Geodesic
Instantons and monopoles
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 2, pp. 305-360 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this survey we present the main notions and constructions of gauge theories, namely, the Donaldson theory, the Seiberg–Witten theory, and the theory of B-monopoles, which connects the previous two theories. In the framework of differential geometry these theories give new invariants of smooth structures in dimension 4. The introduction of these new gauge invariants has helped to solve many problems of modern geometry. The apparatus developed in the framework of these theories leads to new modern methods of investigation both in smooth geometry and in applied problems of mathematical physics. Without striving for the greatest possible generality, the survey aims to present the topic in maximal breadth and accessibility. 
@article{RM_2002_57_2_a2,
     author = {N. A. Tyurin},
     title = {Instantons and monopoles},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {305--360},
     year = {2002},
     volume = {57},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2002_57_2_a2/}
}
TY  - JOUR
AU  - N. A. Tyurin
TI  - Instantons and monopoles
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2002
SP  - 305
EP  - 360
VL  - 57
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/RM_2002_57_2_a2/
LA  - en
ID  - RM_2002_57_2_a2
ER  - 
%0 Journal Article
%A N. A. Tyurin
%T Instantons and monopoles
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2002
%P 305-360
%V 57
%N 2
%U http://geodesic.mathdoc.fr/item/RM_2002_57_2_a2/
%G en
%F RM_2002_57_2_a2
N. A. Tyurin. Instantons and monopoles. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 2, pp. 305-360. http://geodesic.mathdoc.fr/item/RM_2002_57_2_a2/

[1] J. Ambjørn, Quantization of Geometry, Lect. given at Les Houches, Nato Adv. Sci. Inst. Fluctuating Geometries in Statistical Mechanics and Field Theory, Session LXII, 1994

[2] M. F. Atiyah, R. Bott, “The Yang–Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A, 308:1505 (1982), 523–615 | DOI | MR

[3] P. A. M. Dirak, Printsipy kvantovoi mekhaniki, Mir, M., 1960

[4] S. Donaldson, “Symplectic submanifolds and almost-complex geometry”, J. Differential Geom., 44 (1996), 666–705 | MR | Zbl

[5] S. Donaldson, “Lefschetz fibrations in symplectic geometry”, Doc. Math. J. DMV. Extra Volume ICM Berlin, II (1998), 309–314 | MR | Zbl

[6] S. Donaldson, P. Kronheimer, The Geometry of Four-Manifolds, Clarendon Press, Oxford, 1990 | MR | Zbl

[7] R. Donagi, E. Witten, “Supersymmetric Yang–Mills theory and integrable systems”, Nuclear Phys. B, 460:2 (1996), 299–334 ; hep-th/9510101 | DOI | MR | Zbl

[8] R. Fintushel, R. Stern, “Knots, links, and 4-manifolds”, Invent. Math., 134:2 (1998), 363–400 | DOI | MR | Zbl

[9] R. Fintushel, R. Stern, “Constructions of smooth 4-manifolds”, Doc. Math. J. DMV. Extra Volume ICM Berlin, II (1998), 1–10 ; math/9907178 | MR

[10] D. Frid, K. Ulenbek, Instantony i chetyrekhmernye mnogoobraziya, Mir, M., 1988 | MR

[11] M. H. Freedman, F. Luo, Selected Applications of Geometry to Low-Dimensional Topology, Marker lectures in the mathematical sciences held at the Pennsylvania (State University, 1987), Amer. Math. Soc., Providence, RI, 1989 | MR | Zbl

[12] B. Karpov, “S-dvoistvennost i isklyuchitelnye rassloeniya”, Izv. RAN. Ser. matem., 63:1 (1999), 107–122 | MR | Zbl

[13] P. W. Kronheimer, T. S. Mrowka, “The genus of embedded surfaces in the projective plane”, Math. Res. Lett., 1 (1994), 797–808 | MR | Zbl

[14] S. Mukai, “On moduli spaces of bundles on K3 surfaces. II”, Vector Bundles on Algebraic Varieties, Bombay colloquium, Oxford Univ. Press, Oxford, 1984, 341–414 | MR

[15] V. Pidstrigach, A. Tyurin, “Invarianty gladkoi struktury algebraicheskoi poverkhnosti, zadavaemye operatorom Diraka”, Izv. RAN. Ser. matem., 56:2 (1992), 279–371 | MR | Zbl

[16] V. Pidstrigach, A. Tyurin, Localisation of the Donaldson's Invariants along Seiberg–Witten Classes, Preprint, Bielefeld University, Bielefeld, 1995

[17] D. Salamon, Spin geometry and Seiberg–Witten invariants, Preprint, University of Warwick, Warwick, 1995

[18] N. Seiberg, E. Witten, “Electric-magnetic duality, monopole condensation and confinement in $N=2$ supersymmetric Yang–Mills theory”, Nuclear Phys. B, 426:1 (1994), 19–52 ; Corrigendum ibid., 430:2, 485–486 ; hep-th/9407087 | DOI | MR | Zbl | MR | Zbl

[19] C. H. Taubes, “The Seiberg–Witten invariants and symplectic forms”, Math. Res. Lett., 1:6 (1994), 809–822 | MR | Zbl

[20] C. H. Taubes, “More constraints on symplectic forms from Seiberg–Witten invariants”, Math. Res. Lett., 2:1 (1995), 9–13 | MR | Zbl

[21] A. N. Tyurin, “Spin-polinomialnye invarianty gladkikh struktur na algebraicheskikh poverkhnostyakh”, Izv. RAN. Ser. matem., 57:2 (1993), 125–164 | MR | Zbl

[22] A. N. Tyurin, “Algebro-geometricheskie aspekty gladkosti. 1. Polinomy Donaldsona”, UMN, 44:3 (1989), 93–143 | MR | Zbl

[23] A. N. Tyurin, “Spetsialnaya lagranzheva geometriya kak malaya deformatsiya algebraicheskoi geometrii (GQP i zerkalnaya simmetriya)”, Izv. RAN. Ser. matem., 64:2 (2000), 141–224 | MR | Zbl

[24] N. A. Tyurin, “Neobkhodimoe i dostatochnoe usloviya deformatsii B-monopolya v instanton”, Izv. RAN. Ser. matem., 60:1 (1996), 211–224 | MR | Zbl

[25] C. Vafa, E. Witten, “A strong coupling test of $S$-duality”, Nuclear Phys. B, 431:1–2 (1994), 3–77 ; hep-th/9408169 | DOI | MR | Zbl

[26] E. Witten, “Monopoles and four-manifolds”, Math. Res. Lett., 1 (1994), 769–796 | MR | Zbl

[27] E. Witten, Duality in supersymmetric gauge theories, Notes from E. Witten's lectures, Santa Cruz, 1995