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In the beautiful paper by Harlow et Ooguri, they write in section 2.1 about this action

$$ S=-\frac12 \int_M F_a \wedge \star F_b \delta^{ab}\;, $$

with index $a=1,2$. They say that if the gauge symmetry is $\mathbb{R}\times \mathbb{R}$ then this has a $U(1)$ global symmetry rotating the two gauge fields. This is clear.

However, they say that for gauge group $U(1)^2$, this would not be a global symmetry (according to their definition) since it would not respect charge quantization. In particular, it would act non-locally on Wilson lines.

Can someone elaborate on this non-local action on Wilson lines? I don't understand it. Many thanks.

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  • $\begingroup$ I'm not sure, but when they wrote "would not act locally on the Wilson loops, since it wouldn't respect charge quantization" (footnote on page 28), I wonder if they really meant "would not be defined on Wilson loops (for a generic rotation angle), since it wouldn't respect the periodicities of either of the two angle-valued gauge fields." $\endgroup$ – Chiral Anomaly Nov 12 '19 at 3:29

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