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Partial Compactification of Monopoles and Metric Asymptotics
About this Title
Chris Kottke and Michael Singer
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 280, Number 1383
ISBNs: 978-1-4704-5541-5 (print); 978-1-4704-7283-2 (online)
DOI: https://doi.org/10.1090/memo/1383
Published electronically: October 7, 2022
Keywords: Non-abelian magnetic monopole,
moduli space,
compactification,
manifold with corners,
pseudodifferential operator,
gauge theory
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction
- 2. The Bogomolny equations on a scattering manifold
- 3. Formal 1-parameter families
- 4. Moduli of ideal monopoles
- 5. Universal gluing space and parameterized gluing
- 6. The metric
- A. Sobolev spaces
- B. Coulomb gauge
- C. Linear analysis
- D. Pseudodifferential operators
Abstract
We construct a partial compactification of the moduli space, $\mathcal {M}_k$, of $\mathrm {SU}(2)$ magnetic monopoles on $\mathbb {R}^3$, wherein monopoles of charge $k$ decompose into widely separated ‘monopole clusters’ of lower charge going off to infinity at comparable rates. The hyperKähler metric on $\mathcal {M}_k$ has a complete asymptotic expansion up to the boundary, the leading term of which generalizes the asymptotic metric discovered by Bielawski, Gibbons and Manton when each lower charge is 1.- Michael Atiyah and Nigel Hitchin, The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988. MR 934202, DOI 10.1515/9781400859306
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