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August  2019, 39(8): 4613-4646. doi: 10.3934/dcds.2019189

Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations

Kwangseok Choe 1, and  Hyungjin Huh 2,,

1. 

Department of Mathematics, Inha University, Incheon, 22212, Republic of Korea
2. 

Department of Mathematics, Chung-Ang University, Seoul, 06974, Republic of Korea

*Corresponding author: Hyungjin Huh

Received  September 2018 Revised  February 2019 Published  May 2019

We propose the Chern-Simons gauged sigma model from $\mathbb{R}^{1+2}$ into the hyperbolic plane $\mathbb{H}^2$. We seek a static configuration of this model and derive self-dual equations. We also establish some existence results for solutions of the self-dual equations under appropriate boundary conditions near $\infty$

Keywords: Chern-Simons gauged sigma model, hyperbolic plane, self-dual equations.
Mathematics Subject Classification: Primary: 81T13; Secondary: 35B40.
Citation: Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189
References:
[1]

J. P. Antoine and B. Piette, Solutions of Euclidean σ models on noncompact Grassmannian manifolds, J. Math. Phys., 29 (1988), 1687-1697.  doi: 10.1063/1.527917.  Google Scholar

[2]

K. Arthur, D. Tchrakian and Y. Yang, Topological and nontopological self-dual Chern-Simons solitions in a gauged O(3) σ model, Phys. Rev. D, 54 (1996), 5245-5248.  doi: 10.1103/PhysRevD.54.5245.  Google Scholar

[3]

D. Bartolucci, C.-C. Chen, C. S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29 (2004), 1241-1265.  doi: 10.1081/PDE-200033739.  Google Scholar

[4]

D. Bartolucci, Y. Lee, C. S. Lin and M. Onodera, Asymptotic analysis of solutions to a gauged O(3) sigma model, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 32 (2015), 651-685.  doi: 10.1016/j.anihpc.2014.03.001.  Google Scholar

[5]

D. Bartolucci and G. Tarantello, Liouville Type equations with singular data and their applications to periodic multivortices for the electroweak Theory, Comm. Math. Phys., 229 (2000), 3-47.  doi: 10.1007/s002200200664.  Google Scholar

[6]

A. A. Belavin and A. M. Polyakov, Metastable states of two dimensional isotropic ferromagnets, JETP Lett., 22 (1975), 245-247.   Google Scholar

[7]

H. Brezis and J-M. Coron, Large solutions for harmonic maps in two dimensions, Comm. Math. Phys., 92 (1983), 203-215.  doi: 10.1007/BF01210846.  Google Scholar

[8]

H. Brezis and F. Merle, Uniform estimates and blowup behavior for solutions of $-\Delta u = V(x)e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.  Google Scholar

[9]

H. Chan, C.-C. Fu and C.-S. Lin, Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys., 231 (2002), 189-221.  doi: 10.1007/s00220-002-0691-6.  Google Scholar

[10]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[11]

W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $\mathbb{R}^2$, Duke Math. J., 71 (1993), 427-439.  doi: 10.1215/S0012-7094-93-07117-7.  Google Scholar

[12]

X. Chen, S. Hastings, J. B. McLeod and Y. Yang, A nonlinear problem elliptic equation arising from gauge field theory and cosmology, Proc. Roy. Soc. London Ser. A, 446 (1994), 453-478.  doi: 10.1098/rspa.1994.0115.  Google Scholar

[13]

K. Choe, J. Han, C.-S. Lin and T.-C. Lin, Uniqueness and solution structure of nonlinear equations arising from the Chern-Simons gauged O(3) sigma models, J. Differential Equations, 255 (2013), 2136-2166.  doi: 10.1016/j.jde.2013.06.010.  Google Scholar

[14]

K. Choe, J. Han and C.-S. Lin, Bubbling solutions for the Chern-Simons gauged O(3) sigma modelin $\mathbb{R}^2$, Discrete Contin. Dyn. Syst., 34 (2014), 2703-2728.  doi: 10.3934/dcds.2014.34.2703.  Google Scholar

