Let $Sigma$ be a compact Riemann surface and $h_{d,k}(Sigma)$ denote the space of degree $dgeq 1$ holomorphic maps $Sigmara CP^k$. 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 kahler 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 kahler metrics on $h_{1,k}(S^2)$ have finite volume for $kgeq 2$. 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 Baptistas formula for $h_{d,k}(S^2)$ for all $d$, $k$, and $h_{2,1}(T^2)$.
تحميل البحث