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Angle integration gives a dual representation in any lattice dimension. Discrete summation by parts rewrites the phase asEach normalized angle integral is a Kronecker delta. This proves the integer-current representation of the Villain modelThe PDF does not specify dimension or boundaries. The point-vortex vortex Coulomb gas interpretation below is two-dimensional. In three dimensions the defects are vortex lines, and the dual constrained-current representation still holds but is not a gas of point charges with logarithmic interactions.

First take a connected simply connected planar lattice, with free outer angles, so no torus winding sectors are silently discarded. Assign an integer height $\ell$ to each dual face, fixing the exterior face height to zero. A divergence-free integer current is exactly the oriented height difference on the two faces adjoining its bond. To see existence, integrate the rotated current along a dual path: the change around a closed dual path is the total divergence inside, hence zero. Integer currents give integer heights, and fixing the exterior removes their constant-shift ambiguity. Thereforewhere $L$ is the positive dual Graph Laplacian with the exterior height fixed. This gives the dual integer-height representation of the planar Villain model.

Apply the supplied Poisson summation formula separately to every interior height. For $N_f$ interior faces,Complete the Gaussian square, or use its characteristic function, to evaluate the integral asThus the smooth fluctuations and integer defects separate:The Gaussian factor is $Z_{\mathrm{sw}}=(2\pi \beta {)}^{-(N_b-N_f)/2}/\sqrt{\det L}$. It really is the spin wave factor of the original lattice: the planar Euler identity gives $N_b-N_f=N_x-1$, and the matrix-tree theorem identifies $\det L$ with the spanning-tree count of the original graph. Its reduced primal Laplacian has the same determinant, so the noncompact Gaussian angle integral with one angle fixed has exactly this prefactor. The compact common-angle mode has been normalized to one. The remaining $q$ variables are integer vortex windings on the faces, giving the vortex Coulomb gas.

For periodic boundaries, $L$ has a zero mode, and there are also two noncontractible current sectors. Neutral charges use the inverse $G$ on the mean-zero subspace. An integer dual height may have twists $w_x,w_y\in \mathbb{Z}$ across periods $L_x,L_y$. Subtracting $w_{x}x/L_x+w_{y}y/L_y$ makes the remaining height periodic; the gradient energy gains $(L_y{w}_x^2/L_x+L_x{w}_y^2/L_y)/(2\beta )$. Its height Poisson summation supplies the phase $\exp [2\pi i(w_x{P}_x/L_x+w_y{P}_y/L_y)]$, where $P_x=\sum q_{x}x$ and $P_y=\sum q_{x}y$. A second Poisson summation in $w_x,w_y$ yields the positive harmonic-sector factor in periodic Villain dualityUp to charge-independent normalization, the exact periodic result is $Z_V=Z_{\mathrm{sw}}{\sum }_{\sum q=0}{e}^{-2{\pi }^2\beta q^{T}Gq}\mathcal{W}(q)$. Smooth spin waves still factor out, but the finite-volume global winding sectors couple to vortex dipole moments. The familiar neutral bulk formula suppresses this boundary information; it must not be asserted as an exact finite-torus formula without specifying those sectors.