Four vector for dual field-strength tensor

Question

To generate the electromagnetic field strength tensor, one can use the electromagnetic four-vector using by $F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$. Is there a similar four-vector which can generate the dual field-strength tensor, $\tilde{F}^{\alpha\beta}=\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}$, in the same way?

Answer 1

Hint: A dual 4-potential $\tilde{A}$ such that $\tilde{F}=\mathrm{d}\tilde{A}$ would be inconsistent with Maxwell's equations, unless there are no sources.