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Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment
National Institute for Mathematical Sciences, Academic exchanges, KT Daeduk 2 Research Center, 70 Yuseong-daero 1689 beon-gil, Yuseong-gu, Daejeon, 34047, Republic of Korea |
In this paper, we consider a Maxwell-Chern-Simons model with anomalous magnetic moment. Our main goal is to show the existence and uniqueness of topological type solutions to this problem on a flat two torus for any configuration of vortex points. Moreover, we also discuss about the stability of topological solutions.
References:
[1] |
A. A. Abrikosov,
On the magnetic properties of superconductors of the second group, Soviet Phys. JETP, 5 (1957), 1174-1182.
|
[2] |
D. Bartolucci, Y. Lee, C. S. Lin and M. Onodera,
Asymptotic analysis of solutions to a gauged $O(3)$ sigma model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 651-685.
doi: 10.1016/j.anihpc.2014.03.001. |
[3] |
F. S. A. Cavalcante, M. S. Cunha and C. A. S. Almeida,
Vortices in a nonminimal Maxwell-Chern-Simons O(3) sigma model, Phys. Lett. B, 475 (2000), 315-323.
doi: 10.1016/S0370-2693(00)00077-0. |
[4] |
D. Chae and O. Y. Imanuvilov,
The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Commun. Math. Phys., 215 (2000), 119-142.
doi: 10.1007/s002200000302. |
[5] |
D. Chae and O. Y. Imanuvilov,
Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems, J. Funct. Anal., 196 (2002), 87-118.
doi: 10.1006/jfan.2002.3988. |
[6] |
H. Chan, C. C. Fu and C. S. Lin,
Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Commun. Math. Phys., 231 (2002), 189-221.
doi: 10.1007/s00220-002-0691-6. |
[7] |
X. Chen, S. Hastings, J. McLeod and Y. Yang,
A nonlinear elliptic equation arising from gauge field theory and cosmology, Proc. R. Soc. Lond. A, 446 (1994), 453-478.
doi: 10.1098/rspa.1994.0115. |
[8] |
K. Choe,
Self-dual non-topological vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Lett. Math. Phys., 87 (2009), 47-65.
doi: 10.1007/s11005-009-0294-7. |
[9] |
K. Choe, Uniqueness of the topological multivortex solution in the selfdual Chern-Simons theory,
J. Math. Phys., 46 (2005), 012305, 21pp. |
[10] |
K. Choe, J. Han, Y. Lee and C. S. Lin,
Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model on a torus, Calc. Var. Partial Differential Equations, 54 (2015), 1275-1329.
doi: 10.1007/s00526-015-0825-2. |
[11] |
H. R. Christiansen, M. S. Cunha, J. A. Helayël-Neto, L. R. U. Manssur and A. L. M. A. Nogueira,
$N =2$-Maxwell-Chern-Simons model with anomalous magnetic moment coupling via dimensional reduction, Int. J. Mod. Phys. A, 14 (1999), 147-159.
doi: 10.1142/S0217751X99000075. |
[12] |
H. R. Christiansen, M. S. Cunha, J. A. Helayël-Neto, L. R. U. Manssur and A. L. M. A. Nogueira,
Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Int. J. Mod. Phys. A, 14 (1999), 1721-1735.
doi: 10.1142/S0217751X99000877. |
[13] |
W. Ding, J. Jost, J. Li, X. Peng and G. Wang,
Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with 6th order potentials, Comm. Math. Phys., 217 (2001), 383-407.
doi: 10.1007/s002200100377. |
[14] |
G. Dunne,
Self-duality and Chern-Simons theories. Lecture Notes in Physics, Springer, Heidelberg, 1995. |
[15] |
Y. W. Fan, Y. Lee and C. S. Lin,
Mixed type solutions of the SU(3) models on a torus, Comm. Math. Phys., 343 (2016), 233-271.
doi: 10.1007/s00220-015-2532-4. |
[16] |
D. Gilbarg and N. Trudinger,
Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[17] |
J. Han and H. Huh,
Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Lett. Math. Phys., 82 (2007), 9-24.
doi: 10.1007/s11005-007-0193-8. |
[18] |
J. Hong, P. Kim and P. Pac,
Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.
doi: 10.1103/PhysRevLett.64.2230. |
[19] |
G.'t Hooft,
A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B, 153 (1979), 141-160.
doi: 10.1016/0550-3213(79)90595-9. |
[20] |
R. Jackiw and E. Weinberg,
Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.
doi: 10.1103/PhysRevLett.64.2234. |
[21] |
A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkh$ä$user, Boston, 1980.
