Integral eigenvalues in compact rank-2 symmetric $U(1)$ gauge theory I am reading a paper related to rank-2 symmetric $U(1)$ gauge theory:
Fracton topological order from the Higgs and partial-confinement mechanisms of rank-two gauge theory (or arXiv:1802.10108).
My question concerns a skipped calculation in Sec. II. There, the authors write that $A_{\mu\nu}$ being compact (mod $2\pi$) and the canonical commutator $[A_{\mu\nu},E_{\mu\nu}]=-i$ implies that the eigenvalues of $E_{\mu\nu}$ are integers. I don't follow the authors' reasoning here and I don't see why the eigenvalues should be integers. Would someone please enlighten me?
Note: cross-posted on Physics Overflow
 A: The range of $A_{\mu\nu}$ and its commutation relation with $E_{\mu\nu}$ are essentially the same as the those of the azimuthal angle $\phi$ and the angular momentum component $L_{z}$.  $\phi$ covers the compact range from $0$ to $2\pi$, then wraps around on itself, just like $A_{\mu\nu}$.  The corresponding conjugate momentum operator is $L_{z}=\frac{\hbar}{i}\frac{\partial}{\partial\phi}$, and the commutator of the two is $[\phi,L_{z}]=i\hbar$.  This is the same as the commutation relation $[A_{\mu\nu},E_{\mu\nu}]$, apart from an overall factor of $-\hbar$.  It is well known that the operator $L_{z}$ has eigenvalues $\pm m\hbar$, for integer values of $m$; this is required for the system to have a wave function that is single valued under the coordinate redefinition $\phi\rightarrow\phi+2\pi$.  For exactly the same reason, in order that the (physically meaningless) affine shift $A_{\mu\nu}\rightarrow A_{\mu\nu}+2\pi$ not change the wave function, $E_{\mu\nu}$ must be quantized the same way as $L_{z}$ was.  Accounting for the difference of $-\hbar$, this means that the eigenvalues of $E_{\mu\nu}$ are integers.
