ELECTRICALLY CHARGED SOLITONS IN GAUGE FIELD THEORY

doi:10.1016/S0252-9602(10)60186-5

摘要/Abstract

摘要：

Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on monopoles, Schwinger pioneered
the concept of solitons carrying both electric and magnetic charges, called dyons, which are useful in modeling elementary particles. Mathematically, the existence of dyons presents interesting variational partial differential equation problems, subject to topological constraints. This article is a survey on recent progress in the study of dyons.

关键词: gauge field theory, monopoles, vortices, dyons, calculus of variations, topological invariants

Abstract:

Key words: gauge field theory, monopoles, vortices, dyons, calculus of variations, topological invariants

中图分类号:

35J20

杨亦松. ELECTRICALLY CHARGED SOLITONS IN GAUGE FIELD THEORY[J]. 数学物理学报(英文版), 2010, 30(6): 1975-2005.

YANG Yi-Song. ELECTRICALLY CHARGED SOLITONS IN GAUGE FIELD THEORY[J]. Acta mathematica scientia,Series B, 2010, 30(6): 1975-2005.

参考文献

[1] Actor A. Classical solutions of $SU(2)$ Yang--Mills theories. Rev Mod Phys, 1979, 51: 461--525

[2] Adler S L. Monopoles and projective representations: two areas of influence of Yang-Mills theory on mathematics.
Preprint, arXiv:math-ph/0405049

[3] Arafune J, Freund P G O, Goebel C J. Topology of Higgs fields. J Math Phys, 1975, 16: 433--437

[4]  Atiyah M F, Ward R S. Instantons and algebraic geometry. Commun Math Phys, 1977,  55: 117--124

[5] Bayen F, Flato M, Fronsdal C, Lichnerowicz A, Sternheimer D. Deformation theory and quantization. I. Deformations of symplectic structures. Ann Phys, 1978, 111: 61--110

[6]  Bayen F, Flato M, Fronsdal C, Lichnerowicz A, Sternheimer D. Deformation theory and quantization. II. Physical applications. Ann Phys, 1978, 111: 111--151

[7] Belavin A A, Polyakov A M, Schwartz A S, Tyupkin Yu S. Pseudoparticle solutions of the Yang-Mills equations. Phys Lett B, 1975, 59:   85--87

[8]  Bezryadina A, Eugenieva E, Chen Z. Self-trapping and flipping of double-charged vortices in optically induced photonic lattices. Optics Lett, 2006, 31: 2456--2458

[9] Bogomol'nyi E B. The stability of classical solutions. Sov J Nucl Phys, 1976, 24: 449--454

[10] Bramwell S T, Giblin S R, Calder S, Aldus R, Prabhakaran D, Fennell T. Measurement of the charge and current of magnetic monopoles in spin ice. Nature, 2009, 461: 956--959

[11]  Caffarelli L, Yang Y. Vortex condensation in the Chern--Simons Higgs model: an existence theorem. Commun Math Phys, 1995, 168: 321--336

[12] Callias C. Axial anomalies and index theorems on open spaces. Commun Math Phys, 1978, 62: 213--234

[13]  Castelnovo C, Moessner R, Sondhi S L. Magnetic monopoles in spin ice. Nature, 2008, 451: 42--45

[14] Chae D, Yu O. Imanuvilov, The existence of nontopological multivortex solutions in the relativistic self-dual Chern--Simons theory. Commun Math Phys, 2000, 215: 119--142

[15] Chae D, Kim N. Topological multivortex solutions of the self-dual Maxwell--Chern--Simons--Higgs system. J Diff Eqs, 1997, 134: 154--182

[16] Chan H, Fu C C, Lin C S. Non-topological multivortex solutions to the self-dual Chern--Simons--Higgs equation.
Commun Math Phys, 2002, 231: 189--221

[17] Chen C C, Lin C S, Wang G. Concentration phenomena of two-vortex solutions in a Chern--Simons model. Ann Sc Norm Super Pisa Cl Sci, 2004, 3: 367--397

[18] Chen R M, Guo Y, Spirn D, Yang Y. Electrically and magnetically charged vortices in the Chern--Simons--Higgs theory. Proc Roy Soc A, 2009, 465: 3489--3516

[19]  Chern J L, Chen Z Y, Lin C S. Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles. Commun Math Phys, 2010, 296: 323--351

[20] Chern S S, Simons J. Some cohomology classes in principal fiber bundles and their application to Riemannian geometry. Proc Nat Acad Sci USA, 1971, 68: 791--794

[21] Chern S S, Simons J. Characteristic forms and geometric invariants. Ann Math, 1974, 99: 48--69

[22] Cho Y M, Kimm K. Electroweak monopoles. Preprint, arXiv:hep-th/9705213.

