Cyclic Monopoles, Affine Toda and Spectral Curves

TL;DR

This paper demonstrates that cyclically symmetric monopoles can be described using affine Toda equations, linking spectral curves, theta-functions, and Nahm data, thus unifying different mathematical frameworks in monopole theory.

Contribution

It establishes a direct connection between cyclic monopoles, affine Toda equations, and spectral curve theta-functions, providing a new integrative perspective.

Findings

Cyclic monopoles are gauge equivalent to Nahm data from Sutcliffe's ansatz.

The Ercolani-Sinha vector and base point are pull-backs of Toda data.

Theta-functions in monopole solutions reduce to Toda theta-functions.

Abstract

We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe's ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha vector) and base point of the linearising flow in the Jacobian of the spectral curve associated to the Nahm equations arise as pull-backs of Toda data. A theorem of Accola and Fay then means that the theta-functions arising in the solution of the monopole problem reduce to the theta-functions of Toda.