Some remarks on the Ercolani—Sinha construction of monopoles

Abstract

We develop the Ercolani-Sinha construction of SU(2) monopoles, which provides a gauge transform of the Nahm data.

The Construction of Monopoles

“…Hitchin's sections [9] are defined just in terms of the two patches N, S and r = n − 1 in the above. The Baker-Akhiezer construction of [5,7]…”

“…To evaluate this for 0 1,2 we note (from [5]) that α(0 1 ) = −τ /2, α(0 2 ) = −1/2. Using θ 4 (τ /2) = 0 = θ 2 (1/2) we may simplify (…”

“…In the monopole context Ercolani and Sinha were the first to construct the Baker-Akhiezer function and use this to study Nahm data [7]; this theory was (corrected and) extended in [5] with a needed generalization of the Abel-Jacobi map proved in [4]. In the notation above these works explicitly construct the Baker-Akhiezer solutions to (3.8) as well 3 as ḣh −1 .…”

“…Unfortunately Nahm data is hard to construct. Some years ago Ercolani and Sinha [7] showed how, using integrable systems techniques, one could solve for a gauge transform of the Nahm data directly from C; this theory has been further extended by the authors [2,5]. Determining the gauge transformation is equivalent to solving a further ODE and although its solution exists we don't know how to do this explicitly; we shall relate this to Donaldson's treatment [6] of the complex and real moment map description of the Nahm equations.…”

“…Thus in the N S transition function we have r = 0. In what follows we set (5.4) F = Diag e zηi/ζ /ζ l where l = n − 1 when describing Hitchin's choice of section and l = 0 when describing the Baker-Akhiezer sections 5. We will use capital Roman letters {I, J, .…”