Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-11T00:46:30.944Z Has data issue: false hasContentIssue false
  • English
  • Français

BEREZIN–TOEPLITZ QUANTIZATION, HYPERKÄHLER MANIFOLDS, AND MULTISYMPLECTIC MANIFOLDS

Part of: Applications to physics - Differential geometry Symplectic geometry, contact geometry

Published online by Cambridge University Press:  10 June 2016

TATYANA BARRON  and
BARAN SERAJELAHI
TATYANA BARRON
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada e-mail: tatyana.barron@uwo.ca
BARAN SERAJELAHI
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada e-mail: bserajel@uwo.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We suggest a way to quantize, using Berezin–Toeplitz quantization, a compact hyperkähler manifold (equipped with a natural 3-plectic form), or a compact integral Kähler manifold of complex dimension n regarded as a (2n−1)-plectic manifold. We show that quantization has reasonable semiclassical properties.

MSC classification

Primary: 53D99: None of the above, but in this section
Secondary: 53Z05: Applications to physics
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 167 - 187
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Andersen, J., Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups, Ann. Math. (2) 163 (1) (2006), 347–368.Google Scholar
2. Andersen, J., Toeplitz operators and Hitchin's projectively flat connection, in The many facets of geometry (Oxford Univ. Press, Oxford, 2010), 177–209.Google Scholar
3. Artin, M., Algebra (Prentice Hall, Inc., Englewood Cliffs, NJ, 1991).Google Scholar
4. de Azcárraga, J. A., Izquierdo, J. M., n-ary algebras: A review with applications, J. Phys. A: Math. Theor. 43 (2010), 293001.Google Scholar
5. de Azcárraga, J. A., A. M. Perelomov and J. C. Pérez Bueno, New generalized Poisson structures, J. Phys. A 29 (7) (1996), L151–L157.Google Scholar
6. Baez, J., Hoffnung, A. and Rogers, C., Categorified symplectic geometry and the classical string, Comm. Math. Phys. 293 (3) (2010), 701–725.Google Scholar
7. Baez, J. and Rogers, C.. Categorified symplectic geometry and the string Lie 2-algebra, Homology, Homotopy Appl. 12 (1) (2010), 221–236.Google Scholar
8. Barron, T., Ma, X., Marinescu, G. and Pinsonnault, M., Semi-classical properties of Berezin-Toeplitz operators with Ck -symbol, J. Math. Phys. 55 (2014), 042108.Google Scholar
9. Bayen, F. and Flato, M., Remarks concerning Nambu's generalized mechanics, Phys. Rev. D (3) 11 (1975), 3049–3053.Google Scholar
10. Berezin, F., Quantization, Math. USSR-Izv. 8 (5) (1974), 1109–1165 (1975).Google Scholar
11. Besse, A., Einstein manifolds (Springer-Verlag, Berlin, 1987).Google Scholar
12. Bordemann, M., Meinrenken, E. and Schlichenmaier, M., Toeplitz quantization of Kähler manifolds and gl(N), N → ∞ limits, Comm. Math. Phys. 165 (2) (1994), 281–296.Google Scholar
13. Boutet de Monvel, L. and Guillemin, V., The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99 (Princeton University Press, Princeton, NJ, University of Tokyo Press, Tokyo, 1981).Google Scholar
14. Bremner, M., Identities for the ternary commutator, J. Algebra 206 (2) (1998), 615–623.Google Scholar
15. Bursztyn, H., Cabrera, A. and Iglesias, D., Multisymplectic geometry and Lie groupoids, in Geometry, mechanics and dynamics. Fields Inst. Commun., vol. 73 (Springer, New York, 2015), 57–73.Google Scholar
16. Cantrijn, F., Ibort, A. and de León, M., On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. Ser. A 66 (3) (1999), 303–330.Google Scholar
17. Chatterjee, R. and Takhtajan, L., Aspects of classical and quantum Nambu mechanics, Lett. Math. Phys. 37 (4) (1996), 475–482.Google Scholar
18. Curtright, T., Jin, X. and Mezincescu, L., Multi-operator brackets acting thrice, J. Phys. A 42 (46) (2009), 462001.Google Scholar
19. Curtright, T. and Zachos, C., Classical and quantum Nambu mechanics, Phys. Rev. D. 68 (2003), 085001, 1–29.Google Scholar
