Abstract
In these lecture notes, an introduction to topological concepts and methods in studies of gauge field theories is presented. The three paradigms of topological objects, the Nielsen–Olesen vortex of the abelian Higgs model, the ’t Hooft–Polyakov monopole of the non-abelian Higgs model and the instanton of Yang–Mills theory, are discussed. The common formal elements in their construction are emphasized and their different dynamical roles are exposed. The discussion of applications of topological methods to Quantum Chromodynamics focuses on confinement. An account is given of various attempts to relate this phenomenon to topological properties of Yang–Mills theory. The lecture notes also include an introduction to the underlying concept of homotopy with applications from various areas of physics.
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References
1. C. F. Gauß, Werke, Vol. 5, Göttingen, Königliche Gesellschaft der Wissenschaften 1867, p. 605
2. B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry, Part II. Springer Verlag 1985
3. T. Frankel, The Geometry of Physics, Cambridge University Press, 1997
4. P.G. Tait, Collected Scientific Papers, 2 Vols., Cambridge University Press, 1898/1900
5. H. K. Moffat, The Degree of Knottedness of Tangled Vortex Lines, J. Fluid Mech. 35, 117 (1969)
6. P. A. M. Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. A 133, 60 (1931)
7. C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance Phys. Rev. 96, 191 (1954)
8. N. K. Nielsen and P. Olesen, Vortex-Line Models for Dual Strings, Nucl. Phys. B 61, 45 (1973)
9. P. G. de Gennes, Superconductivity of Metals and Alloys, W. A. Benjamin 1966
10. M. Tinkham, Introduction to Superconductivity, McGraw-Hill 1975
11. G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Minokur, Rev. Mod. Phys. 66, 1125 (1994)
12. D. Nelson, Defects and Geometry in Condensed Matter Physics, Cambridge University Press, 2002
13. C. P. Poole, Jr., H. A. Farach and R. J. Creswick, Superconductivity, Academic Press, 1995
14. E. B. Bogomol’nyi, The Stability of Classical Solutions, Sov. J. Nucl. Phys. 24, 449 (1976)
15. R. Jackiw and P. Rossi, Zero Modes of the Vortex-Fermion System, Nucl. Phys. B 252, 343 (1991)
16. E. Weinberg, Index Calculations for the Fermion-Vortex System, Phys. Rev. D 24, 2669 (1981)
17. C. Nash and S. Sen, Topology and Geometry for Physicists, Academic Press 1983
18. M. Nakahara, Geometry, Topology and Physics, Adam Hilger 1990
19. J. R. Munkres, Topology, Prentice Hall 2000
20. O. Jahn, Instantons and Monopoles in General Abelian Gauges, J. Phys. A33, 2997 (2000)
21. T. W. Gamelin and R. E. Greene, Introduction to Topology, Dover 1999
22. V. I. Arnold, B. A. Khesin, Topological Methods in Hydrodynamics, Springer 1998
23. D. J. Thouless, Topological Quantum Numbers in Nonrelativistic Physics, World Scientific 1998
24. N. Steenrod, The Topology of Fiber Bundels, Princeton University Press 1951
25. G. Morandi, The Role of Topology in Classical and Quantum Physics, Springer 1992
26. W. Miller, Jr., Symmetry Groups and Their Applications, Academic Press 1972
27. N. D. Mermin, The Topological Theory of Defects in Ordered Media, Rev. Mod. Phys. 51, 591 (1979)
28. V. P. Mineev, Topological Objects in Nematic Liquid Crystals, Appendix A, in: V. G. Boltyanskii and V. A. Efremovich, Intuitive Combinatorial Topology, Springer 2001
29. S. Chandrarsekhar, Liquid Crystals, Cambridge University Press 1992
30. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press 1993
31. P. Poulin, H. Stark, T. C. Lubensky and D.A. Weisz, Novel Colloidal Interactions in Anisotropic Fluids, Science 275 1770 (1997)
32. H. Georgi and S. Glashow, Unified Weak and Electromagnetic Interactions without Neutral Currents, Phys. Rev. Lett. 28, 1494 (1972)
