Rewriting Equation of Motion in terms of Dual Fields (Chern-Simons)

[thatboi]

I am reading the following notes: https://arxiv.org/pdf/hep-th/9902115.pdf and am trying to make the connection between equations (22) and (24). Specifically, I do not understand how they were able to get (24) from (22) using the dual field prescription. I guess naively I'm not even sure where they get the second derivative term in (24) when (22) is only first derivative terms. Trying to take the differential of (22) is not leading me anywhere either.
Any assistance appreciated.

 

[dx]

∂_μ∂^μ ≙ 1 + κe²/2 ε^ναβF_αβ

This is the relationship between the first and second derivative terms.

 

[julian]

I think I answered this question a while ago. See:

https://www.physicsforums.com/threads/maxwell-chern-simons-action.780013/

In particular post #6.