Symmetric monopoles and identity of Ramanujan

Abstract:
We develop the Ercolani-Sinha constrains of $SU(2)$ monopoles and make this effective for (a five parameter family of centred) charge 3 monopoles.  In particular we show how to solve the transcendental constraints arising on the spectral curve. For a class of curve with $Z_N$ symmetry the transendental constraints become a number theoretic problem and a recently proven identity of Ramanujan provides a solution. The Ercolani-Sinha constraints provides a gauge-transform of the Nahm data.