20. Curtright, T. and Zachos, C., Nambu dynamics, deformation quantization, and superintegrability, in Superintegrability in classical and quantum systems, CRM Proc. Lecture Notes, vol. 37 (Amer. Math. Soc., Providence, RI, 2004), 29–46.Google Scholar
21. DeBellis, J., Sämann, C. and Szabo, R., Quantized Nambu-Poisson manifolds and n-Lie algebras, J. Math. Phys. 51 (12) (2010), 122303.Google Scholar
22. Dito, G., Flato, M., Sternheimer, D. and Takhtajan, L., Deformation quantization and Nambu mechanics, Comm. Math. Phys. 183 (1) (1997), 1–22.Google Scholar
23. Filippov, V., n-Lie algebras, Siberian Math. J. 26 (6) (1985), 879–891.Google Scholar
24. Foth, T. and Uribe, A., The manifold of compatible almost complex structures and geometric quantization, Comm. Math. Phys. 274 (2) (2007), 357–379.Google Scholar
25. Gautheron, P., Some remarks concerning Nambu mechanics, Lett. Math. Phys. 37 (1) (1996), 103–116.Google Scholar
26. Hélein, F., Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory, in Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math., vol. 350 (Amer. Math. Soc., Providence, RI, 2004).Google Scholar
27. Huybrechts, D., Hyperkähler manifolds and sheaves, in Proceedings of the International Congress of Mathematicians, vol. II, (Hindustan Book Agency, New Delhi, 2010), 450–460.Google Scholar
28. Ibàñez, R., de León, M., Marrero, J. and de Diego, D. Martín, Dynamics of generalized Poisson and Nambu-Poisson brackets, J. Math. Phys. 38 (5) (1997), 2332–2344.Google Scholar
29. Verbitsky, M. and Kaledin, D., Hyperkahler manifolds, Mathematical Physics (Somerville), vol. 12 (International Press, Somerville, MA, 1999).Google Scholar
30. Karabegov, A. and Schlichenmaier, M., Identification of Berezin-Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49–76.Google Scholar
31. Ma, X. and Marinescu, G., Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254 (Birkhäuser Verlag, Basel, 2007).Google Scholar
32. Ma, X. and Marinescu, G., Toeplitz operators on symplectic manifolds, J. Geom. Anal. 18 (2) (2008), 565–611.Google Scholar
33. Ma, X. and Marinescu, G., Berezin-Toeplitz quantization and its kernel expansion, Geometry and quantization, Trav. Math., vol. 19 (Univ. Luxemb., Luxembourg, 2011), 125–166.Google Scholar
34. Madsen, T. and Swann, A., Multi-moment maps, Adv. Math. 229 (4) (2012), 2287–2309.Google Scholar
35. Martin, G., A Darboux theorem for multi-symplectic manifolds, Lett. Math. Phys. 16 (2) (1988), 133–138.Google Scholar
36. Mukunda, N. and Sudarshan, E., Relation between Nambu and Hamiltonian mechanics, Phys. Rev. D (3) 13 (10) (1976), 2846–2850.Google Scholar
37. Nambu, Y., Generalized Hamiltonian dynamics, Phys. Rev. D (3) 7 (1973), 2405–2412.Google Scholar
38. Polterovich, L., Quantum unsharpness and symplectic rigidity, Lett. Math. Phys. 102 (3) (2012), 245–264.Google Scholar
39. Rogers, C., m L ∞-algebras from multisymplectic geometry, Lett. Math. Phys. 100 (1) (2012), 29–50.Google Scholar
40. Rogers, C., 2-plectic geometry, Courant algebroids, and categorified prequantization, J. Symplectic Geom. 11 (1) (2013), 53–91.Google Scholar
41. Rubinstein, Y. and Zelditch, S., The Cauchy problem for the homogeneous Monge-Ampère equation, I. Toeplitz quantization, J. Differential Geom. 90 (2) (2012), 303–327.Google Scholar
42. Sämann, C. and Szabo, R., Groupoids, loop spaces and quantization of 2-plectic manifolds, Rev. Math. Phys. 25 (03) (2013), 1330005.Google Scholar
43. Schlichenmaier, M., Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization, Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., vol. 22 (Kluwer Acad. Publ., Dordrecht, 2000), 289–306.Google Scholar
44. Schlichenmaier, M., Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results, Adv. Math. Phys. (2010), Article ID 927280, 38 pages.Google Scholar
45. Takhtajan, L., On foundation of the generalized Nambu mechanics, Comm. Math. Phys. 160 (2) (1994), 295–315.Google Scholar
46. Vaisman, I., A survey on Nambu-Poisson brackets, Acta Math. Univ. Comenian. (N.S.) 68 (2) (1999), 213–241.Google Scholar