# The Nahm Transform for Calorons

# The Nahm Transform for Calorons

One mysterious feature of the self-duality equations on ℝ^{4} is the existence of a quite remarkable non-linear transform, the Nahm transform. It maps solutions to the self-duality equations on ℝ^{4} invariant under a closed translation group G ⊂ ℝ^{4} to solutions to the self-duality equations on (ℝ^{4})✱ 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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