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

Abstract

Let $\Sigma$ be a compact Riemann surface and $\h_{d,k}(\Sigma)$ denote the space of degree $d\geq 1$ holomorphic maps $\Sigma\ra \CP^k$. In theoretical physics this arises as the moduli space of charge $d$ lumps (or instantons) in the $\CP^k$ model on $\Sigma$. 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 $\CP^k$ lumps. In this paper an explicit formula for the $L^2$ metric on of $\h_{d,k}(\Sigma)$ in the special case $d=1$ and $\Sigma=S^2$ is computed. Essential use is made of the k\"ahler property of the $L^2$ metric, and its invariance under a natural action of $G=U(k+1)\times U(2)$. It is shown that {\em all} $G$-invariant k\"ahler metrics on $\h_{1,k}(S^2)$ have finite volume for $k\geq 2$. The volume of $\h_{1,k}(S^2)$ with respect to the $L^2$ metric is computed explicitly and is shown to agree with a general formula for $\h_{d,k}(\Sigma)$ recently conjectured by Baptista. The area of a family of twice punctured spheres in $\h_{d,k}(\Sigma)$ is computed exactly, and a formal argument is presented in support of Baptista's formula for $\h_{d,k}(S^2)$ for all $d$, $k$, and $\h_{2,1}(T^2)$.