Abstract
Non-linear sigma models with scalar fields taking values on \( \mathbb{C}{\mathrm{\mathbb{P}}}^n \) complex manifolds are addressed. In the simplest n = 1 case, where the target manifold is the \( {\mathbb{S}}^2 \) sphere, we describe the scalar fields by means of stereographic maps. In this case when the \( \mathbb{U}(1) \) symmetry is gauged and Maxwell and mass terms are allowed, the model accommodates stable self-dual vortices of two kinds with different energies per unit length and where the Higgs field winds at the cores around the two opposite poles of the sphere. Allowing for dielectric functions in the magnetic field, similar and richer self-dual vortices of different species in the south and north charts can be found by slightly modifying the potential. Two different situations are envisaged: either the vacuum orbit lies on a parallel in the sphere, or one pole and the same parallel form the vacuum orbit. Besides the self-dual vortices of two species, there exist BPS domain walls in the second case. Replacing the Maxwell contribution of the gauge field to the action by the second Chern-Simons secondary class, only possible in (2 + 1)-dimensional Minkowski space-time, new BPS topological defects of two species appear. Namely, both BPS vortices and domain ribbons in the south and the north charts exist because the vacuum orbit consits of the two poles and one parallel. Formulation of the gauged \( \mathbb{C}{\mathrm{\mathbb{P}}}^2 \) model in a reference chart shows a self-dual structure such that BPS semi-local vortices exist. The transition functions to the second or third charts break the \( \mathbb{U}(1)\times \mathbb{S}\mathbb{U}(2) \) semi-local symmetry, but there is still room for standard self-dual vortices of the second species. The same structures encompassing N complex scalar fields are easily generalized to gauged \( \mathbb{C}{\mathrm{\mathbb{P}}}^N \) models.
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References
M.K. Prasad and C.M. Sommerfield, An Exact Classical Solution for the ’t Hooft Monopole and the Julia-Zee Dyon, Phys. Rev. Lett. 35 (1975) 760 [INSPIRE].
E.B. Bogomolny, Stability of Classical Solutions, Sov. J. Nucl. Phys. 24 (1976) 449 [INSPIRE].
E. Witten and D. Olive, Supersymmetry algebras that include topological charges, Phys. Lett. 78 (1978) 97 [INSPIRE].
H.B. Nielsen and P. Olesen, Vortex Line Models for Dual Strings, Nucl. Phys. B 61 (1973) 45 [INSPIRE].
R. Jackiw and E.J. Weinberg, Selfdual Chern-Simons vortices, Phys. Rev. Lett. 64 (1990) 2234 [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].
C.-k. Lee, K.-M. Lee and E.J. Weinberg, Supersymmetry and Selfdual Chern-Simons Systems, Phys. Lett. B 243 (1990) 105 [INSPIRE].
W. García Fuertes and J. Mateos Guilarte, Selfdual solitons in N = 2 supersymmetric Chern-Simons gauge theory, J. Math. Phys. 38 (1997) 6214 [INSPIRE].
A. Yung, Vortices on the Higgs branch of the Seiberg-Witten theory, Nucl. Phys. B 562 (1999) 191 [hep-th/9906243] [INSPIRE].
J.D. Edelstein, W. García Fuertes, J. Mas and J. Mateos Guilarte, Phases of dual superconductivity and confinement in softly broken N = 2 supersymmetric Yang-Mills theories, Phys. Rev. D 62 (2000) 065008 [hep-th/0001184] [INSPIRE].
X.-r. Hou, Abrikosov string in N = 2 supersymmetric QED, Phys. Rev. D 63 (2001) 045015 [hep-th/0005119] [INSPIRE].
A. Achucarro, A.C. Davis, M. Pickles and J. Urrestilla, Vortices in theories with flat directions, Phys. Rev. D 66 (2002) 105013 [hep-th/0109097] [INSPIRE].
K. Konishi and L. Spanu, NonAbelian vortex and confinement, Int. J. Mod. Phys. A 18 (2003) 249 [hep-th/0106175] [INSPIRE].
B.J. Schroers, Bogomolny solitons in a gauged O(3) σ-model, Phys. Lett. B 356 (1995) 291 [hep-th/9506004] [INSPIRE].
G. Nardelli, Magnetic vortices from a nonlinear σ-model with local symmetry, Phys. Rev. Lett. 73 (1994) 2524 [INSPIRE].
J.M. Baptista, Vortex equations in Abelian gauged σ-models, Commun. Math. Phys. 261 (2006) 161 [math/0411517] [INSPIRE].
M. Nitta and W. Vinci, Decomposing Instantons in Two Dimensions, J. Phys. A 45 (2012) 175401 [arXiv:1108.5742] [INSPIRE].
M. Nitta, Fractional instantons and bions in the O(N) model with twisted boundary conditions, arXiv:1412.7681 [INSPIRE].
T. Misumi, M. Nitta and N. Sakai, Neutral bions in the \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) model for resurgence, arXiv:1412.0861 [INSPIRE].
A. Alonso-Izquierdo, M.A.G. Leon and J. Mateos Guilarte, Kinks in a non-linear massive σ-model, Phys. Rev. Lett. 101 (2008) 131602 [arXiv:0808.3052] [INSPIRE].
A. Alonso-Izquierdo, M.A.G. Leon and J. Mateos Guilarte, BPS and non-BPS kinks in a massive non-linear S 2 -σ-model, Phys. Rev. D 79 (2009) 125003 [arXiv:0903.0593] [INSPIRE].
