Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons

Details Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons

2006-08-24

arxiv.org/abs/math/0608597v7

Article:Commun.Math.Phys.280:285-313 (2008); Erratum: Commun.Math.Phys.288:799-800 (2009)

Mathematics

Differential Geometry

29 pages, LaTeX, no figures; Lemma 2.1 corrected

Scientific paper

We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 singularity removal theorem of Sibner--Sibner and Rade we prove that given a smooth, finite energy, self-dual SU(2) connection over a complete ALF space, its energy is congruent to a Chern--Simons invariant of the boundary three-manifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly. Then we introduce framed moduli spaces of self-dual connections over Ricci flat ALF spaces. We prove that the moduli space of smooth, irreducible, rapidly decaying self-dual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold. We calculate its dimension by a variant of the Gromov--Lawson relative index theorem. As an application, we study Yang--Mills instantons over the flat R^3 x S^1, the multi-Taub--NUT family, and the Riemannian Schwarzschild space.

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