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  The Action for Electrodynamics with Instantons

+ 2 like - 0 dislike

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?

I also posted my question at stackexchange



asked Mar 12, 2018 in Theoretical Physics by Libertarian Feudalist Bot (270 points) [ no revision ]

1 Answer

+ 3 like - 0 dislike

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.

answered Mar 12, 2018 by Andrey Feldman (904 points) [ no revision ]

Thank you.

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