A. Alonso-Izquierdo, M.A.G. Leon, J. Mateos Guilarte and M. de la Torre Mayado, On domain walls in a Ginzburg-Landau non-linear S 2 -σ-model, JHEP 08 (2010) 111 [arXiv:1009.0617] [INSPIRE].
J.-H. Lee and S. Nam, Bogomolny equations of Chern-Simons Higgs theory from a generalized Abelian Higgs model, Phys. Lett. B 261 (1991) 437 [INSPIRE].
W. García Fuertes and J. Mateos Guilarte, Self-Dual Vortices in Abelian Higgs Models with Dielectric Function on the Noncommutative Plane, Eur. Phys. J. C 74 (2014) 3002 [arXiv:1404.7740] [INSPIRE].
M. Arai, M. Naganuma, M. Nitta and N. Sakai, Manifest supersymmetry for BPS walls in N =2 nonlinear σ-models, Nucl. Phys. B 652 (2003) 35 [hep-th/0211103] [INSPIRE].
Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, All exact solutions of a 1/4 Bogomol’nyi-Prasad-Sommerfield equation, Phys. Rev. D 71 (2005) 065018 [hep-th/0405129] [INSPIRE].
M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Solitons in the Higgs phase: The moduli matrix approach, J. Phys. A 39 (2006) R315 [hep-th/0602170] [INSPIRE].
T. Vachaspati and A. Achucarro, Semilocal cosmic strings, Phys. Rev. D 44 (1991) 3067 [INSPIRE].
M. Hindmarsh, Existence and stability of semilocal strings, Phys. Rev. Lett. 68 (1992) 1263 [INSPIRE].
G.W. Gibbons, M.E. Ortiz, F. Ruiz Ruiz and T.M. Samols, Semilocal strings and monopoles, Nucl. Phys. B 385 (1992) 127 [hep-th/9203023] [INSPIRE].
M.A. Lohe, Generalized noninteracting vortices, Phys. Rev. D 23 (1981) 2335 [INSPIRE].
W. García Fuertes and J. Mateos Guilarte, Selfdual solitons in N = 2 supersymmetric Chern-Simons gauge theory, J. Math. Phys. 38 (1997) 6214 [INSPIRE].
D. Bazeia, R. Casana, E. da Hora and R. Menezes, Generalized self-dual Maxwell-Chern-Simons-Higgs model, Phys. Rev. D 85 (2012) 125028 [arXiv:1206.0998] [INSPIRE].
D. Bazeia, E. da Hora, C. dos Santos and R. Menezes, BPS Solutions to a Generalized Maxwell-Higgs Model, Eur. Phys. J. C 71 (2011) 1833 [arXiv:1201.2974] [INSPIRE].
E.J. Weinberg, Multivortex Solutions of the Ginzburg-Landau Equations, Phys. Rev. D 19 (1979) 3008 [INSPIRE].
A. Jaffe and C. Taubes, Vortices and Monopoles, Structure of Static Gauge Theories, Birkhäuser, (1980).
R. Wang, The existence of Chern-Simons vortices, Commun. Math. Phys. 137 (1991) 587 [INSPIRE].
L. Jacobs and C. Rebbi, Interaction Energy of Superconducting Vortices, Phys. Rev. B 19 (1979) 4486 [INSPIRE].
A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other topological Defects, Cambridge University Press, (2000).
M. Eto, T. Fujimori, S.B. Gudnason, K. Konishi, T. Nagashima et al., Fractional Vortices and Lumps, Phys. Rev. D 80 (2009) 045018 [arXiv:0905.3540] [INSPIRE].
W. García Fuertes and J. Mateos Guilarte, On the solitons of the Chern-Simons-Higgs model, Eur. Phys. J. C 9 (1999) 167 [hep-th/9812102] [INSPIRE].
G. Dunne, Self-dual Chern-Simons Theories, Springer Verlag, Heidelberg, (1995).
M. Shifman, W. Vinci and A. Yung, Effective World-Sheet Theory for Non-Abelian Semilocal Strings in N = 2 Supersymmetric QCD, Phys. Rev. D 83 (2011) 125017 [arXiv:1104.2077] [INSPIRE].
M. Shifman and A. Yung, Two-dimensional σ-models related to non-abelian strings in super-Yang-Mills, arXiv:1401.7067 [INSPIRE].
A. Alonso-Izquierdo, W. García Fuertes, M. de la Torre Mayado and J. Mateos Guilarte, One loop corrections to the mass of self-dual semi-local planar topological solitons, Nucl. Phys. B 797 (2008) 431 [arXiv:0707.4592] [INSPIRE].
J. Mateos Guilarte, A. Alonso-Izquierdo, W. García Fuertes, M. de la Torre Mayado and M.J. Senosiain, PoS(ISFTG)013.
L.J. Boya, J.F. Cariñena and J. Mateos Guilarte, Homotopy and Solitons, Fortsch. Phys. 26 (1978) 175 [INSPIRE].
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Alonso-Izquierdo, A., Fuertes, W.G. & Guilarte, J.M. Two species of vortices in massive gauged non-linear sigma models. J. High Energ. Phys. 2015, 139 (2015). https://doi.org/10.1007/JHEP02(2015)139
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DOI: https://doi.org/10.1007/JHEP02(2015)139