General solution of second Maxwell equation

Question

The second Maxwell equation can be written as $$partial_{[gamma}F_{munu]}=0$$
We know $F_{munu}=partial_mu A_nu-partial_nu A_mu $ satisfies above equation. My question: is this the only solution of the equation or there can be more solution as well but they're rejected in electrodynamics maybe on physical grounds?

Rd Basha · Accepted Answer

Written in the language of differential-forms, the equation reads $dF=0$ ("$F$ is a closed form"). You're asking if that implies that $F= dA$ for some $A$ ("$F$ is exact"). This is always true locally, but not necessarily globally, and depends on the topology of the spacetime manifold. The subject exploring the difference between the two is called Cohomology theory.
see:
https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field#Differential_forms_approach
https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms
https://en.wikipedia.org/wiki/Cohomology

General solution of second Maxwell equation

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