![]() ![]() |
[22] |
T. Lee and H. Min,
Bogomol'nyi equations for solitons in Maxwell-Chern-Simons gauge theories with the magnetic moment interaction term, Phys. Rev. D, 50 (1994), 7738-7741.
doi: 10.1103/PhysRevD.50.7738. |
[23] |
C. Lee, K. Lee and H. Min,
Self-dual Maxwell-Chern-Simons solitons, Phys. Lett. B, 252 (1990), 79-83.
doi: 10.1016/0370-2693(90)91084-O. |
[24] |
B. H. Lee, C. Lee and H. Min,
Supersymmetric Chern-Simons vortex systems and fermion zero modes, Phys. Rev. D, 45 (1992), 4588-4599.
doi: 10.1103/PhysRevD.45.4588. |
[25] |
P. Navrátil,
$N =2$ supersymmetry in a Chern-Simons system with the magnetic moment interaction, Phys. Lett. B, 365 (1996), 119-124.
doi: 10.1016/0370-2693(95)01254-0. |
[26] |
H. B. Nielsen and P. Olesen,
Vortex-line models for dual-strings, Nucl. Phys. B, 61 (1973), 45-61.
|
[27] |
S. Paul and A. Khare,
Charged vortices in an abelian Higgs model with Chern-Simons term, Phys. Lett. B, 174 (1986), 420-422.
doi: 10.1016/0370-2693(86)91028-2. |
[28] |
J. Spruck and Y. Yang,
The existence of non-topological solitons in the self-dual Chern-Simons theory, Commun. Math. Phys., 149 (1992), 361-376.
doi: 10.1007/BF02097630. |
[29] |
G. Tarantello,
Selfdual Maxwell-Chern-Simons vortices, Milan J. Math., 72 (2004), 29-80.
doi: 10.1007/s00032-004-0030-9. |
[30] |
G. Tarantello,
Uniqueness of self-dual periodic Chern-Simons vortices of topological-type, Calc. Var. P.D.E., 29 (2007), 191-217.
doi: 10.1007/s00526-006-0062-9. |
[31] |
G. Tarantello,
Selfdual gauge field vortices an analytical approach, In Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston, 2008. |
[32] |
M. Torres,
Bogomol'nyi limit for nontopological solitons in a Chern-Simons model with anomalous magnetic moment, Phys. Rev. D, 46 (1992), 2295-2298.
doi: 10.1103/PhysRevD.46.R2295. |
[33] |
Y. Yang,
Solitons in Field Theory and Nonlinear Analysis, Springer Monograph in Mathematics. Springer, New York, 2001. |
show all references
References:
[1] |
A. A. Abrikosov,
On the magnetic properties of superconductors of the second group, Soviet Phys. JETP, 5 (1957), 1174-1182.
|
[2] |
D. Bartolucci, Y. Lee, C. S. Lin and M. Onodera,
Asymptotic analysis of solutions to a gauged $O(3)$ sigma model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 651-685.
doi: 10.1016/j.anihpc.2014.03.001. |
[3] |
F. S. A. Cavalcante, M. S. Cunha and C. A. S. Almeida,
Vortices in a nonminimal Maxwell-Chern-Simons O(3) sigma model, Phys. Lett. B, 475 (2000), 315-323.
doi: 10.1016/S0370-2693(00)00077-0. |
[4] |
D. Chae and O. Y. Imanuvilov,
The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Commun. Math. Phys., 215 (2000), 119-142.
doi: 10.1007/s002200000302. |
[5] |
D. Chae and O. Y. Imanuvilov,
Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems, J. Funct. Anal., 196 (2002), 87-118.
doi: 10.1006/jfan.2002.3988. |
[6] |
H. Chan, C. C. Fu and C. S. Lin,
Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Commun. Math. Phys., 231 (2002), 189-221.
doi: 10.1007/s00220-002-0691-6. |
[7] |
X. Chen, S. Hastings, J. McLeod and Y. Yang,
A nonlinear elliptic equation arising from gauge field theory and cosmology, Proc. R. Soc. Lond. A, 446 (1994), 453-478.
doi: 10.1098/rspa.1994.0115. |
[8] |
K. Choe,
Self-dual non-topological vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Lett. Math. Phys., 87 (2009), 47-65.
doi: 10.1007/s11005-009-0294-7. |
[9] |
K. Choe, Uniqueness of the topological multivortex solution in the selfdual Chern-Simons theory,
J. Math. Phys., 46 (2005), 012305, 21pp. |
[10] |
K. Choe, J. Han, Y. Lee and C. S. Lin,
Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model on a torus, Calc. Var. Partial Differential Equations, 54 (2015), 1275-1329.
