# The Action for Electrodynamics with Instantons

+ 2 like - 0 dislike
393 views

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

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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.