The K\"ahler Potential of Abelian Higgs Vortices

We calculate the K"ahler potential for the Samols metric on the moduli space of Abelian Higgs vortices on $\text{mathbbm}{R}^2$, 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 $\text{mathbbm}{H}^2$ 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 $\text{mathbbm}{R}^2$ in a third way by relating the vortices to SU(2) Yang-Mills instantons on $\text{mathbbm}{R}^2\times S^2$. This approach does not give the correct result, and we offer a possible explanation for this.