[23] Cho Y M, Maison D. Monopole configuration in Weinberg--Salam model. Phys Lett B, 1997, 391:  360--365

[24]  Choe K. Uniqueness of the topological multivortex solution in the self-dual Chern-Simons theory. J Math Phys, 2005, 46: 012305

[25] Choe K. Multiple existence results for the self-dual Chern-Simons-Higgs vortex equation. Comm Partial Diff Eqs, 2009, 34: 1465--1507

[26] Choe K, Kim N. Blow-up solutions of the self-dual Chern--Simons--Higgs vortex equation. Ann Inst H Poincarè--Anal Nonlinèaire, 2008, 25: 313--338

[27] S. Coleman, Le monopôle magnètique cinquante ans aprè The magnetic monopole fifty years later]//Gauge Theories in High Energy Physics. Part I, II (Les Houches, 1981). Amsterdam: North-Holland, 1983: 461--552

[28]  Curie P. On the possible existence of magnetic conductivity and free magnetism. Séances Soc Phys (Paris), 1894: 76--77

[29] Deser S, Jackiw R, Templeton S. Three-dimensional massive gauge theories. Phys Rev Lett, 1982, 48: 975--978

[30] Deser S, Jackiw R, Templeton S. Topologically massive gauge theories. Ann Phys, 1982, 140: 372--411

[31] Dirac P A M. Quantised Singularities in the electromagnetic field. Proc Royal Soc A, 1931, 133: 60--72

[32] Dunne G. Self-Dual Chern--Simons Theories. Lecture Notes in Physics, Vol 36. Berlin: Springer-Verlag, 1995

[33] Dunne G. Mass degeneracies in self-dual models. Phys Lett B, 1995, 345: 452--457

[34] Dunne G. Aspects of Chern-Simons theory. Preprint, arXiv: hep-th/9902115, Les Houches Lectures, 1998

[35] Dziarmaga J. Low energy dynamics of [U(1)]^N Chern--Simons solitons. Phys Rev D, 1994, 49: 5469--5479

[36] Dziarmaga J. Only hybrid anyons can exist in the broken symmetry phase of nonrelativistic [U(1)]² Chern--Simons theory. Phys Rev D, 1994, 50: R2376--R2380

[37] Felsager B. Geometry, Particles, and Fields. Berlin, New York: Springer-Verlag, 1998

[38] Forgacs P Horvath Z, Palla L. Exact multimonopole solutions in the Bogomolny-Prasad-Sommerfied limit. Phys Lett B, 1981, 99: 232--236

[39] Fröhlich J, Marchetti P A. Quantum field theories of vortices and anyons. Comm Math Phys, 1989, 121: 177--223

[40] Georgi H, Glashow S L. Unified weak and electromagnetic interactions without neutral currents. Phys Rev Lett, 1972, 28: 1494--1497

[41] Georgi H, Glashow S L. Unity of all elementary-particle forces. Phys Rev Lett, 1974, 32: 438--441

[42] Gingras M J P. Observing monopoles in a magnetic analog of ice. Science, 2009, 26: 375--376

[43] Ginzburg V L, Landau L D. On the theory of superconductivity//Collected Papers of L. D. Landau (edited by D. Ter Haar), New York: Pergamon, 1965: 546--568

[44] Goddard P, Olive D I. Magnetic monopoles in gauge field theories. Rep Prog Phys, 1978, 41: 1357--1437

[45] Greiner W, Müller B. Quantum Mechanics--Symmetries. 2nd ed. Berlin, New York: Springer-Verlag, 1994

[46] Groenewold H J. On the principles of elementary quantum mechanics. Physica, 1946, 12: 405--460

