The volume of the space of holomorphic maps from S^2 to CP^k

Let $\mathrm{\Sigma }$ be a compact Riemann surface and ${\text{h}}_{d,k}(\mathrm{\Sigma })$ denote the space of degree $d\ge 1$ holomorphic maps $\mathrm{\Sigma }\text{ra}{\text{CP}}^k$. In theoretical physics this arises as the moduli space of charge $d$ lumps (or instantons) in the ${\text{CP}}^k$ model on $\mathrm{\Sigma }$. There is a natural Riemannian metric on this moduli space, called the $L^2$ metric, whose geometry is conjectured to control the low energy dynamics of ${\text{CP}}^k$ lumps. In this paper an explicit formula for the $L^2$ metric on of ${\text{h}}_{d,k}(\mathrm{\Sigma })$ in the special case $d=1$ and $\mathrm{\Sigma }=S^2$ is computed. Essential use is made of the k"ahler property of the $L^2$ metric, and its invariance under a natural action of $G=U(k+1)\times U(2)$\mathrm{.}Itisshownthat\text{em}all$G$-invariantk\ddot{a}hlermetricson$\h_{1,k}(S^2)$havefinitevolumefor$k\geq 2$\mathrm{.}Thevolumeof$\h_{1,k}(S^2)$withrespecttothe$L^2$ metric is computed explicitly and is shown to agree with a general formula for ${\text{h}}_{d,k}(\mathrm{\Sigma })$ recently conjectured by Baptista. The area of a family of twice punctured spheres in ${\text{h}}_{d,k}(\mathrm{\Sigma })$ is computed exactly, and a formal argument is presented in support of Baptista’s formula for ${\text{h}}_{d,k}(S^2)$ for all $d$, $k$, and ${\text{h}}_{2,1}(T^2)$.