Geodesic
Kostant Prequantization of Symplectic Manifolds with Contact Singularities
Matematičeskie zametki, Tome 105 (2019) no. 6, pp. 857-878.

Voir la notice de l'article provenant de la source Math-Net.Ru

The relationship between the Bohr–Sommerfeld quantization condition and the integrality of the symplectic structure in Planck constant units is considered. Constructions of spherical and toric $\Theta$-handles are proposed which allow one to obtain symplectic manifolds with contact singularities, preserve Kostant–Souriau prequantization, and expect interesting topological applications. In particular, the toric $\Theta$-handle glues Liouville foliations, while the spherical handle generates (pre)quantized connected sums of symplectic manifolds. In this way, nonorientable manifolds may arise.
Mots-clés : quantization, Kostant–Souriau quantization
Keywords: Bohr–Sommerfeld conditions, contact singularity, $\Theta$-handle. 
@article{MZM_2019_105_6_a4,
     author = {D. B. Zot'ev},
     title = {Kostant {Prequantization} of {Symplectic} {Manifolds} with {Contact} {Singularities}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {857--878},
     publisher = {mathdoc},
     volume = {105},
     number = {6},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_105_6_a4/}
}
TY  - JOUR
AU  - D. B. Zot'ev
TI  - Kostant Prequantization of Symplectic Manifolds with Contact Singularities
JO  - Matematičeskie zametki
PY  - 2019
SP  - 857
EP  - 878
VL  - 105
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2019_105_6_a4/
LA  - ru
ID  - MZM_2019_105_6_a4
ER  - 
%0 Journal Article
%A D. B. Zot'ev
%T Kostant Prequantization of Symplectic Manifolds with Contact Singularities
%J Matematičeskie zametki
%D 2019
%P 857-878
%V 105
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2019_105_6_a4/
%G ru
%F MZM_2019_105_6_a4
D. B. Zot'ev. Kostant Prequantization of Symplectic Manifolds with Contact Singularities. Matematičeskie zametki, Tome 105 (2019) no. 6, pp. 857-878. http://geodesic.mathdoc.fr/item/MZM_2019_105_6_a4/

[1] A. A. Kirillov, “Geometricheskoe kvantovanie”, Dinamicheskie sistemy – 4, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 4, VINITI, M., 1985, 141–176 | MR

[2] I. E. Segal, “Quantization of non-linear systems”, J. Math. Phys., 1 (1960), 468–488 | DOI | MR | Zbl

[3] V. Giiemin, S. Sternberg, Geometricheskie asimptotiki, Mir, M., 1981

[4] M. V. Karasev, V. P. Maslov, “Kvantovanie simplekticheskikh mnogoobrazii s konicheskimi tochkami”, TMF, 53:3 (1982), 374–387 | MR | Zbl

[5] M. V. Karasev, V. P. Maslov, “Asimptoticheskoe i geometricheskoe kvantovanie”, UMN, 39:6 (240) (1984), 115–173 | MR | Zbl

[6] J. M. Souriau, “Quantification géométrique”, Comm. Math. Phys., 1 (1966), 374–398 | MR | Zbl

[7] B. Kostant, “Kvantovanie i unitarnye predstavleniya”, UMN, 28:1 (169) (1973), 163–225 | MR | Zbl

[8] S. M. Natanzon, Vvedenie v puchki, rassloeniya i klassy Cherna, MTsNMO, M., 2010

[9] V. Geizenberg, Fizicheskie printsipy kvantovoi teorii, GTTI, M., 1932

[10] A. V. Bolsinov, A. T. Fomenko, Integriruemye gamiltonovy sistemy. Topologiya. Geometriya. Klassifikatsiya, T. I, Udmurtskii un-t, Izhevsk, 1999

[11] D. B. Zotev, Topology of Integrable Systems. The Fomenko Theory, Cambridge Sci. Publ., Cambridge, 2010 | MR

[12] J. Czyz, “On a modification of the Kostant–Souriau geometric quantization procedure”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 26:2 (1978), 129–138 | MR | Zbl

[13] H. Hess, “On a geometric quantization scheme generalizing those of Kostant–Souriau and Gzyz”, Differential Geometric Methods in Mathematical Physics, Lecture Notes in Phys., 139, 1981, 1–35 | DOI | MR

[14] D. B. Zotev, “O simplekticheskoi geometrii mnogoobrazii s pochti vsyudu nevyrozhdennoi zamknutoi 2-formoi”, Matem. zametki, 76:1 (2004), 66–77 | DOI | MR | Zbl

[15] D. B. Zotev, “Kontaktnye vyrozhdeniya zamknutykh 2-form”, Matem. sb., 198:4 (2007), 47–78 | DOI | MR | Zbl

[16] Sh. Kobayasi, K. Nomidzu, Osnovy differentsialnoi geometrii, T. 1, 2, Nauka, M., 1981 | MR | MR | Zbl | Zbl

[17] M. Khatchins, K. G. Taubs, “Vvedenie v teoriyu Zaiberga–Vittena dlya simplekticheskikh mnogoobrazii”, Simplekticheskaya geometriya i topologiya, MTsNMO, M., 2008, 121–135

[18] A. T. Fomenko, Simplekticheskaya geometriya. Metody i prilozheniya, Izd-vo Mosk. un-ta, M., 1988