 Solution

At long distances in two dimensions the lattice Green function solves $-{\nabla }^{2}G=\delta$ and has $G(r)=-(2\pi {)}^{-1}\log (r/a)+\text{constant}$. Indeed the flux of $-\nabla G$ through a circle is one, fixing that coefficient. For well-separated vortices at $r_i$ with integer charges $q_i$, the vortex Coulomb gas action becomeswith the self-energy convention, boundary terms and any global sectors retained separately. The factor two in the pair sum comes from the two ordered terms in $q^{T}Gq$. Opposite charges have pair action $+2\pi \beta |q_i{q}_j|\log (r/a)$, so separating them costs energy and they attract. Like charges have a negative logarithmic pair coefficient and repel. The ultraviolet lattice regularizes the core, while multiply charged defects have a quadratic self-energy cost. A separated unit vortex-antivortex pair costs $2{\mu }_{\mathrm{core}}+2\pi \beta \log (r/a)$, the energetic basis for vortex-antivortex binding at low temperature. In three dimensions one instead has interacting closed vortex lines with a three-dimensional Green kernel; the logarithmic point-pair formula is specifically two-dimensional.