I Hall Conductivity and Rotational Invariance

[thatboi]

I am reading up on QHE from: https://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf
and am confused about the comment: "The action (5.5) has no Hall conductivity because this is ruled out in d = 3+1 dimensions on rotational grounds." Can someone explain why this is the case? My only guess is that the action in (5.5) contains no linear terms, whereas if we look at the general case in IQHE where have an electric and magnetic field, the Hamiltonian would contain terms linear in the magnetic field and electric field. But I cannot understand why this would necessarily rule out the IQHE?

 

[thatboi]

Ok, I think the easiest way to see this is to rewrite (5.5) using the Maxwell tensor $F_{\mu\nu}^{2} = 2(\partial_{\mu}A_{\nu})^2-2(\partial_{\mu}A_{\mu})^{2}$. Then we note that taking the functional derivative of the Maxwell action with respect to any specific $A_{\mu}$ necessarily evaluates to $0$ so there is no Hall Conductivity.