Symplektické konexe a kvantování
| Thesis title in Czech: | Symplektické konexe a kvantování |
|---|---|
| Thesis title in English: | Symplectic Connections and Quantization |
| Key words: | symplektická konexe|Fedosovova konexe|deformační kvantování|geometrické kvantování|homologická algebra |
| English key words: | symplectic connection|Fedosov connection|deformation quantization|geometric quantization|homological algebra |
| Academic year of topic announcement: | 2025/2026 |
| Thesis type: | dissertation |
| Thesis language: | |
| Department: | Mathematical Institute of Charles University (32-MUUK) |
| Supervisor: | doc. RNDr. Svatopluk Krýsl, Ph.D. |
| Author: |
| Guidelines |
| Student se seznámí se základy deformačního a geometrického kvantování, symplektickými konexemi a
jejich tenzory křivosti a pokusí se tyto pojmy propojit. Základy BRST-kvantování mohou být součástí studia. The student will study basics on deformation and geometric quantization, symplectic connections and the curvature tensors of such connections. A possible relation of these notions shall be studied. Basics on the so called BRST-quantization can be studied as well. |
| References |
| Blau, M., Symplectic geometry and geometric quantization, preprint, electronically available
www.blau.itp.unibe.ch/lecturesGQ.ps.gz Cahen, M., Schwachhoefer, L., Special symplectic connections J. Differential Geom. 83 (2009), no. 2, 229--271. Cattaneo, A., Felder, G., Tomassini, L., From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115(2), 329--352, doi: 10.1215/S0012-7094-02-11524-5. Chan, K., Leung, N., Li, Q., Quantizable functions on Kaehler manifolds and non-formal quantization, Adv. Math. 433 (2023), Paper No. 109293. Fedosov, B., A simple geometrical construction of deformation quantization. J. Differential Geom. 40 (1994), no. 2, 213–238 Forger, M., Hess, H., Universal metaplectic structures and geometric quantization, Commun. Math. Phys. 67 (1979), 267–278 Gelfand, I., Shubin, M., Retakh, V., Fedosov manifolds, Adv. Math. 136 (1998), no. 1, 104–140 Guillemin, V., Sternberg, S., Symplectic techniques in Physics. Cambridge University Press, Cambridge, 1984. Habermann, K., The Dirac operator on symplectic spinors, Ann. Global Anal. Geom. 13 (1995), no. 2, 155–168. Habermann, K., Klein, A., Lie derivative of symplectic spinor fields, metaplectic represen- tation, and quantization, Rostock. Math. Kolloq. no. 57 (2003), 71–91. Krysl, S., Twistor operators in symplectic geometry, Adv. Appl. Clifford Algebr. 32 (2022), no. 1, Paper No. 14, Krysl, S., Induced C*-complexes in metaplectic geometry, Communications in Mathematical Physics, Volume 365, Issue 1 (2019), 61–91 Krysl, S., Cohomology of the de Rham complex twisted by the oscillatory representation, Diff. Geom. Appl., Vol. 33 (2014), 290–297. Tondeur, P., Affine Zusammenhaenge auf Mannigfaltigkeiten mit fast-symplektischer Struktur. Comment. Math. Helv. 36 (1961), 234–244. Vaisman, I., Symplectic curvature tensors, Monatshefte fuer Mathematik 100 (1985), 299– 327. Waldmann, S., Poisson-Geometrie und Deformationsquantisierung. Eine Einfuehrung, Springer-Verlag, 2007. Wallach, N., Symplectic geometry and Fourier analysis, Math Sci Press, Brookline, Mass., 1977; Lie Groups: History Frontiers and Applications, Vol. V. Weil, A., Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 |
| Preliminary scope of work |
| Jelikož je známo, že nedeformační kvantování je nemožné (tzv. věta Groenewald - van Hovea), je vhodné studovat tzv. deformační kvantování.
V tomto případě, se ukazují být vhodné symplektické konexe, které jsou ploché nebo částečně ploché. |
| Preliminary scope of work in English |
| Since non-deformation quantization is known to be impossible (Groenewald - van Hove theorem), it is desirable to investigate kinds
of deformation quantization. In this case, those symplectic connections are important which are flat or partially flat. |