doi: 10.1007/s00526-015-0825-2. |
[11] |
H. R. Christiansen, M. S. Cunha, J. A. Helayël-Neto, L. R. U. Manssur and A. L. M. A. Nogueira,
$N =2$-Maxwell-Chern-Simons model with anomalous magnetic moment coupling via dimensional reduction, Int. J. Mod. Phys. A, 14 (1999), 147-159.
doi: 10.1142/S0217751X99000075. |
[12] |
H. R. Christiansen, M. S. Cunha, J. A. Helayël-Neto, L. R. U. Manssur and A. L. M. A. Nogueira,
Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Int. J. Mod. Phys. A, 14 (1999), 1721-1735.
doi: 10.1142/S0217751X99000877. |
[13] |
W. Ding, J. Jost, J. Li, X. Peng and G. Wang,
Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with 6th order potentials, Comm. Math. Phys., 217 (2001), 383-407.
doi: 10.1007/s002200100377. |
[14] |
G. Dunne,
Self-duality and Chern-Simons theories. Lecture Notes in Physics, Springer, Heidelberg, 1995. |
[15] |
Y. W. Fan, Y. Lee and C. S. Lin,
Mixed type solutions of the SU(3) models on a torus, Comm. Math. Phys., 343 (2016), 233-271.
doi: 10.1007/s00220-015-2532-4. |
[16] |
D. Gilbarg and N. Trudinger,
Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[17] |
J. Han and H. Huh,
Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Lett. Math. Phys., 82 (2007), 9-24.
doi: 10.1007/s11005-007-0193-8. |
[18] |
J. Hong, P. Kim and P. Pac,
Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.
doi: 10.1103/PhysRevLett.64.2230. |
[19] |
G.'t Hooft,
A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B, 153 (1979), 141-160.
doi: 10.1016/0550-3213(79)90595-9. |
[20] |
R. Jackiw and E. Weinberg,
Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.
doi: 10.1103/PhysRevLett.64.2234. |
[21] |
A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkh$ä$user, Boston, 1980.
![]() ![]() |
[22] |
T. Lee and H. Min,
Bogomol'nyi equations for solitons in Maxwell-Chern-Simons gauge theories with the magnetic moment interaction term, Phys. Rev. D, 50 (1994), 7738-7741.
doi: 10.1103/PhysRevD.50.7738. |
[23] |
C. Lee, K. Lee and H. Min,
Self-dual Maxwell-Chern-Simons solitons, Phys. Lett. B, 252 (1990), 79-83.
doi: 10.1016/0370-2693(90)91084-O. |
[24] |
B. H. Lee, C. Lee and H. Min,
Supersymmetric Chern-Simons vortex systems and fermion zero modes, Phys. Rev. D, 45 (1992), 4588-4599.
doi: 10.1103/PhysRevD.45.4588. |
[25] |
P. Navrátil,
$N =2$ supersymmetry in a Chern-Simons system with the magnetic moment interaction, Phys. Lett. B, 365 (1996), 119-124.
doi: 10.1016/0370-2693(95)01254-0. |
[26] |
H. B. Nielsen and P. Olesen,
Vortex-line models for dual-strings, Nucl. Phys. B, 61 (1973), 45-61.
|
[27] |
S. Paul and A. Khare,
Charged vortices in an abelian Higgs model with Chern-Simons term, Phys. Lett. B, 174 (1986), 420-422.
doi: 10.1016/0370-2693(86)91028-2. |
[28] |
J. Spruck and Y. Yang,
The existence of non-topological solitons in the self-dual Chern-Simons theory, Commun. Math. Phys., 149 (1992), 361-376.
doi: 10.1007/BF02097630. |
[29] |
G. Tarantello,
Selfdual Maxwell-Chern-Simons vortices, Milan J. Math., 72 (2004), 29-80.
doi: 10.1007/s00032-004-0030-9. |
[30] |
G. Tarantello,
Uniqueness of self-dual periodic Chern-Simons vortices of topological-type, Calc. Var. P.D.E., 29 (2007), 191-217.
doi: 10.1007/s00526-006-0062-9. |
[31] |
G. Tarantello,
Selfdual gauge field vortices an analytical approach, In Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston, 2008. |
[32] |
M. Torres,
Bogomol'nyi limit for nontopological solitons in a Chern-Simons model with anomalous magnetic moment, Phys. Rev. D, 46 (1992), 2295-2298.
doi: 10.1103/PhysRevD.46.R2295. |
[33] |
Y. Yang,
Solitons in Field Theory and Nonlinear Analysis, Springer Monograph in Mathematics. Springer, New York, 2001. |
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