[47] Han J. Asymptotic limit for condensate solutions in the abelian Chern--Simons--Higgs model. Proc Amer Math Soc, 2003, 131: 1839--1845

[48] Han J, Huh H. Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling. Lett Math Phys, 2007, 82: 9--24

[49] Hirsch M. Differential Topology. New York, Heidelberg: Springer-Verlag,1976

[50] Hong J, Kim Y, Pac P Y. Multivortex solutions of the Abelian Chern--Simons--Higgs theory. Phys Rev Lett, 1990, 64: 2330--2333

[51] Horvathy P A, Zhang P. Vortices in (abelian) Chern-Simons gauge theory. Phys Rep, 2009, 481: 83--142

[52] Huh H. Local and global solutions of the Chern-Simons-Higgs system. J Funct Anal, 2007, 242: 526--549

[53] Humphreys J E. Introduction to Lie Algebras and Representation Theory. New York, Heidelberg: Springer-Verlag, 1972

[54] Inouye S, Gupta S, Rosenband T, Chikkatur A P, Görlitz A, Gustavson T L, Leanhardt A E, Pritchard D E, Ketterle W. Observation of vortex phase singularities in Bose-Einstein condensates. Phys Rev Lett, 2001, 87: 080402

[55] Jackiw R. Quantum meaning of classical field theory. Rev Mod Phys, 1977, 49: 681--707

[56] Jackiw R, Chern S S. Chern--Simons terms//Differential Geometry and Physics. Nankai Tracts Math 10. Hackensack, NJ: World Sci Publ, 2006: 53--62

[57] Jackiw R, Templeton S. How super-renormalizable interactions cure their infrared divergences. Phys Rev D,1981, 23: 2291--2304

[58] Jackiw R, Weinberg E J. Self-dual Chern--Simons vortices. Phys Rev Lett, 1990, 64: 2334--2337

[59] Jacobs L, Khare A, Kumar C N, Paul S K. The interaction of Chern--Simons vortices. Int J Mod Phys A, 1991, 6: 3441--3466

[60] Jaffe A, Taubes C H. Vortices and Monopoles. Boston: Birkhäuser, 1980

[61] Jost J, Wang G. Analytic aspects of the Toda system. I. A Moser-Trudinger inequality. Comm Pure Appl Math, 2001, 54: 1289--1319

[62] Julia B, Zee A. Poles with both magnetic and electric charges in non-Abelian gauge theory. Phys Rev D, 1975, 11: 2227--2232

[63] Kao H C, Lee K. Self-dual SU(3) Chern--Simons Higgs systems. Phys Rev D, 1994, f50: 6626--6632

[64] Kawaguchi Y, Ohmi T. Splitting instability of a multiply charged vortex in a Bose--Einstein condensate. Phys Rev A, 2004, 70: 043610

[65] Khomskii D I, Freimuth A. Charged vortices in high temperature superconductors. Phys Rev Lett, 1995, 75: 1384--1386

[66] Kim C, Lee C, Kao P, Lee B H, Min H. Schrödinger fields on the plane with [U(1)]^N Chern--Simons interactions and generalized self-dual solitons. Phys Rev D, 1993, 48: 1821--1840

[67] Kim N. Existence of vortices in a self-dual gauged linear sigma model and its singular limit. Nonlinearity, 2006, 19: 721--739

[68] Kontsevich M. Lecture Notes on Deformation Theory. Berkeley, 1995

[69] Kontsevich M. Formality conjecture//Gutt S, Rawnsley J, Sternheimer D, eds. Deformation Theory and Symplectic Geometry. Math Phys Stud 20. Dordrecht: Kluwer Acad Publ, 1997: 139--156

[70] Kontsevich M. Deformation quantization of Poisson manifolds, I. Lett Math Phys, 2003, 66: 157--216

[71] Kubo R. Wigner representation of quantum operators and its applications to electrons in a magnetic field. J Phys Soc Japan, 1964, 19: 2127--2139

[72] Kumar C N, Khare A. Charged vortex of finite energy in nonabelian gauge theories with Chern--Simons term. Phys Lett B, 1986, 178: 395--399

