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Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results. (English) Zbl 1207.81049
Summary: This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deformation quantization for compact quantizable Kähler manifolds. The basic objects, concepts, and results are given. This concerns the correct semiclassical limit behaviour of the operator quantization, the unique Berezin-Toeplitz deformation quantization (star product), covariant and contravariant Berezin symbols, and Berezin transform. Other related objects and constructions are also discussed.

MSC:
81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
32Q15 Kähler manifolds
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
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References:
[1] M. Bordemann, E. Meinrenken, and M. Schlichenmaier, “Toeplitz quantization of Kähler manifolds and gl(n), n\rightarrow \infty limits,” Communications in Mathematical Physics, vol. 165, no. 2, pp. 281-296, 1994. · Zbl 0813.58026
[2] J. Madore, “The fuzzy sphere,” Classical and Quantum Gravity, vol. 9, no. 1, pp. 69-87, 1992. · Zbl 0742.53039
[3] A. P. Balachandran, B. P. Dolan, J. Lee, X. M., and D. O/Connor, “Fuzzy complex projective spaces and their star-products,” Journal of Geometry and Physics, vol. 43, no. 2-3, pp. 184-204, 2002. · Zbl 1007.51007
[4] A. P. Balachandran, S. Kurkcuoglu, and S. Vaidya, Lectures on Fuzzy and Fuzzy SUSY Physics, World Scientific, Singapore, 2007. · Zbl 1132.81001
[5] C. I. Lazaroiu, D. McNamee, and Sämann, “Generalized Berezin quantization, Bergman metric and fuzzy laplacians,” Journal of High Energy Physics. In press. · Zbl 1245.53065
[6] B. P. Dolan, I. Huet, S. Murray, and D. O/Connor, “Noncommutative vector bundles over fuzzy \Bbb C\Bbb PN and their covariant derivatives,” Journal of High Energy Physics, vol. 7, article 007, 2007.
[7] P. Aschieri, J. Madore, P. Manousselis, and G. Zoupanos, “Dimensional reduction over fuzzy coset spaces,” Journal of High Energy Physics, vol. 2004, no. 4, article 034, 2004. · Zbl 1052.81112
[8] H. Grosse, M. Maceda, J. Madore, and H. Steinacker, “Fuzzy instantons,” International Journal of Modern Physics A, vol. 17, no. 15, pp. 2095-2111, 2002. · Zbl 1014.81054
[9] H. Grosse and H. Steinacker, “Finite gauge theory on fuzzy \Bbb C\Bbb P2,” Nuclear Physics B, vol. 704, p. 145, 2005. · Zbl 1160.81410
[10] U. Carow-Watamura, H. Steinacker, and S. Watamura, “Monopole bundles over fuzzy complex projective spaces,” Journal of Geometry and Physics, vol. 54, no. 4, pp. 373-399, 2005. · Zbl 1159.58301
[11] J. E. Andersen, “Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups,” Annals of Mathematics, vol. 163, no. 1, pp. 347-368, 2006. · Zbl 1157.53049
[12] J. E. Andersen, “Deformation quantization and geometric quantization of abelian moduli spaces,” Communications in Mathematical Physics, vol. 255, no. 3, pp. 727-745, 2005. · Zbl 1079.53136
[13] M. Schlichenmaier, “Berezin-Toeplitz quantization of the moduli space of flat SU(N) connections,” Journal of Geometry and Symmetry in Physics, vol. 9, pp. 33-44, 2007. · Zbl 1151.81026
[14] J. E. Andersen, “Mapping class groups do not have property (T),” http://arxiv.org/abs/0706.2184.
[15] J. E. Andersen and N. L. Gammelgaard, “Hitchin/s projectively flat connection, Toeplitz operators and the asymptotic expansion of TQFT curve operators,” http://arxiv.org/abs/0903.4091. · Zbl 1232.53078
[16] J. E. Andersen and K. Ueno, “Abelian conformal field theory and determinant bundles,” International Journal of Mathematics, vol. 18, no. 8, pp. 919-993, 2007. · Zbl 1128.81026
[17] J. E. Andersen and K. Ueno, “Geometric construction of modular functors from conformal field theory,” Journal of Knot Theory and Its Ramifications, vol. 16, no. 2, pp. 127-202, 2007. · Zbl 1123.81041
[18] X. Ma and G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels, vol. 254 of Progress in Mathematics, Birkhäuser, Basel, Switzerland, 2007. · Zbl 1135.32001
[19] X. Ma and G. Marinescu, “Toeplitz operators on symplectic manifolds,” Journal of Geometric Analysis, vol. 18, no. 2, pp. 565-611, 2008. · Zbl 1152.81030
[20] L. Charles, “Toeplitz operators and Hamiltonian torus actions,” Journal of Functional Analysis, vol. 236, no. 1, pp. 299-350, 2006. · Zbl 1099.53059
[21] S. T. Ali and H.-D. Doebner, “Ordering problem in quantum mechanics: prime quantization and a physical interpretation,” Physical Review A, vol. 41, no. 3, pp. 1199-1210, 1990.
