Kähler quantization of vortex moduli

Journal article, 2020

We discuss the Kähler quantization of moduli spaces of vortices in line bundles over compact surfaces Σ. This furnishes a semiclassical framework for the study of quantum vortex dynamics in the Schrödinger–Chern–Simons model. We employ 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 Σ 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 Σ 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.