Crystal Volumes and Monopole Dynamics
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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.
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.
- [Cro19]Cross, R.: Doubly Periodic Monopole Dynamics and Crystal Volumes, Ph.D. thesis, University of Arizona (2019)Google Scholar
- [GKZ08]Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants and multidimensional determinants, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA (2008), Reprint of the 1994 editionGoogle Scholar
- [Moc19]Mochizuki, T.: Doubly Periodic Monopoles and q-difference Modules, arXiv e-prints (2019). arXiv:1902.03551
- [PP88]Pedersen, H., Poon, Y.S.: Hyper-Kähler metrics and a generalization of the bogomolny equations. Commun. Math. Phys. 117(4), 569–580 (1988). http://projecteuclid.org/euclid.cmp/1104161817
- [Swa62]Swan, R.G.: Factorization of polynomials over finite fields. Pac. J. Math. 12(3), 1099–1106 (1962). https://www.projecteuclid.org:443/euclid.pjm/1103036322
- [TWZ18]Treumann, D., Williams, H., Zaslow, E.: Kasteleyn Operators from Mirror Symmetry, arXiv e-prints (2018). arXiv:1810.05985