[22] S. T. Ali and M. Engli\vs, “Quantization methods: a guide for physicists and analysts,” Reviews in Mathematical Physics, vol. 17, no. 4, pp. 391-490, 2005. · Zbl 1075.81038
[23] S. T. Ali, “Quantization techniques: a quick overview,” in Contemporary Problems in Mathematical Physics, pp. 3-78, World Scientific, River Edge, NJ, USA, 2002. · Zbl 1044.81068
[24] D. Sternheimer, “Deformation quantization: twenty years after,” in Particles, Fields, and Gravitation, vol. 453 of AIP Conference Proceedings, pp. 107-145, The American Institute of Physics, Woodbury, NY, USA, 1998. · Zbl 0977.53082
[25] G. Dito and D. Sternheimer, “Deformation quantization: genesis, developments and metamorphoses,” in Deformation Quantization, vol. 1 of IRMA Lectures in Mathematics and Theoretical Physics, pp. 9-54, Walter de Gruyter, Berlin, Germany, 2002. · Zbl 1014.53054
[26] M. Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces, Theoretical and Mathematical Physics, Springer, Berlin, Germany, 2nd edition, 2007. · Zbl 1151.81026
[27] R. Abraham and J. E. Marsden, Foundations of Mechanics, Addison Wesley, Redwood City, Calif, USA, 1985.
[28] L. Boutet de Monvel and V. Guillemin, The Spectral Theory of Toeplitz Operators, Princeton University Press, Princeton, NJ, USA, 1981. · Zbl 0469.47021
[29] A. V. Karabegov and M. Schlichenmaier, “Identification of Berezin-Toeplitz deformation quantization,” Journal für die Reine und Angewandte Mathematik, vol. 540, pp. 49-76, 2001. · Zbl 0997.53067
[30] M. Schlichenmaier, “Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization,” in Proceedings of the Conférence Moshé Flato, G. Dito and D. Sternheimer, Eds., vol. 22 of Mathematical Physics Studies, pp. 289-306, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 1028.53085
[31] M. Rieffel, “Questions on quantization,” in Operator Algebras and Operator Theory, vol. 228 of Contemporary Mathematics, pp. 315-326, American Mathematical Society, Providence, RI, USA, 1998. · Zbl 1083.53506
[32] N. P. Landsman, Mathematical Topics between Classical and Quantum Mechanics, Springer Monographs in Mathematics, Springer, New York, NY, USA, 1998. · Zbl 0923.00008
[33] G. M. Tuynman, “Generalized Bergman kernels and geometric quantization,” Journal of Mathematical Physics, vol. 28, no. 3, pp. 573-583, 1987. · Zbl 0616.58041
[34] M. Bordemann, J. Hoppe, P. Schaller, and M. Schlichenmaier, “gl(\infty ) and geometric quantization,” Communications in Mathematical Physics, vol. 138, no. 2, pp. 209-244, 1991. · Zbl 0735.58020
[35] M. Schlichenmaier, Zwei Anwendungen algebraisch-geometrischer Methoden in der theoretischen Physik: Berezin-Toeplitz-Quantisierung und globale Algebren der zweidimensionalen konformen Feldtheorie, Habiliationsschrift Universität Mannheim, 1996.
[36] F. A. Berezin, “Quantization in complex symmetric spaces,” Izvestiya Akademii Nauk SSSR, vol. 9, no. 2, pp. 341-379, 1975. · Zbl 0324.53049
[37] H. Upmeier, “Toeplitz C\ast -algebras on bounded symmetric domains,” Annals of Mathematics, vol. 119, no. 3, pp. 549-576, 1984. · Zbl 0549.46031
[38] H. Upmeier, “Toeplitz operators on symmetric Siegel domains,” Mathematische Annalen, vol. 271, no. 3, pp. 401-414, 1985. · Zbl 0565.47016
[39] H. Upmeier, “Fredholm indices for Toeplitz operators on bounded symmetric domains,” American Journal of Mathematics, vol. 110, no. 5, pp. 811-832, 1988. · Zbl 0758.47030
[40] H. Upmeier, “Toeplitz C\ast -algebras and noncommutative duality,” Journal of Operator Theory, vol. 26, no. 2, pp. 407-432, 1991. · Zbl 0801.46091
[41] H. Upmeier, Toeplitz Operators and Index Theory in Several Complex Variables, vol. 81 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1996. · Zbl 0957.47023
[42] C. A. Berger and L. A. Coburn, “Toeplitz operators and quantum mechanics,” Journal of Functional Analysis, vol. 68, no. 3, pp. 273-299, 1986. · Zbl 0629.47022
[43] L. A. Coburn, “Deformation estimates for the Berezin-Toeplitz quantization,” Communications in Mathematical Physics, vol. 149, no. 2, pp. 415-424, 1992. · Zbl 0829.46056
[44] S. