Topological Mass in Maxwell-Chern-Simons theory [closed]

Starting from the Maxwell-Chern-Simons lagrangian (in 2+1 dimensions):

$$L_{MCS}=-\frac{1}{4}F^{\mu \nu}F_{\mu\nu}+\frac{g}{2} \epsilon^{\mu \nu \rho}A_\mu\partial_\nu A_\rho$$

I've derived equations of motion for the gauge field $A_\mu$.

$$\partial_\mu F^{\mu \nu}+\frac{g}{2}\epsilon^{ \nu \alpha \beta}F_{\alpha \beta}=0$$

In various texts on Chern-Simons theory [e.g. Dunne's notes] it is stated, that by using the 2+1 dimensional Hodge dual $\widetilde{F}^\mu=\frac{1}{2}\epsilon^{ \nu \alpha \beta}F_{\alpha \beta}$, these equations of motion can be rewritten as a massive Klein-Gordon equation

$$\left(\partial_\nu \partial^\nu+g^2\right)\widetilde{F}^\mu=0$$

In these variables, we can now see that the massless vector field $A_\mu$ becomes massive due to the presence of the Chern-Simons term.

However, I cannot do this change of variables explicitly. I've tried a lot of ways but only what I've found is $\partial_\nu \widetilde{F}^\nu=0$ (just differentiated initial equation of motion with respect to $x_\nu$ and used antisymmetric property of $F_{\mu\nu}$.)

closed as off-topic by ACuriousMind♦, Kyle Oman, Kyle Kanos, JamalS, BMSJan 17 '15 at 6:31

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind, Kyle Oman, Kyle Kanos, JamalS, BMS
If this question can be reworded to fit the rules in the help center, please edit the question.

• Hint: Observe that your starting equation is $(\partial_\mu \epsilon^{\nu\mu\rho} + g\eta^{\mu\nu})\tilde{F}_\nu = 0$. Apply the operator $(\partial_\mu \epsilon^{\nu\mu\rho} + g\eta^{\mu\nu})$ once more to this to get the desired result ($\eta$ is the metric on your spacetime). – ACuriousMind Jan 16 '15 at 18:40
• @ACuriousMind Thx for response, I've got it. :) – Oiale Jan 17 '15 at 11:15