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Moduli of monopole walls and amoebas

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Abstract

We study doubly-periodic monopoles, also called monopole walls, determining their spectral data and computing the dimensions of their moduli spaces. Using spectral data we identify the moduli, and compare our results with a perturbative analysis. We also identify an SL(2, \(\mathbb{Z}\)) action on monopole walls, in which the S transformation corresponds to the Nahm transform.

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References

  1. N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].

  2. G. Chalmers and A. Hanany, Three-dimensional gauge theories and monopoles, Nucl. Phys. B 489 (1997) 223 [hep-th/9608105] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  4. S.A. Cherkis and A. Kapustin, Periodic monopoles with singularities and N = 2 super QCD, Commun. Math. Phys. 234 (2003) 1 [hep-th/0011081] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. S. Bolognesi, Multi-monopoles, magnetic bags, bions and the monopole cosmological problem, Nucl. Phys. B 752 (2006) 93 [hep-th/0512133] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. K.-M. Lee and E.J. Weinberg, BPS magnetic monopole bags, Phys. Rev. D 79 (2009) 025013 [arXiv:0810.4962] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  7. D. Harland, The large-N limit of the Nahm transform, Commun. Math. Phys. 311 (2012) 689 [arXiv:1102.3048] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. S. Bolognesi and D. Tong, Monopoles and holography, JHEP 01 (2011) 153 [arXiv:1010.4178] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. P. Sutcliffe, Monopoles in AdS, JHEP 08 (2011) 032 [arXiv:1104.1888] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. N. Manton, Monopole planets and galaxies, Phys. Rev. D 85 (2012) 045022 [arXiv:1111.2934] [INSPIRE].

    ADS  Google Scholar 

  11. J. Evslin and S.B. Gudnason, High Q BPS monopole bags are urchins, arXiv:1111.3891 [INSPIRE].

  12. C.M. Linton, Rapidly convergent representations for Green’s functions for Laplace’s equation, Proc. R. Soc. Lond. A 455 (1999) 1767.

    MathSciNet  ADS  Google Scholar 

  13. D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220 [hep-th/9608163] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  14. S.A. Cherkis, C. O’Hara and C. Samann, Super Yang-Mills theory with impurity walls and instanton moduli spaces, Phys. Rev. D 83 (2011) 126009 [arXiv:1103.0042] [INSPIRE].

    ADS  Google Scholar 

  15. E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. S.A. Cherkis, Instantons on gravitons, Commun. Math. Phys. 306 (2011) 449 [arXiv:1007.0044] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. B. Haghighat and S. Vandoren, Five-dimensional gauge theory and compactification on a torus, JHEP 09 (2011) 060 [arXiv:1107.2847] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. S.A. Cherkis and A. Kapustin, Nahm transform for periodic monopoles and N = 2 super Yang-Mills theory, Commun. Math. Phys. 218 (2001) 333 [hep-th/0006050] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. S. Donaldson, Nahm’s equations and the classification of monopoles, Commun. Math. Phys. 96 (1984) 387 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. S. Jarvis, Construction of Euclidian monopoles, Proc. London Math. Soc. 77 (1998) 193.

    Article  MathSciNet  Google Scholar 

  21. S. Jarvis, Euclidian monopoles and rational maps, Proc. London Math. Soc. 77 (1998) 170.

    Article  MathSciNet  Google Scholar 

  22. R. Ward, Periodic monopoles, Phys. Lett. B 619 (2005) 177 [hep-th/0505254] [INSPIRE].

    ADS  Google Scholar 

  23. I. Newton, Letters to Oldenburg, 13 June 1676 and 24 October 1676.

  24. I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhauser, Boston U.S.A. (1994).

    Book  MATH  Google Scholar 

  25. G. Mikhalkin and H. Rullgård, Amoebas of maximal area, Int. Math. Res. Not. 9 (2001) 441

    Article  Google Scholar 

  26. A.G. Khovanskii, Newton polyhedra and toroidal varieties, Func. Anal. Appl. 11 (1977) 289.

    Article  Google Scholar 

  27. W. Nahm, The construction of all self-dual multimonopoles by the ADHM method, in Monopoles in quantum field theory, N.S. Craigie, P. Goddard and W. Nahm eds., World Scientific, Singapore (1982).

    Google Scholar 

  28. W. Nahm, Self-dual monooles and calorons, in Group theoretical methods in physics, Springer Lecture Notes in Physics 201, Springer, Berlin Germany (1984).

    Google Scholar 

  29. R. Ward, A monopole wall, Phys. Rev. D 75 (2007) 021701 [hep-th/0612047] [INSPIRE].

    ADS  Google Scholar 

  30. K.-M. Lee, Sheets of BPS monopoles and instantons with arbitrary simple gauge group, Phys. Lett. B 445 (1999) 387 [hep-th/9810110] [INSPIRE].

    ADS  Google Scholar 

  31. F.J.W. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST handbook of mathematical functions, NIST and Cambridge University Press, Cambridge U.K. (2010).

    Google Scholar 

  32. M.F. Atiyah, N.J. Hitchin and I.M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. London A 362 (1978) 425.

    MathSciNet  ADS  Google Scholar 

  33. J.M. Arms, J.E. Marsden and V. Moncrief, Symmetry and bifurcations of momentum mappings, Comm. Math. Phys. 78 (1980/81) 455.

  34. J. Huebschmann, The singularities of Yang-Mills connections for bundles on a surface. I. The local model, Math. Z. 220 (1995) 595 [dg-ga/9411006].

    Article  MathSciNet  MATH  Google Scholar 

  35. N. Manton, A remark on the scattering of BPS monopoles, Phys. Lett. B 110 (1982) 54 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, 85721-0089, U.S.A.

    Sergey A. Cherkis

  2. Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, U.K.

    Richard S. Ward

Authors
  1. Sergey A. Cherkis
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  2. Richard S. Ward
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Correspondence to Sergey A. Cherkis.

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ArXiv ePrint: 1202.1294

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Cherkis, S.A., Ward, R.S. Moduli of monopole walls and amoebas. J. High Energ. Phys. 2012, 90 (2012). https://doi.org/10.1007/JHEP05(2012)090

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