33. H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel Verlag 1928.
34. R. Jackiw, Introduction to the Yang–Mills Quantum Theory, Rev. Mod. Phys. 52, 661 (1980)
35. F. Lenz, H. W. L. Naus and M. Thies, QCD in the Axial Gauge Representation, Ann. Phys. 233, 317 (1994)
36. F. Lenz and S. Wörlen, Compact variables and Singular Fields in QCD, in: at the frontier of Particle Physics, handbook of QCD edited by M. Shifman, Vol. 2, p. 762, World Scientific 2001
37. G.’t Hooft, Magnetic Monopoles in Unified Gauge Models, Nucl. Phys. B 79, 276 (1974)
38. A.M. Polyakov, Particle Spectrum in Quantum Field Theory, JETP Lett. 20, 194 (1974); Isometric States in Quantum Fields, JETP Lett. 41, 988 (1975)
39. F. Lenz, H. W. L. Naus, K. Ohta, and M. Thies, Quantum Mechanics of Gauge Fixing, Ann. Phys. 233, 17 (1994)
40. A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press 1994
41. S.L.Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Continuous Quantum Phase Transitions, Rev.Mod.Phys. 69 315, (1997)
42. R. Rajaraman, Solitons and Instantons, North Holland 1982
43. B. Julia and A. Zee, Poles with Both Electric and Magnetic Charges in Nonabelian Gauge Theory, Phys. Rev. D 11, 2227 (1975)
44. E. Tomboulis and G. Woo, Soliton Quantization in Gauge Theories, Nucl. Phys. B 107, 221 (1976); J. L. Gervais, B. Sakita and S. Wadia, The Surface Term in Gauge Theories, Phys. Lett. B 63 B, 55 (1999)
45. C. Callias, Index Theorems on Open Spaces, Commun. Mat. Phys. 62, 213 (1978)
46. R. Jackiw and C. Rebbi, Solitons with Fermion Number 1/2, Phys. Rev. D 13, 3398 (1976)
47. R. Jackiw and C. Rebbi, Spin from Isospin in Gauge Theory, Phys. Rev. Lett. 36, 1116 (1976)
48. P. Hasenfratz and G. ’t Hooft, Fermion-Boson Puzzle in a Gauge Theory, Phys. Rev. Lett. 36, 1119 (1976)
49. E. W. Kolb and M. S. Turner, The Early Universe, Addison-Wesley 1990
50. J. A. Peacock, Cosmological Physics, Cambridge University Press 1999
51. V. N. Gribov, Quantization of Non-Abelian Gauge Theories, Nucl. Phys. B 139, 1 (1978)
52. I. M. Singer, Some Remarks on the Gribov Ambiguity, Comm. Math. Phys. 60, 7 (1978)
53. T. T. Wu and C. N. Yang, Concept of Non-Integrable Phase Factors and Global Formulations of Gauge Fields, Phys. Rev. D 12, 3845 (1975)
54. A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Yu. S. Tyupkin, Pseudoparticle solutions of the Yang–Mills equations, Phys. Lett. B 59, 85 (1975)
55. J. D. Bjorken, in: Lectures on Lepton Nucleon Scattering and Quantum Chromodynamics, W. Atwood et al. , Birkhäuser 1982
56. R. Jackiw, Topological Investigations of Quantized Gauge theories, in: Current Algebra and Anomalies, edt. by S. Treiman et al., Princeton University Press, 1985
57. A. S. Schwartz, Quantum Field Theory and Topology, Springer 1993
58. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vol. 2, Cambridge University Press 1987
59. G. ’t Hooft, Computation of the Quantum Effects Due to a Four Dimensional Quasiparticle, Phys. Rev. D 14, 3432 (1976)
60. G. Esposito, Dirac Operators and Spectral Geometry, Cambridge University Press 1998
61. T. Schäfer and E. V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70, 323 (1998)
62. C. G. Callan, R. F. Dashen and D. J. Gross, Toward a Theory of Strong Interactions, Phys. Rev. D 17, 2717 (1978)
63. V. de Alfaro, S. Fubini and G. Furlan, A New Classical Solution Of The Yang–Mills Field Equations, Phys. Lett. B 65, 163 (1976).
64. F. Lenz, J. W. Negele and M. Thies, Confinement from Merons, hep-th/0306105 to appear in Phys. Rev. D
65. H. K. Moffat and A. Tsinober, Helicity in Laminar and Turbulent Flow, Ann. Rev. Fluid Mech. 24 281, (1992)
66. P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001
67. E. Witten, Some Geometrical Applications of Quantum Field Theory, in *Swansea 1988, Proceedings of the IX th International Congr. on Mathematical Physics, p.77
68. L. H. Kauffman, Knots and Physics, World Scientific 1991
69. A. M. Polyakov, Fermi-Bose Transmutation Induced by Gauge Fields, Mod. Phys. Lett. A 3, 325 (1988)
70. B. Svetitsky, Symmetry Aspects of Finite Temperature Confinement Transitions, Phys. Rep. 132, 1 (1986)
71. D. J. Toms, Casimir Effect and Topological Mass, Phys. Rev. D 21, 928 (1980)
72. F. Lenz and M. Thies, Polyakov Loop Dynamics in the Center Symmetric Phase, Ann. Phys. 268, 308 (1998)
