Curvature of Hyperkähler Quotients

We prove estimates for the sectional curvature of hyperkähler quotients and give applications to moduli spaces of solutions to Nahm’s equations and Hitchin’s equations. This note was motivated by the following observation: the sectional curvature of the moduli space of charge k SU(2)-monopoles is bounded (by an explicit constant depending on normalisations). Unlike most statements about monopole metrics, this one has a remarkably easy proof which led us to investigate estimates on sectional curvature of general (finite or infinite-dimensional) Kähler and hyperkähler quotients. We recall that there is an explicit formula, due to J. Jost and X.-W. Peng [7], for the sectional curvature of a large class of quotients, which include hyperkähler quotients. The quotients in [7] are formed by taking a Riemannian Banach manifold (M, g) with a smooth and isometric action of Banach Lie group G which is free on an invariant the level set φ−1(c) of a suitable smooth map φ. Jost and Peng compute the curvature of φ−1(c)/G by giving variational formulae for the second fundamental form of the embedding φ−1(c) ↪→M and for the O’Neill tensor of the submersion φ−1(c)→ φ−1(c)/G. Our aim is to give only pointwise estimates on the sectional curvature of hyperkähler quotients and so our proofs are much simpler than in [7]. We conclude that for 1and 2-dimensional gauge theories, i.e. moduli spaces of solutions to Nahm’s equations and to Hitchin’s equations, one gets bounds on the curvature for free, i.e. without seeking any apriori bounds on solutions of relevant differential equations. In the 1-dimensional case, this is a consequence of the Sobolev embedding W (a, b) → L∞(a, b), while in dimension 2 this follows from an analogous embedding of W (Z) into the Orlicz space Let2−1(Z). We also give a simple criterion for a hyperkähler quotient of a finite-dimensional vector space to have asymptotically null curvature. 1. Infinite-dimensional hyperkähler quotients 1.1. Riemannian Banach manifolds. Let M be a smooth Banach manifold modelled on a Banach space E (see [8] for basics on Banach manifolds). We have a well-defined tangent bundle TM and the cotangent bundle T ∗M (bundle of continuous linear functionals). Both of these are Banach manifolds. Since E∗ ⊗ E∗ is not necessarily complete (with the norm ‖α‖ = sup{α(x, y); ‖x‖E = ‖y‖E = 1}), we consider its completion E∗⊗̂E∗ and the corresponding bundle T ∗M⊗̂T ∗M . Definition 1.1. A weak Riemannian metric onM is a smooth section g of T ∗M⊗̂T ∗M which induces a (continuous) positive definite symmetric bilinear form on each tangent space TmM . The metric g is called strong if the topology induced by g on each fibre is equivalent to the topology of the model Banach space E.