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Non-Abelian A4 vortices in SO(3) gauge theory and noninvertible symmetries

Yoshihiko Abe1,2,3,*, Tetsutaro Higaki4,5,†, Kazuya Murakami4,‡, Muneto Nitta6,5,7,§, and Ryo Yokokura6,5,∥

  • 1Graduate School of Science and Technology, Keio University, Yokohama, Kanagawa 223-8522, Japan
  • 2Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University, Tokyo 108-8345, Japan
  • 3Quantum Computing Center, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan
  • 4Department of Physics, Keio University, Yokohama 223-8533, Japan
  • 5Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
  • 6Department of Physics, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
  • 7International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2), Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8531, Japan

  • *Contact author: yabe3@keio.jp
  • †Contact author: thigaki@rk.phys.keio.ac.jp
  • ‡Contact author: kazuyamurakami@keio.jp
  • §Contact author: mune.nitta@gmail.com
  • ∥Contact author: ryokokur@keio.jp
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Phys. Rev. D 114, 105005 – Published 4 November, 2026

DOI: https://doi.org/10.1103/bz2v-mls7

Abstract

We construct finite-tension non-Abelian vortex solutions in a renormalizable (3+1)-dimensional SO(3) gauge theory Higgsed to the tetrahedral group A4 by a Higgs field in the spin-3 representation. Since the vacuum manifold is SO(3)/A4, the vortices are characterized by the non-Abelian fundamental group π1(SO(3)/A4)≃A˜4, the binary tetrahedral group. We obtain explicit axisymmetric vortex solutions carrying holonomies corresponding to the order-two and order-three conjugacy classes of A4, determine their tensions numerically, and show that they exhibit type-I, type-II, and Bogomol’nyi-Prasad-Sommerfield-like behavior depending on the Higgs and gauge boson mass ratios. The vortices are classified by conjugacy classes of A˜4, while their infrared descriptions are labeled by conjugacy classes of A4. We further demonstrate that the smooth finite-tension vortices reduce in the infrared to Gukov-Witten surface operators of the A4 discrete gauge theory, thereby establishing a finite-energy ultraviolet completion of noninvertible defects in a renormalizable gauge-Higgs theory.

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References (71)

