On two geometric problems arising in mathematical physics
Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 157-166.

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We consider two mathematical problems that can be ascribed to the category pointed out in the title. The first one relates to the geometric quantization and deals with the twistor approach to the quantization of smooth strings. The second one concerns the adiabatic limit in the Ginzburg–Landau and Seiberg–Witten equations.
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A. G. Sergeev. On two geometric problems arising in mathematical physics. Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 157-166. http://geodesic.mathdoc.fr/item/FPM_2015_20_2_a10/

[1] Domrin A. V., “Analogi vikhrei Ginzburga–Landau”, Teor. i matem. fiz., 124:1 (2000), 18–35 | DOI | MR | Zbl

[2] Kirillov A. A., “Geometricheskoe kvantovanie”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 4, 1985, 141–178 | MR | Zbl

[3] Kirillov A. A., “Kelerova struktura na $K$-orbitakh gruppy diffeomorfizmov okruzhnosti”, Funkts. analiz i ego pril., 21:2 (1987), 42–45 | MR | Zbl

[4] Kirillov A. A., Yurev D. V., “Kelerova geometriya beskonechnomernogo odnorodnogo prostranstva $M=\operatorname{Diff}_+(S^1)/\mathrm{Rot}(S^1)$”, Funkts. analiz i ego pril., 21:4 (1987), 35–46 | MR | Zbl

[5] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, Chast 2, v. 9, Statisticheskaya fizika, Fizmatlit, M., 2000

[6] Palvelev R. V., “Rasseyanie vikhrei v abelevoi modeli Khiggsa”, Teor. i matem. fiz., 156:1 (2008), 77–91 | DOI | MR | Zbl

[7] Palvelev R. V., “Obosnovanie adiabaticheskogo printsipa v abelevoi modeli Khiggsa”, Tr. MMO, 72, 2011, 281–314 | Zbl

[8] Palvelev R. V., Sergeev A. G., “Obosnovanie adiabaticheskogo predela dlya giperbolicheskikh uravnenii Ginzburga–Landau”, Tr. MIAN, 277, 2012, 199–214 | MR | Zbl

[9] Pressli A., Sigal G., Gruppy petel, Mir, M., 1990 | MR

[10] Sergeev A. G., Geometricheskoe kvantovanie prostranstv petel, MIAN, M., 2009

[11] Sergeev A. G., Chechin S. V., “Rasseyanie medlenno dvizhuschikhsya vikhrei v abelevoi $(2+1)$-mernoi modeli Khiggsa”, Teor. i matem. fiz., 85:3 (1990), 397–411 | MR

[12] Bowick M. J., Lahiri A., “The Ricci curvature of $\operatorname{Diff}S^1/\mathrm{SL}(2,\mathbb R)$”, J. Math. Phys., 29 (1988), 1979–1981 | DOI | MR | Zbl

[13] Bowick M. J., Rajeev S. G., “The holomorphic geometry of closed bosonic string theory and $\operatorname{Diff}S^1/S^1$”, Nuclear Phys. B, 293 (1987), 348–384 | DOI | MR

[14] Guillemin V., Sternberg S., Geometric Asymptotics, Amer. Math. Soc., Providence, 1977 | MR | Zbl

[15] Jaffe A., Taubes C. H., Vortices and Monopoles, Birkhäuser, Boston, 1980 | MR | Zbl

[16] Manton N. S., “A remark on the scattering of BPS monopoles”, Phys. Lett. B, 110 (1982), 54–56 | DOI | MR | Zbl

[17] Morgan J. W., The Seiberg–Witten Equations and Applications to the Topology of Smooth Four- Manifolds, Princeton Univ. Press, Princeton, 1996 | MR | Zbl

[18] Salamon D., Spin geometry and Seiberg–Witten invariants, Preprint, Univ. of Warwick, 1996 | MR

[19] Seiberg N., Witten E., “Electro-magnetic duality, monopole condensation and confinement in $N=2$ supersymmetric Yang–Mills theory”, Nuclear Phys. B, 426 (1994), 19–52 | DOI | MR | Zbl

[20] Seiberg N., Witten E., “Monopoles, duality and chiral symmetry breaking in $N=2$ supersymmetric Yang–Mills theory”, Nuclear Phys. B, 426 (1994), 581–640 | DOI | MR

[21] Sergeev A. G., Vortices and Seiberg–Witten Equations, Nagoya Univ., Nagoya, 2002

[22] Sergeev A., Kähler Geometry of Loop Spaces, World Scientific, Singapore, 2010 | MR

[23] Souriau J.-M., Structure des Systémes Dynamiques, Dunod, Paris, 1970 | MR

[24] Sniatycki J., Geometric Quantization and Quantum Mechanics, Springer, Berlin, 1980 | MR | Zbl

[25] Taubes C. H., “Arbitrary $N$-vortex solutions to the first-order Ginzburg–Landau equations”, Comm. Math. Phys., 72 (1980), 277–292 | DOI | MR | Zbl

[26] Taubes C. H., “$\mathrm{SW}\to\mathrm{Gr}$: From the Seiberg–Witten equations to pseudo-holomorphic curves”, J. Amer. Math. Soc., 9 (1996), 845–918 | DOI | MR | Zbl

[27] Taubes C. H., “$\mathrm{Gr}\to\mathrm{SW}$: From pseudo-holomorphic curves to Seiberg–Witten solutions”, J. Differential Geom., 51 (1999), 203–334 | MR | Zbl

[28] Taubes C. H., “$\mathrm{Gr}=\mathrm{SW}$: Counting curves and connections”, J. Differential Geom., 52 (1999), 453–609 | MR | Zbl

[29] Tuynman G. M., Geometric Quantization, CWI Syllabus, Amsterdam, 1985 | MR | Zbl

[30] Woodhouse N. M. J., Geometric Quantization, Clarendon Press, Oxford, 1992 | MR | Zbl