= Solution
Angle integration gives a dual representation in any lattice dimension. Discrete summation by parts rewrites the phase as
$$
\sum_{x,\mu}n_{x,\mu}(\theta_{x+\hat\mu}-\theta_x)
=-\sum_x\theta_x\sum_\mu(n_{x,\mu}-n_{x-\hat\mu,\mu}).
$$
Each normalized angle integral is a Kronecker delta. This proves the <integer-current representation of the Villain model>
$$
Z_V=(2\pi\beta)^{-N_b/2}\sum_{\operatorname{div}n=0}\exp\left[-\frac1{2\beta}\sum_en_e^2\right].
$$
The 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. Therefore
$$
\sum_en_e^2=\sum_{\langle x,y\rangle_{\rm dual}}(\ell_x-\ell_y)^2=\ell^TL\ell,
$$
where $L$ 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,
$$
Z_V=(2\pi\beta)^{-N_b/2}\sum_{q\in\mathbb Z^{N_f}}
\int_{\mathbb R^{N_f}}d\phi\,
\exp\left[-\frac1{2\beta}\phi^TL\phi+2\pi i q^T\phi\right].
$$
Complete the Gaussian square, or use its <characteristic function>, to evaluate the integral as
$$
\frac{(2\pi\beta)^{N_f/2}}{\sqrt{\det L}}\exp[-2\pi^2\beta q^TL^{-1}q].
$$
Thus the smooth fluctuations and integer defects separate:
$$
\boxed{Z_V=Z_{\rm sw}Z_{\rm v},\qquad
Z_{\rm v}=\sum_{q\in\mathbb Z^{N_f}}e^{-2\pi^2\beta q^TGq},\quad G=L^{-1}.}
$$
The Gaussian factor is $Z_{\rm sw}=(2\pi\beta)^{-(N_b-N_f)/2}/\sqrt{\det L}$. 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 $w_xx/L_x+w_yy/L_y$ makes the remaining height periodic; the <gradient> energy gains $(L_yw_x^2/L_x+L_xw_y^2/L_y)/(2\beta)$. Its height <Poisson summation> supplies the phase $\exp[2\pi i(w_xP_x/L_x+w_yP_y/L_y)]$, where $P_x=\sum q_xx$ and $P_y=\sum q_xy$. A second <Poisson summation> in $w_x,w_y$ yields the positive <harmonic-sector factor in periodic Villain duality>
$$
\mathcal W(q)=\sum_{W_x,W_y\in\mathbb Z}\exp\left[-2\pi^2\beta\left\{
\frac{L_x}{L_y}\left(W_x-\frac{P_x}{L_x}\right)^2+
\frac{L_y}{L_x}\left(W_y-\frac{P_y}{L_y}\right)^2\right\}\right].
$$
Up to charge-independent normalization, the exact periodic result is $Z_V=Z_{\rm sw}\sum_{\sum q=0}e^{-2\pi^2\beta q^TGq}\mathcal W(q)$. 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.
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