[73] Lee K, Weinberg E J. Nontopological magnetic monopoles and new magnetically charged black holes. Phys Rev Lett, 1994, 73: 1203--1206

[74] Lai C H, ed. Selected Papers on Gauge Theory of Weak and Electromagnetic Interactions. Singapore: World Scientific, 1981

[75] Lin C S, Ponce A C, Yang Y. A system of elliptic equations arising in Chern--Simons field theory. J Funct Anal, 2007, 247: 289--350

[76] Lin C S, Prajapat J V. Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus. Commun Math Phys, 2009, 288: 311--347

[77] Lucia M, Nolasco M. SU(3) Chern-Simons vortex theory and Toda systems. J Diff Eqs, 2002, 184: 443--474

[78] Macri M, Nolasco M, Ricciardi T. Asymptotics for self-dual vortices on the torus and on the plane: a gluing technique. SIAM J Math Anal, 2005, 37: 1--16

[79] Manton N. Complex structure of monopoles. Nucl Phys B, 1978, 135: 319--332

[80] Manton N, Sutcliffe P. Topological Solitons. Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge Univ Press, 2004

[81] Matsuda Y, Nozakib K, Kumagaib K. Charged vortices in high temperature superconductors probed by nuclear magnetic resonance. J Phys Chem Solids, 2002, 63: 1061--1063

[82] Mehra J, Rechenberg H. The Historical Development of Quantum Theory. Volume 6: The Completion of Quantum Mechanics 1926--1941. New York: Springer-Verlag, 2000

[83] Milnor J W. Topology from the Differentiable Viewpoint. Charlottesville: University Press of Virginia, 1965

[84] Milton K A. Theoretical and experimental status of magnetic monopoles. Rep Progress Phys, 2006, 69: 1637--1711

[85] Morris DJ P, Tennant D A, Grigera S A, Klemke B, Castelnovo C, Moessner R, Czternasty C, Meissner M, Rule K C, Hoffmann J U, Kiefer K, Gerischer S, Slobinsky D, Perry R S. Dirac strings and magnetic monopoles in the spin ice Dy₂Ti₂O₇. Science, 2009, 326: 411--414

[86]  Moyal J E. Quantum mechanics as a statistical theory. Proc Cambridge Philosophical Soc, 1949, 45: 99--124

[87]  Nielsen H B, Olesen P. Vortex-line models for dual strings. Nucl Phys B, 1973, 61: 45--61

[88] Nolasco M. Nontopological N-vortex condensates for the self-dual Chern--Simons theory. Comm Pure Appl Math,
2003, 56: 1752--1780

[89] Nolasco M, Tarantello G. Double vortex condensates in the Chern--Simons--Higgs theory. Calc Var & PDE's, 1999, 9: 31--91

[90] Nolasco M, Tarantello G. Vortex condensates for the SU(3) Chern--Simons theory. Commun Math Phys, 2000, 213: 599--639

[91]  Paul S, Khare A. Charged vortices in an Abelian Higgs model with Chern--Simons term. Phys Lett B, 1986, 17: 420--422

[92] Polyakov A M. Particle spectrum in quantum field theory. JETP Lett, 1974, 20: 194

[93] Prasad M K, Sommerfield C M. Exact classical solutions for the 't Hooft monopole and the Julia--Zee dyon. Phys Rev Lett, 1975, 35: 760--762

[94] Preskill J. Vortices and monopoles//Architecture of Fundamental Interactions at Short Distances. Les Houches, Session XLIV, 1985. Elsevier, 1987: 235--337

[95]  Ricciardi T, Tarantello G. Vortices in the Maxwell--Chern--Simons theory. Comm Pure Appl Math, 2000, 53: 811--851

[96]  Ryder L H. Quantum Field Theory. 2nd ed. Cambridge, U K: Cambridge Univ Press, 1996

[97]  Schechter M, Weder R. A theorem on the existence of dyon solutions. Ann Phys, 1981, 132: 293--327

[98] Schonfeld J S. A massive term for three-dimensional gauge fields. Nucl Phys B, 1981, 185: 157--171

