= Dual integer-height representation of the planar Villain model
{title2=$Z_V\propto\sum_{\ell\in\mathbb Z^{\rm faces}}e^{-\ell^TL\ell/(2\beta)}$}
On a finite simply connected planar graph, fix the height of the exterior face to zero. Every divergence-free integer bond current is uniquely the oriented difference of the two neighboring face heights. The positive <Graph Laplacian> $L$ on the remaining faces gives its energy. Applying the <Poisson summation formula> to these heights and completing the square gives
$$
Z_V=Z_{\rm sw}\sum_{q\in\mathbb Z^{\rm faces}}e^{-2\pi^2\beta q^TL^{-1}q}.
$$
The Gaussian prefactor $Z_{\rm sw}$ equals the <spin wave> determinant of the original graph, up to angle zero-mode normalization. One way to verify this is that the <matrix-tree theorem> equates the spanning-tree counts of a planar graph and its dual. Boundaries can admit nonzero total vortex charge; neutrality is an additional periodic or finite-energy condition.
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