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The Many Facets of GeometryA Tribute to Nigel Hitchin$
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Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon

Print publication date: 2010

Print ISBN-13: 9780199534920

Published to Oxford Scholarship Online: September 2010

DOI: 10.1093/acprof:oso/9780199534920.001.0001

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The Nahm Transform for Calorons

The Nahm Transform for Calorons

(p.34) IV The Nahm Transform for Calorons
The Many Facets of Geometry

Benoit Charbonneau

Jacques Hurtubise

Oxford University Press

One mysterious feature of the self-duality equations on ℝ4 is the existence of a quite remarkable non-linear transform, the Nahm transform. It maps solutions to the self-duality equations on ℝ4 invariant under a closed translation group G ⊂ ℝ4 to solutions to the self-duality equations on (ℝ4)✱ invariant under the dual group G✱. This transform uses spaces of solutions to the Dirac equation, it is quite sensitive to boundary conditions, which must be defined with care, and it is not straightforward: for example, it tends to interchange rank and degree. This chapter is organized as follows. Section 4.2 summarizes the work of Nye and Singer towards showing that the Nahm transform is an equivalence between calorons and appropriate solutions to Nahm's equations. Section 4.3 describes the complex geometry (‘spectral data’) that encodes a caloron. Section 4.4 studies the process by which spectral data also correspond to solutions to Nahm's equations. Section 4.5 shows that the two Nahm transforms are inverses. Section 4.6 gives a description of moduli, expounded in Charbonneau and Hurtubise (2007).

Keywords:   Nahm transform, calorons, spectral data, complex geometry, Dirac equation

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