  The gauged sigma model with target $\mathbb{P}^1$, defined on a Riemann
surface $\Sigma$, supports static solutions in which $k_+$ vortices coexist in
stable equilibrium with $k_-$ antivortices. Their moduli space is a noncompact
complex manifold $M_{(k_+,k_-)}(\Sigma)$ of dimension $k_++k_-$ which inherits
a natural K\"ahler metric $g_{L^2}$ governing the model's low energy dynamics.
This paper presents the first detailed study of $g_{L^2}$, focussing on the
geometry close to the boundary divisor $D=\partial M_{(k_+,k_-)}(\Sigma)$.
  On $\Sigma=S^2$, rigorous estimates of $g_{L^2}$ close to $D$ are obtained
which imply that $M_{(1,1)}(S^2)$ has finite volume and is geodesically
incomplete. On $\Sigma=\mathbb{R}^2$, careful numerical analysis and a
point-vortex formalism are used to conjecture asymptotic formulae for $g_{L^2}$
in the limits of small and large separation. All these results make use of a
localization formula, expressing $g_{L^2}$ in terms of data at the (anti)vortex
positions, which is established for general $M_{(k_+,k_-)}(\Sigma)$.
  For arbitrary compact $\Sigma$, a natural compactification of the space
$M_{(k_+,k_-)}(\Sigma)$ is proposed in terms of a certain limit of gauged
linear sigma models, leading to formulae for its volume and total scalar
curvature. The volume formula agrees with the result established for
$Vol(M_{(1,1)}(S^2))$, and allows for a detailed study of the thermodynamics of
vortex-antivortex gas mixtures. It is found that the equation of state is
independent of the genus of $\Sigma$, and that the entropy of mixing is always
positive.