[15]

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.  Google Scholar

[16]

K. Choe, N. Kim and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 837-852.  doi: 10.1016/j.anihpc.2011.06.003.  Google Scholar

[17]

P. K. Ghosh and S. K. Ghosh, Topological and nontopological solitons in a gauged O(3) sigma model with Chern-Simons term, Phys. Lett. B, 366 (1996), 199-204.  doi: 10.1016/0370-2693(95)01365-2.  Google Scholar

[18]

A. C. Huang, Harmonic maps of punctured surfaces to hyperbolic plane, preprint, arXiv: 1605.07715v1. Google Scholar

[19]

H. Huh and G. Jin, Local and global solutions of Chern-Simons gauged O(3) sigma equations in one space dimension, J. Math. Phys., 57 (2016), 081511, 11pp. doi: 10.1063/1.4960744.  Google Scholar

[20]

E. Hulett, Harmonic superconformal maps of surfaces in $\Bbb{H}^n$, J. Geo. Phys., 42 (2002), 139-165.  doi: 10.1016/S0393-0440(01)00082-1.  Google Scholar

[21]

A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhäuser Boston, 1980.  Google Scholar

[22]

S.-S. Kim and P. Oh, On the gauged noncompact spin system, International Journal of Modern Physics A, 13 (1998), 5503-5517.  doi: 10.1142/S0217751X9800250X.  Google Scholar

[23]

K. Kimm, K. Lee and T. Lee, Anyonic Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Rev. D, 53 (1996), 4436-4440.   Google Scholar

[24]

J. Krieger, Global regularity of wave maps from $\mathbb{R}^{2+1}$ to $\mathbb{H}^2$. Small energy, Comm. Math. Phys., 250 (2004), 507-580.  doi: 10.1007/s00220-004-1088-5.  Google Scholar

[25]

A. Kundu, On σ-models with noncompact groups, Lett. Math. Phys., 6 (1982), 479-485.  doi: 10.1007/BF00405869.  Google Scholar

[26]

A. Kundu, Gauge equivalence of sigma models with non-compact Grassmannian manifolds, J. Phys. A: Math. Gen., 19 (1986), 1303-1314.  doi: 10.1088/0305-4470/19/8/012.  Google Scholar

[27]

P. O. Mazur, A relationship between the electrovacuum Ernst equations and nonlinear σ-model, Acta Phys. Polon. B, 14 (1983), 219-234.   Google Scholar

[28]

P. O. Mazur, A global identity for nonlinear σ-models, Phys. Lett. A, 100 (1984), 341-344.  doi: 10.1016/0375-9601(84)91084-3.  Google Scholar

[29]

R.-C. McOwen, Conformal metrics in $\mathbb{R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J., 34 (1985), 97-104.  doi: 10.1512/iumj.1985.34.34005.  Google Scholar

[30]

L. Nirenberg, Topics in Nonlinear Functional Analysis, American Mathematical Society, 2001. doi: 10.1090/cln/006.  Google Scholar

[31]

P. Oh, Bogomol'nyi solitons and Hermitian symmetric spaces, Rep. Math. Phys., 43 (1999), 271-281.  doi: 10.1016/S0034-4877(99)80035-4.  Google Scholar

[32]

I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical O(3) σ-model, Ann. of Math., 172 (2010), 187-242.  doi: 10.4007/annals.2010.172.187.  Google Scholar

[33]

B. J. Schroers, Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Lett. B, 356 (1995), 291-296.  doi: 10.1016/0370-2693(95)00833-7.  Google Scholar

[34]

J. Shatah and A. Shadi Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719-754.  doi: 10.1002/cpa.3160470507.  Google Scholar

[35]

M. Struwe, Radially symmetric wave maps from (1+2)-dimensional Minkowski space to the sphere, Math. Z., 242 (2002), 407-414.  doi: 10.1007/s002090100345.  Google Scholar

[36]

H. J. de Vega and N. Sanchez, Exact integrability of strings in D-dimensional de Sitter spacetime, Phys. Rev. D, 47 (1993), 3394-3404.  doi: 10.1103/PhysRevD.47.3394.  Google Scholar

