Dual integer-height representation of the planar Villain model ($Z_V\propto {\sum }_{\ell \in {\mathbb{Z}}^{\mathrm{faces}}}{e}^{-{\ell }^{T}L\ell /(2\beta )}$)

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 givesThe Gaussian prefactor $Z_{\mathrm{sw}}$ 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.