= Integer-current representation of the Villain model
{title2=$Z_V\propto\sum_{n_e\in\mathbb Z,\,\operatorname{div}n=0}e^{-\sum_en_e^2/(2\beta)}$}
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.
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