The volume of the space of holomorphic maps from S2S2 to CPkCPk

Let ΣΣ be a compact Riemann surface and Hd,k(Σ)Hd,k(Σ) denote the space of degree d≥1d≥1 holomorphic maps Σ→CPkΣ→CPk. In theoretical physics this arises as the moduli space of charge dd lumps (or instantons) in the CPkCPk model on ΣΣ. There is a natural Riemannian metric on this moduli space, called the L2L2 metric, whose geometry is conjectured to control the low-energy dynamics of CPkCPk lumps. In this paper an explicit formula for the L2L2 metric on Hd,k(Σ)Hd,k(Σ) in the special case d=1d=1 and Σ=S2Σ=S2 is computed. Essential use is made of the kähler property of the L2L2 metric, and its invariance under a natural action of G=U(k+1)×U(2)G=U(k+1)×U(2). It is shown that all  GG-invariant kähler metrics on H1,k(S2)H1,k(S2) have finite volume for k≥2k≥2. The volume of H1,k(S2)H1,k(S2) with respect to the L2L2 metric is computed explicitly and is shown to agree with a general formula for Hd,k(Σ)Hd,k(Σ) recently conjectured by Baptista. The area of a family of twice punctured spheres in Hd,k(Σ)Hd,k(Σ) is computed exactly, and a formal argument is presented in support of Baptista’s formula for Hd,k(S2)Hd,k(S2) for all dd, kk, and H2,1(T2)H2,1(T2).