  We discuss the K\"ahler quantization of moduli spaces of vortices in line
bundles over compact surfaces $\Sigma$. This furnishes a semiclassical
framework for the study of quantum vortex dynamics in the
Schr\"odinger-Chern-Simons model. We follow Deligne's approach to Quillen's
metric in determinants of cohomology to construct all the quantum Hilbert
spaces in this context. An alternative description of the quantum wavesections,
in terms of multiparticle states of spinors on $\Sigma$ itself (valued in a
prequantization of a multiple of its area form), is also obtained. This
viewpoint sheds light on the nature of the quantum solitonic particles that
emerge from the gauge theory. We find that in some cases (where the area of
$\Sigma$ is small enough in relation to its genus) the dimensions of the
quantum Hilbert spaces may be sensitive to the input data required by the
quantization scheme, and also address the issue of relating different choices
of such data geometrically.
