New Yang–Mills instantons on multi-centered gravitational instantons.pdfVIP

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New Yang–Mills instantons on multi-centered gravitational instantons.pdf

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New Yang–Mills instantons on multi-centered gravitational instantons

New Yang–Mills instantons on multi-centered gravitational instantons Ga?bor Etesi Alfre?d Re?nyi Institute of Mathematics, Rea?ltanoda u. 13-15, Budapest, H-1053 Hungary etesi@math-inst.hu Tama?s Hausel Miller Institute for Basic Research in Science and Department of Mathematics, University of California at Berkeley, Berkeley CA 94720, USA hausel@math.berkeley.edu July 22, 2002 Abstract In this paper we explicitly calculate the analogue of the ’t Hooft SU(2) Yang–Mills instantons on Gibbons–Hawking multi-centered gravitational instantons, which come in two parallel families: the multi-Eguchi–Hanson, or Ak ALE gravitational instantons and the multi-Taub–NUT spaces, or Ak ALF gravitational instantons. We calculate their energy and find the reducible ones. Following Kronheimer we also exploit the U(1) invariance of our solutions and study the corresponding explicit singular SU(2) magnetic monopole solutions of the Bogomolny equations on flat R3. 1 Introduction This paper is a continuation of the project of finding new SU(2) Yang–Mills instantons on ALF gravitational instantons started in [7] and [8]. In [7] we identified all the reducible SU(2) instantons on the Euclidean Schwarzschild manifold (which is ALF and Ricci flat, though not self-dual), and showed that these solutions, albeit not their reducibility, were already known to [3]. Then in [8] we went on and studied SU(2) instantons on the Taub–NUT space (which is an ALF gravitational instanton). Following [13] we exploited the self-duality of the metric, to obtain a new family of SU(2) instantons, which could be considered as the analogue of the ’t Hooft solutions on R4. (For more historical remarks the reader is referred to [8].) G. Etesi, T. Hausel: New instantons over multi-centered gravitational instantons 2 We carry out the generalization of [8] to the more general case of multi-Taub–NUT spaces (which are also ALF gravitational instantons). An interesting aspect of our paper is that what we do goes almost ver