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


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