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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.

Replace each XY model bond weight by its periodic Gaussian approximation. With angles integrated using $d\theta /(2\pi )$, the Villain model iswhere $N_b$ is the number of bonds. Changing the angle-measure normalization multiplies $Z_V$ by an explicit constant only. The Poisson summation formula applied to one bond gives the equivalent winding representationThus the same Villain model can be written $Z_V=\int {\prod }_{x}d{\theta }_x/(2\pi ){\sum }_{\{p_e\}}\exp [-\beta {\sum }_e({\Delta }_e\theta -2\pi p_e{)}^2/2]$. The coefficient $1/\sqrt{2\pi \beta }$ is the PDF's value; the converted TeX incorrectly places $\beta$ outside the square root. The cosine-to-Gaussian replacement is a large-$\beta$ approximation, but the rest of the calculation treats $Z_V$ as the exact model for every positive $\beta$.

For a periodic two-dimensional lattice the vortex charge neutrality constraint is $\boxed{\sum _x{q}_x=0}$. In the winding form of the Villain model, $q$ is the oriented plaquette curl of the integer bond variable $p$, up to an orientation sign. Summing all plaquette curls cancels each bond against its neighboring plaquette. This proves neutrality directly. Equivalently, restoring the common dual-height mode modulo its integer shift gives ${\int }_0^{1}d{\varphi }_0{e}^{2\pi i{\varphi }_0\sum q}$, which vanishes unless the total charge is zero; only then is the inverse lattice Laplacian well-defined on the charge source.

On an infinite plane, the same condition is required for a localized finite-energy configuration with constant phase at infinity. Net charge $Q$ produces $\theta \sim Q\arg x$ far away, so its exterior action iswhich diverges unless $Q=0$. With an open boundary the total charge instead equals the boundary winding, and the finite planar duality in part (b) can sum nonneutral configurations. Thus neutrality follows from periodicity or the stated finite-energy boundary condition, not from arbitrary boundaries. In three dimensions the analogous local constraint is that vortex lines cannot end in the bulk.

For the Villain model on a periodic two-dimensional lattice, the vortex charge is the oriented curl of its integer bond variable. Summing over all faces cancels every bond twice, proving neutrality. In an infinite plane, a configuration with net winding $Q$ has far-field spin wave action $\pi \beta Q^2\log (L/a)$, so finite-energy localized configurations must have $Q=0$. At an open boundary the total winding instead equals the boundary circulation; neutrality cannot be imposed without a boundary condition.