[99] Schwinger J. Sources and magnetic charge. Phys Rev, 1968, 173: 1536--1544

[100] Schwinger J. A magnetic model of matter. Science, 1969, 165: 757--761

[101] Sokoloff J B. Charged vortex excitations in quantum Hall systems. Phys Rev B, 1985, 31: 1924--1928

[102]  Spruck J, Yang Y. Topological solutions in the self-dual Chern--Simons theory: existence and approximation. Ann Inst H Poincaré--Anal non linéaire, 1995, 12: 75--97

[103]  Spruck J, Yang Y. The existence of non-topological solitons in the self-dual Chern--Simons theory. Commun Math Phys, 1992, 149: 361--376

[104]  Spruck J, Yang Y. Proof of the Julia--Zee theorem. Commun Math Phys, 2009, 291: 347--356

[105] Sutcliffe P. BPS monopoles. Internat J Mod Phys A, 1997, 12: 4663--4706

[106]  Tarantello G. Multiple condensate solutions for the Chern--Simons--Higgs theory. J Math Phys, 1996, 37: 3769--3796

[107] Tarantello G. Self-Dual Gauge Field Vortices. Progress in Nonlinear Differential Equations and Their Applications 72. Boston: Birkhäuser, 2008

[108] Taubes C H. The existence of multimonopole solutions to the nonabelian, Yang-Mills-Higgs equations for arbitrary simple gauge groups. Commun Math Phys, 1981, 80: 343--367

[109]  Taubes C H. The existence of a non-minimal solution to the SU(2) Yang--Mills--Higgs equations on R³, Part I.
Commun Math Phys, 1982, 86: 257--298

[110]  Taubes C H. Min-max theory for the Yang-Mills-Higgs equations. Commun Math Phys, 1985, 97: 473--540

[111] Tchrakian D H. The 't Hooft electromagnetic tensor for Higgs fields of arbitrary isospin. Phys Lett B, 1980, 91: 415--416

[112] t Hooft G. Magnetic monopoles in unified gauge theories. Nucl Phys B, 1974, 79: 276--284

[113] de Vega H J, Schaposnik F. Electrically charged vortices in non-Abelian gauge theories with Chern--Simons term. Phys Rev Lett, 1986, 56: 2564--2566

[114] de Vega H J, Schaposnik F. Vortices and electrically charged vortices in non-Abelian gauge theories. Phys Rev D, 1986, 34: 3206--3213

[115] Weinberg E J. Parameter counting for multimonopole solutions. Phys Rev D, 1979, 20: 936--944

[116] Wells R O. Differential Analysis on Complex Manifolds. New York, Berlin: Springer-Verlag, 1980

[117] Wilczek F. Quantum mechanics of fractional-spin particles. Phys Rev Lett, 1982, 49: 957--959

[118] Wilczek F. Fractional Statistics and Anyon Superconductors. Singapore: World Scientific, 1990

[119] Yang Y. The relativistic non-Abelian Chern--Simons equations. Commun Math Phys, 1997, 186: 199--218

[120] Yang Y. Dually charged particle-like solutions in the Weinberg--Salam theory. Proc Roy Soc A, 1998, 454: 155--178

[121] Yang Y. The Lee--Weinberg magnetic monopole of unit charge: existence and uniqueness. Physica D, 1998, 117: 215--240

[122] ang Y. Coexistence of vortices and antivortices in an Abelian gauge theory. Phys Rev Lett, 1998, 80: 26--29

[123] Yang Y. Strings of opposite magnetic charges in a gauge field theory. Proc Roy Soc A, 1999, 455: 601--629

[124] Yang Y. Solitons in Field Theory and Nonlinear Analysis. Springer Monographs in Mathematics. New York: Springer-Verlag, 2001

[125] Yang Y. On the deformation of some Lie algebras. Acta Math Sci, 1984, 4: 509--517

[126] Zachos C, Fairlie D, Curtright T. Quantum Mechanics in Phase Space. Singapore: World Scientific, 2005

[127] Zwanziger D. Exactly soluble nonrelativistic model of particles with both electric and magnetic charges. Phys Rev, 1968, 176: 1480--1488

[128] Zwanziger D. Quantum field theory of particles with both electric and magnetic charges. Phys Rev, 1968, 176: 1489--1495