 Integer-current representation of the Villain model

Integrating every angle in the bond Fourier series of the Villain model imposes the Kronecker constraint ${\sum }_{\mu }[n_{x,\mu }-n_{x-\hat{\mu },\mu }]=0$. Thus its dual variables are divergence-free integer currents. On a planar lattice such currents are rotated gradients of integer heights on faces. Noncontractible current sectors must additionally be retained on a torus.