  We calculate the K\"ahler potential for the Samols metric on the moduli space
of Abelian Higgs vortices on $\mathbbm{R}^{2}$, in two different ways. The
first uses a scaling argument. The second is related to the Polyakov conjecture
in Liouville field theory. The K\"ahler potential on the moduli space of
vortices on $\mathbbm{H}^{2}$ is also derived, and we are led to a geometrical
reinterpretation of these vortices. Finally, we attempt to find the K\"ahler
potential for vortices on $\mathbbm{R}^{2}$ in a third way by relating the
vortices to SU(2) Yang-Mills instantons on $\mathbbm{R}^{2}\times S^{2}$. This
approach does not give the correct result, and we offer a possible explanation
for this.