  1. A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, England, 2000).
  2. N. S. Manton and P. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2004).
  3. E. J. Weinberg, Classical solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2012).
  4. A. A. Abrikosov, On the Magnetic Properties of Superconductors of the Second Group, Sov. Phys. JETP 5, 1174 (1957), https://jetp.ras.ru/cgi-bin/e/index/e/5/6/p1174?a=list.
  5. H. B. Nielsen and P. Olesen, Vortex line models for dual strings, Nucl. Phys. B61, 45 (1973).
  6. F. A. Bais, Flux metamorphosis, Nucl. Phys. B170, 32 (1980).
  7. M. G. Alford, K. Benson, S. R. Coleman, J. March-Russell, and F. Wilczek, The interactions and excitations of nonabelian vortices, Phys. Rev. Lett. 64, 1632 (1990); 65, 668(E) (1990).
  8. F. Wilczek and Y.-S. Wu, Space-time approach to holonomy scattering, Phys. Rev. Lett. 65, 13 (1990).
  9. M. G. Alford, K. Benson, S. R. Coleman, J. March-Russell, and F. Wilczek, Zero modes of nonabelian vortices, Nucl. Phys. B349, 414 (1991).
  10. M. G. Alford, K.-M. Lee, J. March-Russell, and J. Preskill, Quantum field theory of nonAbelian strings and vortices, Nucl. Phys. B384, 251 (1992).
  11. M. Bucher, K.-M. Lee, and J. Preskill, On detecting discrete Cheshire charge, Nucl. Phys. B386, 27 (1992).
  12. H.-K. Lo and J. Preskill, NonAbelian vortices and nonAbelian statistics, Phys. Rev. D 48, 4821 (1993).
  13. L. Brekke, H. Dykstra, S. J. Hughes, and T. D. Imbo, Knots and links of nonabelian string, Phys. Lett. B 288, 273 (1992).
  14. L. Brekke, S. J. Collins, and T. D. Imbo, NonAbelian vortices on surfaces and their statistics, Nucl. Phys. B500, 465 (1997).
  15. P. McGraw, Evolution of a nonAbelian cosmic string network, Phys. Rev. D 57, 3317 (1998).
  16. A. Hanany and D. Tong, Vortices, instantons and branes, J. High Energy Phys. 07 (2003) 037.
  17. R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi, and A. Yung, NonAbelian superconductors: Vortices and confinement in N=2 SQCD, Nucl. Phys. B673, 187 (2003).
  18. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, and N. Sakai, Moduli space of non-Abelian vortices, Phys. Rev. Lett. 96, 161601 (2006).
  19. M. Eto, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci, and N. Yokoi, Non-Abelian vortices of higher winding numbers, Phys. Rev. D 74, 065021 (2006).
  20. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, and N. Sakai, Solitons in the Higgs phase: The Moduli matrix approach, J. Phys. A 39, R315 (2006).
  21. M. Shifman and A. Yung, Supersymmetric Solitons (Cambridge University Press, Cambridge, England, 2009).
  22. A. P. Balachandran, S. Digal, and T. Matsuura, Semi-superfluid strings in high density QCD, Phys. Rev. D 73, 074009 (2006).
  23. E. Nakano, M. Nitta, and T. Matsuura, Non-Abelian strings in high density QCD: Zero modes and interactions, Phys. Rev. D 78, 045002 (2008).
  24. M. Eto and M. Nitta, Color magnetic flux tubes in dense QCD, Phys. Rev. D 80, 125007 (2009).
  25. M. Eto, E. Nakano, and M. Nitta, Effective world-sheet theory of color magnetic flux tubes in dense QCD, Phys. Rev. D 80, 125011 (2009).
  26. M. Eto, M. Nitta, and N. Yamamoto, Instabilities of non-Abelian vortices in dense QCD, Phys. Rev. Lett. 104, 161601 (2010).
  27. M. Eto, Y. Hirono, M. Nitta, and S. Yasui, Vortices and other topological solitons in dense quark matter, Prog. Theor. Exp. Phys. 2014, 012D01 (2014).
  28. N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51, 591 (1979).
  29. V. Poenaru and G. Toulouse, The crossing of defects in ordered media and the topology of 3-manifolds, J. Phys. (Paris) 38, 887 (1977).
  30. G. E. Volovik and V. P. Mineev, Investigation of singularities in superfluid He3 and in liquid crystals by the homotopic topology methods, Zh. Eksp. Teor. Fiz. 72, 2256 (1977) [Sov. Phys. JETP 45, 1186 (1977)], https://jetp.ras.ru/cgi-bin/e/index/e/45/6/p1186?a=list.
  31. A. P. Balachandran, F. Lizzi, and V. G. J. Rodgers, Topological symmetry breakdown in cholesterics, nematics, and He3, Phys. Rev. Lett. 52, 1818 (1984).
  32. O. D. Lavrentovich and M. Kleman, Cholesteric liquid crystals: Defects and topology, in Chirality in Liquid Crystals, edited by H.-S. Kitzerow and C. Bahr (Springer, New York, 2001), pp. 115–158.
  33. G. W. Semenoff and F. Zhou, Discrete symmetries and 1/3-quantum vortices in condensates of F=2 cold atoms, Phys. Rev. Lett. 98, 100401 (2007).
  34. M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda, Collision dynamics and rung formation of non-Abelian vortices, Phys. Rev. Lett. 103, 115301 (2009).
  35. Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates, Phys. Rep. 520, 253 (2012).
  36. M. O. Borgh and J. Ruostekoski, Core structure and non-Abelian reconnection of defects in a biaxial nematic spin-2 Bose-Einstein condensate, Phys. Rev. Lett. 117, 275302 (2016).
  37. T. Mawson, T. Petersen, J. Slingerland, and T. Simula, Braiding and fusion of non-Abelian vortex anyons, Phys. Rev. Lett. 123, 140404 (2019).
