Hyperkaehler Limits and the Atiyah-Hitchin Manifold

摘要

Atiyah and Hitchin made a detailed study of the moduli space of centered chargetwo SU(2) monopoles on R(sup 3). The moduli space, which the author shall refer to as the Atiyah-Hitchin manifold, is a noncompact manifold of real dimension four, with the homotopy type of RP(sup 2). It carries a naturally defined SO(3)-invariant Riemannian metric, which is hyperkahler and hence Ricci-flat. In previous papers, the author showed the existence of a family of hyperkahler deformations of the double cover of the Atiyah-Hitchin manifold, parameterized by an nonnegative real number lambda. If lambda=0, one obtains the double cover of the Atiyah-Hitchin space. These deformations are obtained as hyperkahler quotients of certain moduli spaces of solutions to the Nahm equations. The authors shall show that there is a natural limit of these manifolds as lambda tends to infinity, and that this limit may be identified with the Atiyah-Hitchin space (not its double cover).