Abstract
We continue the investigation of supersymmetric extensions of baby Skyrme models in d = 2 + 1 dimensions. In a first step, we show that the CP(1) form of the baby Skyrme model allows for the same N = 1 SUSY extension as its O(3) formulation. Then we construct the N = 1 SUSY extension of the gauged baby Skyrme model, i.e., the baby Skyrme model coupled to Maxwell electrodynamics. In a next step, we investigate the issue of N = 2 SUSY extensions of baby Skyrme models. We find that all gauged and ungauged submodels of the baby Skyrme model which support BPS soliton solutions allow for an N = 2 extension such that the BPS solutions are one-half BPS states (i.e., annihilated by one-half of the SUSY charges). In the course of our investigation, we also derive the general BPS equations for completely general N = 2 supersymmetric field theories of (both gauged and ungauged) chiral superfields, and apply them to the gauged nonlinear sigma model as a further, concrete example.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T.H.R. Skyrme, A nonlinear field theory, Proc. Roy. Soc. Lon. A 260 (1961) 127 [INSPIRE].
T.H.R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys. 31 (1962) 556 [INSPIRE].
T.H.R. Skyrme, Kinks and the Dirac equation, J. Math. Phys. 12 (1971) 1735 [INSPIRE].
G.S. Adkins, C.R. Nappi and E. Witten, Static properties of nucleons in the Skyrme model, Nucl. Phys. B 228 (1983) 552 [INSPIRE].
G.S. Adkins and C.R. Nappi, The Skyrme model with pion masses, Nucl. Phys. B 233 (1984) 109 [INSPIRE].
E. Braaten and L. Carson, The deuteron as a toroidal skyrmion, Phys. Rev. D 38 (1988) 3525 [INSPIRE].
O.V. Manko, N.S. Manton and S.W. Wood, Light nuclei as quantized skyrmions, Phys. Rev. C 76 (2007) 055203 [arXiv:0707.0868] [INSPIRE].
R.A. Battye, N.S. Manton, P.M. Sutcliffe and S.W. Wood, Light nuclei of even mass number in the Skyrme model, Phys. Rev. C 80 (2009) 034323 [arXiv:0905.0099] [INSPIRE].
R.A. Battye and P.M. Sutcliffe, Symmetric skyrmions, Phys. Rev. Lett. 79 (1997) 363 [hep-th/9702089] [INSPIRE].
R.M. Battye and P.M. Sutcliffe, Solitonic fullerenes, Phys. Rev. Lett. 86 (2001) 3989 [hep-th/0012215] [INSPIRE].
R.A. Battye and P.M. Sutcliffe, Skyrmions, fullerenes and rational maps, Rev. Math. Phys. 14 (2002) 29 [hep-th/0103026] [INSPIRE].
R. Battye and P. Sutcliffe, Skyrmions and the pion mass, Nucl. Phys. B 705 (2005) 384 [hep-ph/0410157] [INSPIRE].
R. Battye and P. Sutcliffe, Skyrmions with massive pions, Phys. Rev. C 73 (2006) 055205 [hep-th/0602220] [INSPIRE].
N. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press, Cambridge U.K. (2007).
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
E. Witten, Baryons in the 1/n expansion, Nucl. Phys. B 160 (1979) 57 [INSPIRE].
E. Witten, Current algebra, baryons and quark confinement, Nucl. Phys. B 223 (1983) 433 [INSPIRE].
B. Piette, B. Schroers and W. Zakrzewski, Multi-solitons in a two-dimensional Skyrme model, Z. Phys. C 65 (1995) 165 [hep-th/9406160] [INSPIRE].
B. Piette, B. Schroers and W. Zakrzewski, Dynamics of baby skyrmions, Nucl. Phys. B 439 (1995) 205 [hep-ph/9410256] [INSPIRE].
R.A. Leese, M. Peyrard and W.J. Zakrzewski, Soliton scatterings in some relativistic models in (2 + 1) dimensions, Nonlinearity 3 (1990) 773.
B.M.A.G. Piette and W.J. Zakrzewski, Skyrmion dynamics in (2 + 1) dimensions, Chaos, Solitons and Fractals 5 (1995) 2495.
P.M. Sutcliffe, The interaction of Skyrme-like lumps in (2 + 1) dimensions, Nonlinearity 4 (1991) 1109.
T. Weidig, The baby Skyrme models and their multi-skyrmions, Nonlinearity 12 (1999) 1489 [hep-th/9811238] [INSPIRE].
P. Eslami, M. Sarbishaei and W.J. Zakrzewski, Baby Skyrme models for a class of potentials, Nonlinearity 13 (2000) 1867 [hep-th/0001153] [INSPIRE].
A.E. Kudryavtsev, B. Piette and W. Zakrzewski, Mesons, baryons and waves in the baby Skyrmion model, Eur. Phys. J. C 1 (1998) 333 [hep-th/9611217] [INSPIRE].
M. Karliner and I. Hen, Rotational symmetry breaking in baby Skyrme models, Nonlinearity 21 (2008) 399 [arXiv:0710.3939] [INSPIRE].
M. Karliner and I. Hen, Review of rotational symmetry breaking in baby Skyrme models, arXiv:0901.1489 [INSPIRE].
C. Adam, P. Klimas, J. Sanchez-Guillen and A. Wereszczynski, Compact baby skyrmions, Phys. Rev. D 80 (2009) 105013 [arXiv:0909.2505] [INSPIRE].
J. Jaykka, M. Speight and P. Sutcliffe, Broken baby skyrmions, Proc. Roy. Soc. Lond. A 468 (2012) 1085 [arXiv:1106.1125] [INSPIRE].
J. Jaykka and M. Speight, Easy plane baby skyrmions, Phys. Rev. D 82 (2010) 125030 [arXiv:1010.2217] [INSPIRE].
D. Foster, Baby skyrmion chains, Nonlinearity 23 (2010) 465 [arXiv:0904.3846] [INSPIRE].
S. Sondhi, A. Karlhede, S. Kivelson and E. Rezayi, Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies, Phys. Rev. B 47 (1993) 16419 [INSPIRE].
O. Schwindt and N.R. Walet, Towards a phase diagram of the 2D Skyrme model, Europhys. Lett. 55 (2001) 633 [hep-ph/0104229] [INSPIRE].
Y. Kodama, K. Kokubu and N. Sawado, Localization of massive fermions on the baby-skyrmion branes in 6 dimensions, Phys. Rev. D 79 (2009) 065024 [arXiv:0812.2638] [INSPIRE].
Y. Brihaye, T. Delsate, N. Sawado and Y. Kodama, Inflating baby-Skyrme branes in six dimensions, Phys. Rev. D 82 (2010) 106002 [arXiv:1007.0736] [INSPIRE].
T. Delsate and N. Sawado, Localizing modes of massive fermions and a U(1) gauge field in the inflating baby-skyrmion branes, Phys. Rev. D 85 (2012) 065025 [arXiv:1112.2714] [INSPIRE].
T. Gisiger and M.B. Paranjape, Solitons in a baby Skyrme model with invariance under volume/area preserving diffeomorphisms, Phys. Rev. D 55 (1997) 7731 [hep-ph/9606328] [INSPIRE].
M. de Innocentis and R.S. Ward, Skyrmions on the 2-sphere, Nonlinearity 14 (2001) 663 [hep-th/0103046] [INSPIRE].
C. Adam, T. Romanczukiewicz, J. Sanchez-Guillen and A. Wereszczynski, Investigation of restricted baby Skyrme models, Phys. Rev. D 81 (2010) 085007 [arXiv:1002.0851] [INSPIRE].
J. Speight, Compactons and semi-compactons in the extreme baby Skyrme model, J. Phys. A 43 (2010) 405201 [arXiv:1006.3754] [INSPIRE].
C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A Skyrme-type proposal for baryonic matter, Phys. Lett. B 691 (2010) 105 [arXiv:1001.4544] [INSPIRE].
C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A BPS Skyrme model and baryons at large-N c , Phys. Rev. D 82 (2010) 085015 [arXiv:1007.1567] [INSPIRE].
C. Adam, C. Fosco, J. Queiruga, J. Sanchez-Guillen and A. Wereszczynski, Symmetries and exact solutions of the BPS Skyrme model, J. Phys. A 46 (2013) 135401 [arXiv:1210.7839] [INSPIRE].
E. Bonenfant and L. Marleau, Nuclei as near BPS-skyrmions, Phys. Rev. D 82 (2010) 054023 [arXiv:1007.1396] [INSPIRE].
E. Bonenfant, L. Harbour and L. Marleau, Near-BPS skyrmions: non-shell configurations and Coulomb effects, Phys. Rev. D 85 (2012) 114045 [arXiv:1205.1414] [INSPIRE].
