The quantization of the reduced first-order dynamics of the nonrelativistic model for Chern–Simons vortices introduced by Manton is studied on a sphere of given radius. We perform geometric quantization on the moduli space of static solutions, using a Kähler polarization, to construct the quantum Hilbert space. Its dimension is related to the volume of the moduli space in the usual classical limit. The angular momenta associated with the rotational SO(3) symmetry of the model are determined for both the classical and the quantum systems. The results obtained are consistent with the interpretation of the solitons in the model as interacting bosonic particles.

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. *

We investigate the geometry of the moduli space of N -vortices on line bundles over a closed Riemann surface Σ of genus g > 1, in the little explored situation where 1 ≤ N < g. In the regime where the area of the surface is just large enough to accommodate N vortices (which we call the dissolving limit), we describe the relation between the geometry of the moduli space and the complex geometry of the Jacobian variety of Σ. For N = 1, we show that the metric on the moduli space converges to a natural Bergman metric on Σ. When N > 1, the vortex metric typically degenerates as the dissolving limit is approached, the degeneration occurring precisely on the critical locus of the Abel-Jacobi map of Σ at degree N . We describe consequences of this phenomenon from the point of view of multivortex dynamics.