On the Constraints Defining BPS Monopoles

Abstract

We discuss the explicit formulation of the transcendental constraints defining spectral curves of SU (2) BPS monopoles in the twistor approach of Hitchin, following Ercolani and Sinha. We obtain an improved version of the Ercolani-Sinha constraints, and show that the Corrigan-Goddard conditions for constructing monopoles of arbitrary charge can be regarded as a special case of these. As an application, we study the spectral curve of the tetrahedrally symmetric 3-monopole, an example where the Corrigan-Goddard conditions need to be modified. A particular 1-cycle on the spectral curve plays an important rôle in our analysis. *

Line bundles on spectral curves and the generalised Legendre transform construction of hyperkähler metrics

“…defining a spectral curve S. Thus, the derivative with respect to w i a (2i−2 ≥ a ≥ 2) is 0 unless i = 2, and comparing with (7.3), we see, as in [24], that (5.6) are satisfied at S if and only if Observe now that the reality property of each H l translates into the following reality property of E l :…”

“…[24]): Lemma 6.1. A function F defined by (6.5) is real, provided each G p and c p satisfy the following two conditions:…”

Untitled

“…ref. [45] 20) where the contour C 0 goes around the origin, whereas the contour C r encircles the roots of a quadratic equation, …”

“…In the purely gauge 3d, N=4 theory with N c = 2 the quantum moduli space metric is known to be given by the SO(3) invariant Atiyah-Hitchin metric [9,16]. The asymptotical metrics on the classical multi-monopole moduli spaces for well-separated monopoles in 4d are available in their explicit form [17,18], whereas the exact metrics are only known up to certain algebraic (Ercolani-Sinha) constraints [19,20].…”

On charge-3 cyclic monopoles

“…The first of these (H2) may be expressed in the following manner [HMR00]: given a canonical homology basis {â µ ,b µ } for the curveĈ there exists a 1-cycle es = n·â+m·b such that for every holomorphic differential…”