= Vortex charge neutrality
{title2=$\sum_xq_x=0$}
For the <Villain model> on a periodic two-dimensional lattice, the vortex charge is the oriented curl of its integer bond variable. Summing over all faces cancels every bond twice, proving neutrality. In an infinite plane, a configuration with net winding $Q$ has far-field <spin wave> action $\pi\beta Q^2\log(L/a)$, so finite-energy localized configurations must have $Q=0$. At an open boundary the total winding instead equals the boundary circulation; neutrality cannot be imposed without a boundary condition.
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