Notes on a new construction of hyperkahler metrics

In joint work with Davide Gaiotto and Greg Moore [1] we recently proposed a new connection between hyperkähler geometry and the counting of BPS states in supersymmetric field theory. While the story is motivated by physics, it leads to a concrete new recipe for constructing complete hyperkähler metrics on the total spaces of certain complex integrable systems. The aim of this note is briefly to describe what this recipe is, and to comment on some of the issues involved in converting it into an actual theorem. Let us briefly describe some of the highlights. • We begin with a collection of “integrable system data ” described in Section 2.1 below. These data include a complex manifold B containing a divisor D. For example, B could be the complex plane, and D some collection of points. The data also include a local system of lattices Γ over B ′ = B \D, from which we build a 2r-torus bundle M′ over B′, with nontrivial monodromy around D. Finally, we have a “central charge” homomorphism Z: Γ → C, varying holomorphically over B′. From these data we build a simple explicit hyperkähler metric gsf onM′. However, the metric gsf is incomplete, and our main interest is in complete metrics. • Naively we might hope to complete gsf by adding some degenerate torus fibers over D, thus extendingM ′ toM⊃M′, in such a way that gsf will extend toM. However, it seems that this is impossible: roughly speaking, gsf is too homogeneous to have such an extension. Instead, we construct a new metric g on M′, which differs from gsf by certain “quantum corrections.” • The quantum corrections are obtained by solving a certain explicit integral equation, (4.8) below. The main new ingredient in this equation is a set of integer “invariants” Ω(γ), which should be examples of generalized Donaldson-Thomas invariants in the sense of [2, 3]. In particular, the Kontsevich-Soibelman wall-crossing formula for gen-eralized Donaldson-Thomas invariants, as written in [2], plays an important role in the construction. Indeed the original motivation for this construction was an attempt to understand the physical meaning of the formula of [2]. 1 a