P. Di Vecchia and S. Ferrara, Classical solutions in two-dimensional supersymmetric field theories, Nucl. Phys. B 130 (1977) 93 [INSPIRE].
E. Witten and D.I. Olive, Supersymmetry algebras that include topological charges, Phys. Lett. B 78 (1978) 97 [INSPIRE].
A. D’Adda, R. Horsley and P. Di Vecchia, Supersymmetric magnetic monopoles and dyons, Phys. Lett. B 76 (1978) 298 [INSPIRE].
A. D’Adda and P. Di Vecchia, Supersymmetry and instantons, Phys. Lett. B 73 (1978) 162 [INSPIRE].
Z. Hlousek and D. Spector, Why topological charges imply extended supersymmetry, Nucl. Phys. B 370 (1992) 143 [INSPIRE].
C.-K. Lee, K.-M. Lee and E.J. Weinberg, Supersymmetry and selfdual Chern-Simons systems, Phys. Lett. B 243 (1990) 105 [INSPIRE].
J.D. Edelstein, C. Núñez and F. Schaposnik, Supersymmetry and Bogomolny equations in the Abelian Higgs model, Phys. Lett. B 329 (1994) 39 [hep-th/9311055] [INSPIRE].
E.A. Bergshoeff, R.I. Nepomechie and H.J. Schnitzer, Supersymmetric skyrmions in four-dimensions, Nucl. Phys. B 249 (1985) 93 [INSPIRE].
L. Freyhult, The supersymmetric extension of the Faddeev model, Nucl. Phys. B 681 (2004) 65 [hep-th/0310261] [INSPIRE].
D. Bazeia, R. Menezes and A.Y. Petrov, Supersymmetric extensions of k-field models, Phys. Lett. B 683 (2010) 335 [arXiv:0910.2827] [INSPIRE].
C. Adam, J. Queiruga, J. Sanchez-Guillen and A. Wereszczynski, N = 1 supersymmetric extension of the baby Skyrme model, Phys. Rev. D 84 (2011) 025008 [arXiv:1105.1168] [INSPIRE].
C. Adam, J. Queiruga, J. Sanchez-Guillen and A. Wereszczynski, Supersymmetric k-field theories and defect structures, Phys. Rev. D 84 (2011) 065032 [arXiv:1107.4370] [INSPIRE].
C. Adam, J. Queiruga, J. Sanchez-Guillen and A. Wereszczynski, BPS bounds in supersymmetric extensions of k-field theories, Phys. Rev. D 86 (2012) 105009 [arXiv:1209.6060] [INSPIRE].
J. Khoury, J.-L. Lehners and B. Ovrut, Supersymmetric P(X,ϕ) and the ghost condensate, Phys. Rev. D 83 (2011) 125031 [arXiv:1012.3748] [INSPIRE].
J. Khoury, J.-L. Lehners and B.A. Ovrut, Supersymmetric galileons, Phys. Rev. D 84 (2011) 043521 [arXiv:1103.0003] [INSPIRE].
M. Koehn, J.-L. Lehners and B.A. Ovrut, Higher-derivative chiral superfield actions coupled to N = 1 supergravity, Phys. Rev. D 86 (2012) 085019 [arXiv:1207.3798] [INSPIRE].
F. Farakos, C. Germani, A. Kehagias and E.N. Saridakis, A new class of four-dimensional N = 1 supergravity with non-minimal derivative couplings, JHEP 05 (2012) 050 [arXiv:1202.3780] [INSPIRE].
F. Farakos and A. Kehagias, Emerging potentials in higher-derivative gauged chiral models coupled to N = 1 supergravity, JHEP 11 (2012) 077 [arXiv:1207.4767] [INSPIRE].
L.-X. Liu and M. Nitta, Non-abelian vortex-string dynamics from nonlinear realization, Int. J. Mod. Phys. A 27 (2012) 1250097 [arXiv:0912.1292] [INSPIRE].
M. Eto, T. Fujimori, M. Nitta, K. Ohashi and N. Sakai, Higher derivative corrections to non-abelian vortex effective theory, Prog. Theor. Phys. 128 (2012) 67 [arXiv:1204.0773] [INSPIRE].
M. Nitta, Knotted instantons from annihilations of monopole-instanton complex, arXiv:1206.5551 [INSPIRE].
M. Dias, A.Y. Petrov, J. Senise, C.R. and A. da Silva, Effective potential for a SUSY Lee-Wick model: the Wess-Zumino case, arXiv:1212.5220 [INSPIRE].
