Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 071, 30 pages      arXiv:2303.02623
Contribution to the Special Issue on Topological Solitons as Particles

Geometry of Gauged Skyrmions

Josh Cork a and Derek Harland b
a) School of Computing and Mathematical Sciences,University of Leicester, University Road, Leicester, UK
b) School of Mathematics, University of Leeds, Woodhouse Lane, Leeds, UK

Received March 12, 2023, in final form September 14, 2023; Published online October 01, 2023

A work of Manton showed how skymions may be viewed as maps between riemannian manifolds minimising an energy functional, with topologically non-trivial global minimisers given precisely by isometries. We consider a generalisation of this energy functional to gauged skyrmions, valid for a broad class of space and target 3-manifolds where the target is equipped with an isometric $G$-action. We show that the energy is bounded below by an equivariant version of the degree of a map, describe the associated BPS equations, and discuss and classify solutions in the cases where $G={\rm U}(1)$ and $G={\rm SU}(2)$.

Key words: skyrmions; topological solitons; BPS equations.

pdf (602 kb)   tex (41 kb)  


  1. Adam C., Nappi C.R., The Skyrme model with pion masses, Phys. Lett. B 233 (1984), 109-115.
  2. Adam C., Oles K., Wereszczynski A., The dielectric Skyrme model, Phys. Lett. B 807 (2020), 135560, 6 pages, arXiv:2005.00018.
  3. Adam C., Sanchez-Guillen J., Wereszczynski A., A Skyrme-type proposal for baryonic matter, Phys. Lett. B 691 (2010), 105-110, arXiv:1001.4544.
  4. Arthur K., Tchrakian D.H., $\mathrm{SO}(3)$ gauged soliton of an $\mathrm{O}(4)$ sigma model on $\mathbb{R}^3$, Phys. Lett. B 378 (1996), 187-193, arXiv:hep-th/9601053.
  5. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  6. Atiyah M.F., Bott R., The moment map and equivariant cohomology, Topology 23 (1984), 1-28.
  7. Bär C., Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509-521.
  8. Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren Text Ed., Springer, Berlin, 2004.
  9. Biswas I., Hurtubise J., Monopoles on Sasakian three-folds, Comm. Math. Phys. 339 (2015), 1083-1100, arXiv:1412.4050.
  10. Brihaye Y., Hartmann B., Tchrakian D.H., Monopoles and dyons in ${\rm SO}(3)$ gauged Skyrme models, J. Math. Phys. 42 (2001), 3270-3281, arXiv:hep-th/0010152.
  11. Callan Jr. C.G., Witten E., Monopole catalysis of skyrmion decay, Nuclear Phys. B 239 (1984), 161-176.
  12. Callies M., Frégier Y., Rogers C.L., Zambon M., Homotopy moment maps, Adv. Math. 303 (2016), 954-1043, arXiv:1304.2051.
  13. Cieliebak K., Gaio A.R., Mundet i Riera I., Salamon D.A., The symplectic vortex equations and invariants of Hamiltonian group actions, J. Symplectic Geom. 1 (2002), 543-645, arXiv:math.SG/0111176.
  14. Cork J., Skyrmions from calorons, J. High Energy Phys. 2018 (2018), no. 11, 137, 43 pages, arXiv:1810.04143.
  15. Cork J., Harland D., Winyard T., A model for gauged skyrmions with low binding energies, J. Phys. A 55 (2022), 015204, 37 pages, arXiv:2109.06886.
  16. Criado J.C., Khoze V.V., Spannowsky M., The emergence of electroweak skyrmions through Higgs bosons, J. High Energy Phys. 2021 (2021), no. 3, 162, 23 pages, arXiv:2012.07694.
  17. D'Hoker E., Farhi E., Decoupling a fermion in the standard electro-weak theory, Nuclear Phys. B 248 (1984), 77-89.
  18. Faddeev L.D., Some comments on the many-dimensional solitons, Lett. Math. Phys. 1 (1976), 289-293.
  19. Gudnason S.B., Nitta M., Baryonic torii: Toroidal baryons in a generalized Skyrme model, Phys. Rev. D 91 (2015), 045018, 9 pages, arXiv:1410.8407.
  20. Harland D., Topological energy bounds for the Skyrme and Faddeev models with massive pions, Phys. Lett. B 728 (2014), 518-523, arXiv:1311.2403.
  21. Livramento L.R., Radu E., Shnir Y., Solitons in the gauged Skyrme-Maxwell model, SIGMA 19 (2023), 042, 17 pages, arXiv:2301.12848.
  22. Manton N.S., Geometry of skyrmions, Comm. Math. Phys. 111 (1987), 469-478.
  23. Manton N.S., Skyrmions. A theory of nuclei, World Scientific Publishing, London, 2022.
  24. Mundet i Riera I., Teoría de Yang-Mills-Higgs para fibraciones simplécticas, Ph.D. Thesis, Universidad Autónoma de Madrid, 1999, math.SG/9912150.
  25. Navarro-Lérida F., Radu E., Tchrakian D.H., On the topological charge of $SO(2)$ gauged Skyrmions in $2+1$ and $3+1$ dimensions, Phys. Lett. B 791 (2019), 287-292, arXiv:1811.09535.
  26. Piette B.M.A.G., Tchrakian D.H., Static solutions in the ${\rm U}(1)$ gauged Skyrme model, Phys. Rev. D 62 (2000), 025020, 10 pages, arXiv:hep-th/9709189.
  27. Radu E., Tchrakian D.H., Spinning $\mathrm{U}(1)$ gauged skyrmions, Phys. Lett. B 632 (2006), 109-113, arXiv:hep-th/0509014.
  28. Romão N.M., Speight J.M., The geometry of the space of BPS vortex-antivortex pairs, Comm. Math. Phys. 379 (2020), 723-772, arXiv:1807.00712.
  29. Skyrme T.H.R., A unified field theory of mesons and baryons, Nuclear Phys. 31 (1962), 556-569.
  30. Tchrakian D.H., Some aspects of Skyrme-Chern-Simons densities, J. Phys. A 55 (2022), 245401, 19 pages, arXiv:2111.13097.
  31. Walton E., On the geometry of magnetic Skyrmions on thin films, J. Geom. Phys. 156 (2020), 103802, 20 pages, arXiv:1908.08428.
  32. Witten E., Baryons in the $1/N$ expansion, Nuclear Phys. B 160 (1979), 57-115.

Previous article  Next article  Contents of Volume 19 (2023)