[37]

Z. Wang, Symmetries and the calculations of degree, Chin. Ann. of Math. B, 10 (1989), 520-536.   Google Scholar

[38]

R. S. Ward and A. E. Winn, Integrable systems admitting topological solitons, J. Phys. A, 31 (1998), L261–L266. doi: 10.1088/0305-4470/31/13/003.  Google Scholar

show all references

References:
[1]

J. P. Antoine and B. Piette, Solutions of Euclidean σ models on noncompact Grassmannian manifolds, J. Math. Phys., 29 (1988), 1687-1697.  doi: 10.1063/1.527917.  Google Scholar

[2]

K. Arthur, D. Tchrakian and Y. Yang, Topological and nontopological self-dual Chern-Simons solitions in a gauged O(3) σ model, Phys. Rev. D, 54 (1996), 5245-5248.  doi: 10.1103/PhysRevD.54.5245.  Google Scholar

[3]

D. Bartolucci, C.-C. Chen, C. S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29 (2004), 1241-1265.  doi: 10.1081/PDE-200033739.  Google Scholar

[4]

D. Bartolucci, Y. Lee, C. S. Lin and M. Onodera, Asymptotic analysis of solutions to a gauged O(3) sigma model, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 32 (2015), 651-685.  doi: 10.1016/j.anihpc.2014.03.001.  Google Scholar

[5]

D. Bartolucci and G. Tarantello, Liouville Type equations with singular data and their applications to periodic multivortices for the electroweak Theory, Comm. Math. Phys., 229 (2000), 3-47.  doi: 10.1007/s002200200664.  Google Scholar

[6]

A. A. Belavin and A. M. Polyakov, Metastable states of two dimensional isotropic ferromagnets, JETP Lett., 22 (1975), 245-247.   Google Scholar

[7]

H. Brezis and J-M. Coron, Large solutions for harmonic maps in two dimensions, Comm. Math. Phys., 92 (1983), 203-215.  doi: 10.1007/BF01210846.  Google Scholar

[8]

H. Brezis and F. Merle, Uniform estimates and blowup behavior for solutions of $-\Delta u = V(x)e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.  Google Scholar

[9]

H. Chan, C.-C. Fu and C.-S. Lin, Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys., 231 (2002), 189-221.  doi: 10.1007/s00220-002-0691-6.  Google Scholar

[10]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[11]

W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $\mathbb{R}^2$, Duke Math. J., 71 (1993), 427-439.  doi: 10.1215/S0012-7094-93-07117-7.  Google Scholar

[12]

X. Chen, S. Hastings, J. B. McLeod and Y. Yang, A nonlinear problem elliptic equation arising from gauge field theory and cosmology, Proc. Roy. Soc. London Ser. A, 446 (1994), 453-478.  doi: 10.1098/rspa.1994.0115.  Google Scholar

[13]

K. Choe, J. Han, C.-S. Lin and T.-C. Lin, Uniqueness and solution structure of nonlinear equations arising from the Chern-Simons gauged O(3) sigma models, J. Differential Equations, 255 (2013), 2136-2166.  doi: 10.1016/j.jde.2013.06.010.  Google Scholar

[14]

K. Choe, J. Han and C.-S. Lin, Bubbling solutions for the Chern-Simons gauged O(3) sigma modelin $\mathbb{R}^2$, Discrete Contin. Dyn. Syst., 34 (2014), 2703-2728.  doi: 10.3934/dcds.2014.34.2703.  Google Scholar

[15]

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.  Google Scholar

[16]

K. Choe, N. Kim and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 837-852.  doi: 10.1016/j.anihpc.2011.06.003.  Google Scholar

[17]

P. K. Ghosh and S. K. Ghosh, Topological and nontopological solitons in a gauged O(3) sigma model with Chern-Simons term, Phys. Lett. B, 366 (1996), 199-204.  doi: 10.1016/0370-2693(95)01365-2.  Google Scholar

