that the dual potential $C$ couples to another electric current as a consequence of the dualization procedure. This second current, while implicitly related to $J$, say $\widetilde j_e=F^*(J)$, is not necessarily given by the divergence of the boundary current. Hence, a priori these two currents are not related to each other in most theories encompassed by our procedure. However, an exception to the rule is found in the limiting case $p=d-1$. In such a case, one can see that the two currents are given by the explicit expressions: $j_e^ {\mu_2\dots\mu_d} =\epsilon^{\mu_1\mu_2\dots\mu_d}\partial_{\mu_1} J^*$ and $\widetilde j_e^\mu =\partial^\mu\, J^*$, where $J^*$ represents a zero-form⁸. This explicit representation of the two currents leads to the identification $\widetilde j_e^\mu= j_e^{*\, \mu}$ which shows that, in the limiting case, they are, in fact, related by the Hodge duality operation.