Twistor geometry and gauge fields
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A. G. Sergeev
Translated by: the author - Trans. Moscow Math. Soc. 2018, 135-175
- DOI: https://doi.org/10.1090/mosc/277
- Published electronically: November 29, 2018
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Abstract:
The main topic of this survey article is an exposition of basics of the theory of twistors and of applications of this theory to solving equations of gauge field theory, such as, e.g., Yang–Mills equations, monopole equations, etc.References
- M. F. Atiyah, Instantons in two and four dimensions, Comm. Math. Phys. 93 (1984), no. 4, 437–451. MR 763752
- M. F. Atiyah, Geometry on Yang-Mills fields, Scuola Normale Superiore, Pisa, 1979. MR 554924
- M. F. Atiyah and I. M. Singer, The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484–530. MR 236950, DOI 10.2307/1970715
- M. F. Atiyah, N. J. Hitchin, V. G. Drinfel′d, and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187. MR 598562, DOI 10.1016/0375-9601(78)90141-X
- M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
- M. F. Atiyah and R. S. Ward, Instantons and algebraic geometry, Comm. Math. Phys. 55 (1977), no. 2, 117–124. MR 494098
- A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975), no. 1, 85–87. MR 434183, DOI 10.1016/0370-2693(75)90163-X
- E. B. Bogomol′nyĭ, The stability of classical solutions, Jadernaja Fiz. 24 (1976), no. 4, 861–870 (Russian); English transl., Soviet J. Nuclear Phys. 24 (1976), no. 4, 449–454. MR 0443719
- S. K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), no. 2, 269–277. MR 710055, DOI 10.1215/S0012-7094-04-12822-2
- S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), no. 4, 453–460. MR 763753
- J. Eells and S. Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 4, 589–640 (1986). MR 848842
- N. J. Hitchin, Monopoles and geodesics, Comm. Math. Phys. 83 (1982), no. 4, 579–602. MR 649818
- N. J. Hitchin, On the construction of monopoles, Comm. Math. Phys. 89 (1983), no. 2, 145–190. MR 709461
- N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. MR 887284, DOI 10.1112/plms/s3-55.1.59
- Arthur Jaffe and Clifford Taubes, Vortices and monopoles, Progress in Physics, vol. 2, Birkhäuser, Boston, Mass., 1980. Structure of static gauge theories. MR 614447
- W. Nahm, All self-dual multimonopoles for arbitrary gauge groups, Structural elements in particle physics and statistical mechanics (Freiburg, 1981) NATO Adv. Study Inst. Ser. B: Physics, vol. 82, Plenum, New York, 1983, pp. 301–310. MR 807414
- M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI 10.2307/1970710
- Roger Penrose, The twistor programme, Rep. Mathematical Phys. 12 (1977), no. 1, 65–76. MR 465032, DOI 10.1016/0034-4877(77)90047-7
- M. K. Prasad, C. M. Sommerfield, Exact classical solutions of the t’Hooft monopole and the Julia–Zee dyon, Phys. Rev. Letters 35 (1975), 760–762.
- John H. Rawnsley, $f$-structures, $f$-twistor spaces and harmonic maps, Geometry seminar “Luigi Bianchi” II—1984, Lecture Notes in Math., vol. 1164, Springer, Berlin, 1985, pp. 85–159. MR 829229, DOI 10.1007/BFb0081911
- A. G. Sergeev, Harmonic spheres conjecture, Theor. Mathem. Phys. 164 (2010), 1140–1150.
- Clifford Henry Taubes, Stability in Yang-Mills theories, Comm. Math. Phys. 91 (1983), no. 2, 235–263. MR 723549
- Clifford Henry Taubes, Min-max theory for the Yang-Mills-Higgs equations, Comm. Math. Phys. 97 (1985), no. 4, 473–540. MR 787116
- Karen K. Uhlenbeck, Removable singularities in Yang-Mills fields, Comm. Math. Phys. 83 (1982), no. 1, 11–29. MR 648355
- Karen K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. MR 648356
- V. S. Vladimirov and A. G. Sergeev, Compactification of Minkowski space and complex analysis in the future tube, Ann. Polon. Math. 46 (1985), 439–454 (Russian). MR 841848
- R. S. Ward, On self-dual gauge fields, Phys. Lett. A 61 (1977), no. 2, 81–82. MR 443823, DOI 10.1016/0375-9601(77)90842-8
- Richard A. Wentworth, Higgs bundles and local systems on Riemann surfaces, Geometry and quantization of moduli spaces, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2016, pp. 165–219. MR 3675465
- Graeme Wilkin, Morse theory for the space of Higgs bundles, Comm. Anal. Geom. 16 (2008), no. 2, 283–332. MR 2425469
Bibliographic Information
- A. G. Sergeev
- Affiliation: Steklov Mathematical Institute, ul. Gubkina 8, Moscow 117966, GSP-1, Russia
- Email: sergeev@mi.ras.ru
- Published electronically: November 29, 2018
- Additional Notes: While preparing this article the author was partially supported by RRFR grants 16-01-00117 and 16-52-12012, and by the program “Nonlinear Dynamics” of the Presidium of the Russian Academy of Sciences.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2018, 135-175
- MSC (2010): Primary 70S15, 81T13
- DOI: https://doi.org/10.1090/mosc/277
- MathSciNet review: 3881462