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Moduli Spaces of Self-Dual Connections over Asymptotically Locally Flat Gravitational Instantons

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An Erratum to this article was published on 28 March 2009

Abstract

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 Råde 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 \({\mathbb{R}}^3 \times S^1\) , the multi-Taub–NUT family, and the Riemannian Schwarzschild space.

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Authors and Affiliations

  1. Department of Geometry, Mathematical Institute, Faculty of Science, Budapest University of Technology and Economics, Egry J. u. 1, H ép., 1111, Budapest, Hungary

    Gábor Etesi

  2. Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, C.P. 6065, 13083-859, Campinas, SP, Brazil

    Gábor Etesi & Marcos Jardim

Authors
  1. Gábor Etesi
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  2. Marcos Jardim
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Correspondence to Marcos Jardim.

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Communicated by G.W. Gibbons

An erratum to this article can be found online at http://dx.doi.org/http://dx.doi.org/10.1007/s00220-009-0797-1.

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Etesi, G., Jardim, M. Moduli Spaces of Self-Dual Connections over Asymptotically Locally Flat Gravitational Instantons. Commun. Math. Phys. 280, 285–313 (2008). https://doi.org/10.1007/s00220-008-0466-9

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  • DOI: https://doi.org/10.1007/s00220-008-0466-9

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