K\"ahler quantization of vortex moduli

TL;DR

This paper develops a K"ahler quantization framework for vortex moduli spaces on compact surfaces, providing insights into quantum vortex dynamics and the nature of quantum solitonic particles within gauge theories.

Contribution

It introduces a novel semiclassical quantization approach for vortex moduli spaces using Deligne's and Quillen's methods, and offers an alternative multiparticle spinor description of quantum states.

Findings

Quantum Hilbert space dimensions depend on surface area and genus.

A new multiparticle spinor representation of quantum wavefunctions is proposed.

Relations between different quantization data choices are explored.

Abstract

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.