73. J. I. Kapusta, Finite-temperature field theory, Cambridge University Press 1989
74. F. Lenz, H. W. L. Naus, K. Ohta, and M. Thies, Zero Modes and Displacement Symmetry in Electrodynamics, Ann. Phys. 233, 51 (1994)
75. F. Lenz, J. W. Negele, L. O’Raifeartaigh and M. Thies, Phases and Residual Gauge Symmetries of Higgs Models, Ann. Phys. 285, 25 (2000)
76. M. Le Bellac, Thermal field theory, Cambridge University Press 1996
77. H. Reinhardt, M. Engelhardt, K. Langfeld, M. Quandt, and A. Schäfke, Magnetic Monopoles, Center Vortices, Confinement and Topology of Gauge Fields, hep-th/ 9911145
78. J. Greensite, The Confinement Problem in Lattice Gauge Theory, hep-lat/ 0301023
79. H. J. de Vega and F. A. Schaposnik, Electrically Charged Vortices in Non-Abelian Gauge Theories, Phys. Rev. Lett. 56, 2564 (1986)
80. G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138, 1 (1978)
81. S. Samuel, Topological Symmetry Breakdown and Quark Confinement, Nucl. Phys. B 154, 62 (1979)
82. A. Kovner, Confinement, Z N Symmetry and Low-Energy Effective Theory of Gluodynamicsagnetic, in: at the frontier of Particle Physics, handbook of QCD edited by M. Shifman, Vol. 3, p. 1778, World Scientific 2001
83. J. Fingberg, U. Heller, and F. Karsch, Scaling and Asymptotic Scaling in the SU(2) Gauge Theory, Nucl. Phys. B 392, 493 (1993)
84. B. Grossman, S. Gupta, U. M. Heller, and F. Karsch, Glueball-Like Screening Masses in Pure SU(3) at Finite Temperatures, Nucl. Phys. B 417, 289 (1994)
85. M. Ishii, H. Suganuma and H. Matsufuru, Scalar Glueball Mass Reduction at Finite Temperature in SU(3) Anisotropic Lattice QCD, Phys. Rev. D 66, 014507 (2002); Glueball Properties at Finite Temperature in SU(3) Anisotropic Lattice QCD, Phys. Rev. D 66, 094506 (2002)
86. S. Rastogi, G. W. Höhne and A. Keller, Unusual Pressure-Induced Phase Behavior in Crystalline Poly(4-methylpenthene-1): Calorimetric and Spectroscopic Results and Further Implications, Macromolecules 32 8897 (1999)
87. N. Avraham, B. Kayhkovich, Y. Myasoedov, M. Rappaport, H. Shtrikman, D. E. Feldman, T. Tamegai, P. H. Kes, Ming Li, M. Konczykowski, Kees van der Beek, and Eli Zeldov, ‘ Inverse’ Melting of a Vortex Lattice, Nature 411, 451, (2001)
88. F. Lenz, E. J. Moniz and M. Thies, Signatures of Confinement in Axial Gauge QCD, Ann. Phys. 242, 429 (1995)
89. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley Publishing Company, 1995
90. T. Reisz, Realization of Dimensional Reduction at High Temperature, Z. Phys. C 53, 169 (1992)
91. V. L. Eletsky, A. C. Kalloniatis, F. Lenz, and M. Thies, Magnetic and Thermodynamic Stability of SU(2) Yang–Mills Theory, Phys. Rev. D 57, 5010 (1998)
92. F. Karsch, E. Laermann, and A. Peikert, The Pressure in 2, 2 + 1 and 3 Flavor QCD, Phys. Lett. B 478, 447 (2000)
93. J. Engels, F. Karsch and K. Redlich, Scaling Properties of the Energy Density in SU(2) Lattice Gauge Theory, Nucl. Phys. B435, 295 (1995)
94. N. Seiberg, E. Witten, Monopole Condensation, and Confinement in N = 2 Supersymmetric QCD, Nucl. Phys. B 426, 19 (1994); Monopoles, Duality and Chiral Symmetry Breaking in N = 2 supersymmetric QCD Nucl. Phys. B 431, 484 (1995)
95. M. Quandt, H. Reinhardt and A. Schäfke, Magnetic Monopoles and Topology of Yang–Mills Theory in Polyakov Gauge, Phys. Lett. B 446, 290 (1999)
96. C. Ford, T. Tok and A. Wipf, SU(N) Gauge Theories in Polyakov Gauge on the Torus, Phys. Lett. B 456, 155 (1999)
97. O. Jahn and F. Lenz, Structure and Dynamics of Monopoles in Axial Gauge QCD, Phys. Rev. D 58, 85006 (1998)
98. B. J. Harrington and H. K. Shepard, Periodic Euclidean Solutions and the Finite-Temperature Yang–Mills Gas, Phys. Rev. D 17, 2122 (1978)
99. G. ’t Hooft, Topology of the Gauge Condition and New Confinement Phases in Non-Abelian Gauge Theories, Nucl. Phys. B 190, 455 (1981)
100. J. Kogut and L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D 11, 395 (1975)
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Lenz, F. Topological Concepts in Gauge Theories. In: Bick, E., Steffen, F.D. (eds) Topology and Geometry in Physics. Lecture Notes in Physics, vol 659. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31532-2_2
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