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Kähler quantization of vortex moduli

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Abstract

We discuss the Kähler 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 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 \(\Sigma \) 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.

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Notes

  1. Note that the factor of i that is standard in this definition is absent, as we follow the convention of identifying curvatures with real forms, i.e. valued in a copy of \({\mathbb {R}}\) that is identified with the Lie algebra \(\mathfrak {u}(1)\). See also Eq. (19), where both curvature and symplectic form are real, as in [62].

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Acknowledgements

This project was started as part of the activities of a Junior Trimester Program on “Mathematical Physics” hosted at the Hausdorff Research Institute for Mathematics (HIM), University of Bonn, in 2012. We would like to thank HIM for hospitality, as well as Marcel Bökstedt (Aarhus), Kai Cieliebak (Augsburg), Daniel Huybrechts (Bonn), Nick Manton (Cambridge) and two anonymous referees for useful comments.

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  1. Matematiska Vetenskaper, Chalmers Tekniska Högskola and Göteborgs Universitet, 41296, Göteborg, Sweden

    Dennis Eriksson

  2. Institut für Mathematik, Universität Augsburg, 86135, Augsburg, Germany

    Nuno M. Romão

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  1. Dennis Eriksson
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Correspondence to Nuno M. Romão.

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Eriksson, D., Romão, N.M. Kähler quantization of vortex moduli. Lett Math Phys 110, 659–693 (2020). https://doi.org/10.1007/s11005-019-01235-2

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