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2

This is, as you say, analogous to Chern-Simons (CS) theory on a manifold $M$ with boundary. In this case, the non-gauge invariance is not interpreted as an anomaly, but rather an explicit breaking of the symmetry. The $3d$ Chern-Simons action under a gauge transformation $\delta A=dv+[A,v]$ reads \delta\int_M\text{tr}\big(AF-\frac{1}{3}A^3\big)=\int_{\...

1

I think it appears due to a redefinition of the scale, $f_a = v_a/(2N)$ (see section 2.6 in this review by Di Luzio+: https://arxiv.org/abs/2003.01100).

0

I found at least two mistakes in my reasoning. First, I wasn't paying close enough attention to the difference between compact QED, whose gauge field is based on the compact group $U(1)$, and noncompact QED, whose gauge field is based on the noncompact group $\mathbb{R}$. Most texts gloss over this because they look the same in perturbation theory, but ...

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