Integral eigenvalues in compact rank-2 symmetric $U(1)$ gauge theory

The range of $A_{munu}$ and its commutation relation with $E_{munu}$ 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 $2pi$, then wraps around on itself, just like $A_{munu}$. The corresponding conjugate momentum operator is $L_{z}=frac{hbar}{i}frac{partial}{partialphi}$, and the commutator of the two is $[phi,L_{z}]=ihbar$. This is the same as the commutation relation $[A_{munu},E_{munu}]$, apart from an overall factor of $-hbar$. It is well known that the operator $L_{z}$ has eigenvalues $pm mhbar$, for integer values of $m$; this is required for the system to have a wave function that is single valued under the coordinate redefinition $phirightarrowphi+2pi$. For exactly the same reason, in order that the (physically meaningless) affine shift $A_{munu}rightarrow A_{munu}+2pi$ not change the wave function, $E_{munu}$ must be quantized the same way as $L_{z}$ was. Accounting for the difference of $-hbar$, this means that the eigenvalues of $E_{munu}$ are integers.