- \nabla^{2} G = δ

and has

G (r) = - (2 π)^{- 1} lo g (r / a) + 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 becomes

S_{v} = μ_{core} i \sum q_{i}^{2} - 2 π β i < j \sum q_{i} q_{j} lo g (r_{ij} / a),

(1)

with 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 π β ∣ q_{i} q_{j} ∣ lo g (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 μ_{core} + 2 π β lo g (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.