Crystal Volumes and Monopole Dynamics


The low velocity dynamic of a doubly periodic monopole, also called a monopole wall or monowall for short, is described by geodesic motion on its moduli space. This moduli space is hyperkähler and non-compact. We establish a relation between the Kähler potential of this moduli space and the volume of a region in Euclidean three-space cut out by a plane arrangement associated with each monowall.

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  1. 1.

    As in [GKZ08], we normalize the area of a basic triangle with vertices (0, 0), (1, 0),  and (0, 1) to be one, instead of a half.

  2. 2.

    A spine face is the projection of a face of the cut crystal.

  3. 3.

    Each pairwise interaction contributes once.

  4. 4.

    Here we use our conventions of the footnote on page 9, i.e. \(\delta _{123}\) is twice the conventional triangle area.

  5. 5.

    We suppose for concreteness that \(n_2>n_1>n_3.\)


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SCh is grateful to the organizers of the 2019 workshop “Microlocal Methods in Analysis and Geometry” at CIRM–Luminy and to the Institute des Hautes Études Scientifiques, Bures-sur-Yvette where the final stages of this work were completed. SCh received funding from the European Research Council under the European Union Horizon 2020 Framework Programme (h2020) through the ERC Starting Grant QUASIFT (QUantum Algebraic Structures In Field Theories) nr. 677368. RC thanks the Marshall Foundation for her Dissertation Fellowship funding.

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Correspondence to Sergey A. Cherkis.

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Cherkis, S.A., Cross, R. Crystal Volumes and Monopole Dynamics. Commun. Math. Phys. 377, 503–529 (2020).

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