$Z_n$ gauge theory from $U(1)$

Question

In Appendix A of the paper, "Generalized Global Symmetries" by Gaiotto et al., they have considered an action, which for the purpose of the question, can be taken to be

$$S=\frac{n}{2\pi}\int B dA$$

where $B$ is a periodic scalar and $A$ is a $d-1$ form. Then they went on to construct the operators

$$U=e^{i\oint A} \ \ \ and \ \ \ V=e^{iB}$$

They claim that from the equation of motion, it follows that $U^n=1$ and $V^n=1$. I do not quite understand why this is so. The equations of motions are $dA=0$ and $dB=0$, which says that $B$ is a constant and $A$ is locally a pure gauge. Why should $B$ equal to $2\pi/n$? Why should,

$$\oint A \ \ \ be \ equal \ to \ \ \ 2\pi/n \ ?$$

Naively, it seems that $B$ can be some arbitrary constant, and for example, if we take the space to be two dimensional, $\oint A$ should be equal to $2\pi Z$ for non-contractible loops. It seems that we need some additional considerations, other than just EOM to obtain the result they quote.

Answer 1

You need to think about what "$B$ is a periodic scalar" means: When $B$ is $2\pi$-periodic, then since $V$ acts on $B$ as $B\mapsto B + \frac{2\pi}{n}$, you necessarily have $V^n = 1$ since $B+2\pi$ is identified with $B$ so $V^n$ is a "do nothing" operator, i.e. the identity.

Similarly, when Gaiotto et. al. say that the transformations act on $A$ as $$ A\mapsto A + \frac{1}{n}\zeta$$ with $\zeta$ a "properly normalized flat gauge field", the normalization they're talking about is such that $S[B = B_0,A = \zeta] = 1$, i.e. the transformation $A\mapsto A+\zeta$ is a do-nothing transformation in terms of $\mathrm{e}^{2\pi\mathrm{i}S[B,A]}$ or, in another diction, the shift in $A$ is "absorbed" by the periodicity of $B$.