C. Armendariz-Picon, T. Damour and V.F. Mukhanov, k-inflation, Phys. Lett. B 458 (1999) 209 [hep-th/9904075] [INSPIRE].
C. Armendariz-Picon, V.F. Mukhanov and P.J. Steinhardt, A dynamical solution to the problem of a small cosmological constant and late time cosmic acceleration, Phys. Rev. Lett. 85 (2000) 4438 [astro-ph/0004134] [INSPIRE].
C. Armendariz-Picon, V.F. Mukhanov and P.J. Steinhardt, Essentials of k-essence, Phys. Rev. D 63 (2001) 103510 [astro-ph/0006373] [INSPIRE].
E. Babichev, V. Mukhanov and A. Vikman, k-essence, superluminal propagation, causality and emergent geometry, JHEP 02 (2008) 101 [arXiv:0708.0561] [INSPIRE].
E. Babichev, Global topological k-defects, Phys. Rev. D 74 (2006) 085004 [hep-th/0608071] [INSPIRE].
C. Adam, N. Grandi, J. Sanchez-Guillen and A. Wereszczynski, k-fields, compactons and thick branes, J. Phys. A 41 (2008) 212004 [Erratum ibid. A 42 (2009) 159801] [arXiv:0711.3550] [INSPIRE].
C. Adam, N. Grandi, P. Klimas, J. Sanchez-Guillen and A. Wereszczynski, Compact self-gravitating solutions of quartic (k) fields in brane cosmology, J. Phys. A 41 (2008) 375401 [arXiv:0805.3278] [INSPIRE].
M. Olechowski, k-stabilization in brane models, Phys. Rev. D 78 (2008) 084036 [arXiv:0801.1605] [INSPIRE].
D. Bazeia, L. Losano and R. Menezes, First-order framework and generalized global defect solutions, Phys. Lett. B 668 (2008) 246 [arXiv:0807.0213] [INSPIRE].
D. Bazeia, A. Gomes, L. Losano and R. Menezes, Braneworld models of scalar fields with generalized dynamics, Phys. Lett. B 671 (2009) 402 [arXiv:0808.1815] [INSPIRE].
Y.-X. Liu, Y. Zhong and K. Yang, Scalar-kinetic branes, Europhys. Lett. 90 (2010) 51001 [arXiv:0907.1952] [INSPIRE].
M. Andrews, M. Lewandowski, M. Trodden and D. Wesley, Distinguishing k-defects from their canonical twins, Phys. Rev. D 82 (2010) 105006 [arXiv:1007.3438] [INSPIRE].
V. Dzhunushaliev, V. Folomeev and M. Minamitsuji, Thick brane solutions, Rept. Prog. Phys. 73 (2010) 066901 [arXiv:0904.1775] [INSPIRE].
J. Gladikowski, B. Piette and B. Schroers, Skyrme-Maxwell solitons in (2 + 1)-dimensions, Phys. Rev. D 53 (1996) 844 [hep-th/9506099] [INSPIRE].
C. Adam, C. Naya, J. Sanchez-Guillen and A. Wereszczynski, The gauged BPS baby Skyrme model, Phys. Rev. D 86 (2012) 045010 [arXiv:1205.1532] [INSPIRE].
B. Schroers, Bogomolny solitons in a gauged O(3) σ-model, Phys. Lett. B 356 (1995) 291 [hep-th/9506004] [INSPIRE].
P.K. Tripathy, Supersymmetric gauged O(3) σ-model and selfdual Born-Infeld theory, Phys. Rev. D 59 (1999) 085004 [hep-th/9811186] [INSPIRE].
C. Adam, J. Sanchez-Guillen and A. Wereszczynski, Strongly coupled Skyrme-Faddeev-Niemi hopfions, J. Phys. A 43 (2010) 345402 [arXiv:0911.3673] [INSPIRE].
D. Foster, Massive hopfions, Phys. Rev. D 83 (2011) 085026 [arXiv:1012.2595] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1304.0774
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Adam, C., Queiruga, J.M., Sanchez-Guillen, J. et al. Extended supersymmetry and BPS solutions in baby Skyrme models. J. High Energ. Phys. 2013, 108 (2013). https://doi.org/10.1007/JHEP05(2013)108
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2013)108