We discuss the Kähler quantization of moduli spaces of vortices in line bundles over compact surfaces $$\Sigma $$ 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 $$\Sigma $$ 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 $$\Sigma $$ 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.

中文翻译：

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涡旋模量的 Kähler 量化

我们讨论了紧致表面 $$\Sigma $$ Σ 上线丛中涡旋模空间的 Kähler 量化。这为薛定谔-陈省身-西蒙斯模型中的量子涡旋动力学研究提供了一个半经典框架。我们在上同调的行列式中采用 Deligne 对 Quillen 度量的方法来构造此上下文中的所有量子希尔伯特空间。还获得了量子波截面的另一种描述，即在 $$\Sigma $$ Σ 本身（以其面积形式的倍数的预量化值）上的旋量的多粒子状态方面。这一观点阐明了规范理论中出现的量子孤子粒子的性质。