  38. T. Annala, R. Zamora-Zamora, and M. Möttönen, Topologically protected vortex knots and links, Communications Physics 5, 309 (2022).
  39. H. Rajamäki, T. Annala, and M. Möttönen, Topologically protected vortex knots in an experimentally realizable system, Phys. Rev. Lett. 133, 236604 (2024).
  40. M. Kobayashi, Y. Nozaki, Y. Koda, and M. Nitta, Quantum knots that never come untied, arXiv:2410.07470.
  41. K. Masuda and M. Nitta, Half-quantized non-Abelian vortices in neutron P23 superfluids inside magnetars, Prog. Theor. Exp. Phys. 2020, 013D01 (2020).
  42. Y. Masaki, T. Mizushima, and M. Nitta, Non-Abelian half-quantum vortices in P32 topological superfluids, Phys. Rev. B 105, L220503 (2022).
  43. Y. Masaki, T. Mizushima, and M. Nitta, Non-Abelian anyons and non-Abelian vortices in topological superconductors, in Encyclopedia of Condensed Matter Physics, vol. 2 (Elsevier, New York, 2024), 2nd ed., pp. 755–794.
  44. M. Kobayashi and M. Nitta, Core structures of vortices in Ginzburg-Landau theory for neutron P23 superfluids, Phys. Rev. C 105, 035807 (2022).
  45. M. Kobayashi and M. Nitta, Proximity effects of vortices in neutron P32 superfluids in neutron stars: Vortex core transitions and covalent bonding of vortex molecules, Phys. Rev. C 107, 045801 (2023).
  46. G. Marmorini, S. Yasui, and M. Nitta, Pulsar glitches from quantum vortex networks, Sci. Rep. 14, 7857 (2024).
  47. T. Hattori, M. Nitta, and K. Sekizawa, Formation of bound composite vortices of a singly-quantized S10 vortex and half-quantized P32 vortices in the S10-P32 coexisting phase in neutron stars, arXiv:2605.28718.
  48. P. A. Horvathy, The nonabelian Aharonov-Bohm effect, Phys. Rev. D 33, 407 (1986).
  49. R. Sundrum and L. J. Tassie, Nonabelian Aharonov-Bohm effects, Feynman paths, and topology, J. Math. Phys. (N.Y.) 27, 1566 (1986).
  50. M. G. Alford, J. March-Russell, and F. Wilczek, Discrete quantum hair on black holes and the nonabelian Aharonov-Bohm effect, Nucl. Phys. B337, 695 (1990).
  51. J. Preskill and L. M. Krauss, Local discrete symmetry and quantum mechanical hair, Nucl. Phys. B341, 50 (1990).
  52. F. A. Bais, P. van Driel, and M. de Wild Propitius, Quantum symmetries in discrete gauge theories, Phys. Lett. B 280, 63 (1992).
  53. F. A. Bais, P. van Driel, and M. de Wild Propitius, Anyons in discrete gauge theories with Chern-Simons terms, Nucl. Phys. B393, 547 (1993).
  54. L. M. Krauss and F. Wilczek, Discrete gauge symmetry in continuum theories, Phys. Rev. Lett. 62, 1221 (1989).
  55. G. Altarelli and F. Feruglio, Discrete flavor symmetries and models of neutrino mixing, Rev. Mod. Phys. 82, 2701 (2010).
  56. F. Feruglio, Are Neutrino Masses Modular Forms? (World Scientific, Singapore, 2019), pp. 227–266.
  57. H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, and M. Tanimoto, Non-Abelian discrete symmetries in particle physics, Prog. Theor. Phys. Suppl. 183, 1 (2010).
  58. B. A. Ovrut, Isotropy subgroups of SO(3) and Higgs potentials, J. Math. Phys. (N.Y.) 19, 418 (1978).
  59. G. Etesi, Spontaneous symmetry breaking in SO(3) gauge theory to discrete subgroups, J. Math. Phys. (N.Y.) 37, 1596 (1996).
  60. S. F. King and Y.-L. Zhou, Spontaneous breaking of SO(3) to finite family symmetries with supersymmetry—an A4 model, J. High Energy Phys. 11 (2018) 173.
  61. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, J. High Energy Phys. 02 (2015) 172.
  62. S. Gukov and E. Witten, Rigid surface operators, Adv. Theor. Math. Phys. 14, 87 (2010).
  63. B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius, and I. Valenzuela, Non-invertible global symmetries and completeness of the spectrum, J. High Energy Phys. 09 (2021) 203.
  64. S. Kobayashi, M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda, Abe homotopy classification of topological excitations under the topological influence of vortices, Nucl. Phys. B856, 577 (2012).
  65. C. Chatterjee and M. Nitta, BPS Alice strings, J. High Energy Phys. 09 (2017) 046.
  66. C. Chatterjee and M. Nitta, The effective action of a BPS Alice string, Eur. Phys. J. C 77, 809 (2017).
  67. U. Leonhardt and G. E. Volovik, How to create Alice string (half quantum vortex) in a vector Bose-Einstein condensate, Pis’ma Zh. Eksp. Teor. Fiz. 72, 66 (2000).
  68. R. Sato, F. Takahashi, and M. Yamada, Unified origin of axion and monopole dark matter, and solution to the domain-wall problem, Phys. Rev. D 98, 043535 (2018).
  69. C. Chatterjee, T. Higaki, and M. Nitta, Note on a solution to domain wall problem with the Lazarides-Shafi mechanism in axion dark matter models, Phys. Rev. D 101, 075026 (2020).
  70. A. S. Schwarz, Field theories with no local conservation of the electric charge, Nucl. Phys. B208, 141 (1982).
  71. Y. Hidaka, M. Nitta, and R. Yokokura, Selection rules of topological solitons from non-invertible symmetries in axion electrodynamics, J. High Energy Phys. 09 (2025) 082.

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Phys. Rev. D 114, 105005– Published 4 November, 2026
  • Received 21 July 2026
  • Accepted 1 October 2026
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