The Action for Electrodynamics with Instantons

Question

In the paper "A Duality Web in 2+1 Dimensions and Condensed Matter Physics"  

https://arxiv.org/abs/1606.01989

on page 34, the action for electromagnetic field in Lorentzian signature is given by 

$$S=\int d^{4}x\sqrt{-g}\left(\frac{-1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{\theta}{32\pi^{2}}\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\right)$$

I am confused by the metric for the second term, because I used to think that the $\theta$-term should topological. The action should be 

$$S=-\frac{1}{2}\int F\wedge\star F+\frac{\theta}{8\pi^{2}}\int F\wedge F,$$
where there should be no metric dependence in the second term.

Am I misunderstanding anything or is that a mistake in the paper?

I also posted my question at stackexchange

https://physics.stackexchange.com/questions/391685/the-action-for-electric-magnetic-duality

 

Answer 1

The $\epsilon$-tensor in the second term contains the $1/{\sqrt g}$- factor, which cancels the corresponding one from the measure. What is left is the tensor density with $\pm 1$ components.

Comments

Thank you.