[18]

A. C. Huang, Harmonic maps of punctured surfaces to hyperbolic plane, preprint, arXiv: 1605.07715v1. Google Scholar

[19]

H. Huh and G. Jin, Local and global solutions of Chern-Simons gauged O(3) sigma equations in one space dimension, J. Math. Phys., 57 (2016), 081511, 11pp. doi: 10.1063/1.4960744.  Google Scholar

[20]

E. Hulett, Harmonic superconformal maps of surfaces in $\Bbb{H}^n$, J. Geo. Phys., 42 (2002), 139-165.  doi: 10.1016/S0393-0440(01)00082-1.  Google Scholar

[21]

A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhäuser Boston, 1980.  Google Scholar

[22]

S.-S. Kim and P. Oh, On the gauged noncompact spin system, International Journal of Modern Physics A, 13 (1998), 5503-5517.  doi: 10.1142/S0217751X9800250X.  Google Scholar

[23]

K. Kimm, K. Lee and T. Lee, Anyonic Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Rev. D, 53 (1996), 4436-4440.   Google Scholar

[24]

J. Krieger, Global regularity of wave maps from $\mathbb{R}^{2+1}$ to $\mathbb{H}^2$. Small energy, Comm. Math. Phys., 250 (2004), 507-580.  doi: 10.1007/s00220-004-1088-5.  Google Scholar

[25]

A. Kundu, On σ-models with noncompact groups, Lett. Math. Phys., 6 (1982), 479-485.  doi: 10.1007/BF00405869.  Google Scholar

[26]

A. Kundu, Gauge equivalence of sigma models with non-compact Grassmannian manifolds, J. Phys. A: Math. Gen., 19 (1986), 1303-1314.  doi: 10.1088/0305-4470/19/8/012.  Google Scholar

[27]

P. O. Mazur, A relationship between the electrovacuum Ernst equations and nonlinear σ-model, Acta Phys. Polon. B, 14 (1983), 219-234.   Google Scholar

[28]

P. O. Mazur, A global identity for nonlinear σ-models, Phys. Lett. A, 100 (1984), 341-344.  doi: 10.1016/0375-9601(84)91084-3.  Google Scholar

[29]

R.-C. McOwen, Conformal metrics in $\mathbb{R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J., 34 (1985), 97-104.  doi: 10.1512/iumj.1985.34.34005.  Google Scholar

[30]

L. Nirenberg, Topics in Nonlinear Functional Analysis, American Mathematical Society, 2001. doi: 10.1090/cln/006.  Google Scholar

[31]

P. Oh, Bogomol'nyi solitons and Hermitian symmetric spaces, Rep. Math. Phys., 43 (1999), 271-281.  doi: 10.1016/S0034-4877(99)80035-4.  Google Scholar

[32]

I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical O(3) σ-model, Ann. of Math., 172 (2010), 187-242.  doi: 10.4007/annals.2010.172.187.  Google Scholar

[33]

B. J. Schroers, Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Lett. B, 356 (1995), 291-296.  doi: 10.1016/0370-2693(95)00833-7.  Google Scholar

[34]

J. Shatah and A. Shadi Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719-754.  doi: 10.1002/cpa.3160470507.  Google Scholar

[35]

M. Struwe, Radially symmetric wave maps from (1+2)-dimensional Minkowski space to the sphere, Math. Z., 242 (2002), 407-414.  doi: 10.1007/s002090100345.  Google Scholar

[36]

H. J. de Vega and N. Sanchez, Exact integrability of strings in D-dimensional de Sitter spacetime, Phys. Rev. D, 47 (1993), 3394-3404.  doi: 10.1103/PhysRevD.47.3394.  Google Scholar

[37]

Z. Wang, Symmetries and the calculations of degree, Chin. Ann. of Math. B, 10 (1989), 520-536.   Google Scholar

[38]

R. S. Ward and A. E. Winn, Integrable systems admitting topological solitons, J. Phys. A, 31 (1998), L261–L266. doi: 10.1088/0305-4470/31/13/003.  Google Scholar

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