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Asymptotic behaviour and the moduli space of doubly-periodic instantons

Asymptotic behaviour and the moduli space of doubly-periodic instantons We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line ℂ, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T×ℙ1. The converse statement is also true, namely a holomorphic bundle on T×ℙ1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperkähler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperkähler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

Asymptotic behaviour and the moduli space of doubly-periodic instantons

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References (23)

Publisher
Springer Journals
Copyright
Copyright © 2001 by Springer-Verlag Berlin Heidelberg & EMS
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/s100970100032
Publisher site
See Article on Publisher Site

Abstract

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line ℂ, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T×ℙ1. The converse statement is also true, namely a holomorphic bundle on T×ℙ1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperkähler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperkähler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus.

Journal

Journal of the European Mathematical Society – Springer Journals

Published: Nov 1, 2001

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