= Vortex Coulomb gas
{title2=$Z_{\rm v}=\sum_{\{q\}}e^{-2\pi^2\beta\sum_{x,y}q_xG(x-y)q_y}$}
In two dimensions, the integer defects $q_x$ on dual faces are <vortices> of winding $2\pi q_x$. Here $G$ is the inverse positive lattice <Laplacian> with its boundary or zero-mode convention specified. At separations large compared with the lattice spacing, $G(r)=-(2\pi)^{-1}\log(r/a)+\text{constant}$, yielding the pair action $-2\pi\beta\sum_{i<j}q_iq_j\log(r_{ij}/a)$ and core self-energies. Like charges repel and opposite charges attract. With periodic boundaries the total charge is zero and the inverse is taken on the mean-zero subspace; harmonic winding sectors require their own factors.
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