Crystal Volumes and Monopole Dynamics

  • Sergey A. CherkisEmail author
  • Rebekah Cross


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.


  1. [CDZS19]
    Closset, C., Del Zotto, M., Saxena, V.: Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA Perspective. SciPost Phys. 6, 052 (2019). arXiv:1812.10451
  2. [Che07]
    Cherkis, S.A.: A journey between two curves. SIGMA 3, 043 (2007). arXiv:hep-th/0703108 MathSciNetzbMATHGoogle Scholar
  3. [Che14]
    Cherkis, S.A.: Phases of five-dimensional theories, monopole walls, and melting crystals. JHEP 06, 027 (2014). arXiv:1402.7117 ADSCrossRefGoogle Scholar
  4. [CPNS77]
    Coleman, S.R., Parke, S.J., Neveu, A., Sommerfield, C.M.: Can one dent a dyon? Phys. Rev. D 15, 544 (1977). ADSCrossRefGoogle Scholar
  5. [Cro15]
    Cross, R.: Asymptotic dynamics of monopole walls. Phys. Rev. D92(4), 045029 (2015). arXiv:1506.07606
  6. [Cro19]
    Cross, R.: Doubly Periodic Monopole Dynamics and Crystal Volumes, Ph.D. thesis, University of Arizona (2019)Google Scholar
  7. [CW12]
    Cherkis, S.A., Ward, R.S.: Moduli of monopole walls and amoebas. JHEP 05, 090 (2012). arXiv:1202.1294
  8. [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
  9. [GM95]
    Gibbons, G.W., Manton, N.S.: The moduli space metric for well separated BPS monopoles. Phys. Lett. B 356, 32–38 (1995). arXiv:hep-th/9506052 ADSMathSciNetCrossRefGoogle Scholar
  10. [Hit82]
    Hitchin, N.J.: Monopoles and geodesics. Commun. Math. Phys. 83, 579–602 (1982). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [HKLR87]
    Hitchin, N.J., Karlhede, A., Lindström, U., Roček, M.: Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108, 535 (1987). ADSCrossRefzbMATHGoogle Scholar
  12. [HKM14]
    Hamanaka, M., Kanno, H., Muranaka, D.: Hyper-Kähler metrics from monopole walls. Phys. Rev. D 89(6), 065033 (2014). arXiv:1311.7143 ADSCrossRefGoogle Scholar
  13. [Law91]
    Lawrence, J.: Polytope volume computation. Math. Comput. 57(195), 259–271 (1991). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [Lee99]
    Lee, K.-M.: Sheets of BPS monopoles and instantons with arbitrary simple gauge group. Phys. Lett. B 445, 387–393 (1999). arXiv:hep-th/9810110 ADSCrossRefGoogle Scholar
  15. [LR88]
    Lindström, U.: New hyperkähler metrics and new supermultiplets. Commun. Math. Phys. 115, 21 (1988). ADSCrossRefzbMATHGoogle Scholar
  16. [Moc19]
    Mochizuki, T.: Doubly Periodic Monopoles and q-difference Modules, arXiv e-prints (2019). arXiv:1902.03551
  17. [MW14]
    Maldonado, R., Ward, R.S.: Dynamics of monopole walls. Phys. Lett. B B734, 328–332 (2014). arXiv:1405.4646 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [PP88]
    Pedersen, H., Poon, Y.S.: Hyper-Kähler metrics and a generalization of the bogomolny equations. Commun. Math. Phys. 117(4), 569–580 (1988).
  19. [Sci17]
    Sciarappa, A.: Exact relativistic toda chain eigenfunctions from separation of variables and gauge theory. JHEP 10, 116 (2017). arXiv:1706.05142
  20. [Swa62]
    Swan, R.G.: Factorization of polynomials over finite fields. Pac. J. Math. 12(3), 1099–1106 (1962).
  21. [TWZ18]
    Treumann, D., Williams, H., Zaslow, E.: Kasteleyn Operators from Mirror Symmetry, arXiv e-prints (2018). arXiv:1810.05985
  22. [War07]
    Ward, R.S.: A monopole wall. Phys. Rev. D 75, 021701 (2007). arXiv:hep-th/0612047 ADSMathSciNetCrossRefGoogle Scholar
  23. [War11]
    Ward, R.S.: Skyrmions and monopoles: isolated and arrayed. J. Phys. Conf. Ser. 284, 012005 (2011). CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of PhysicsUniversity of ArizonaTucsonUSA

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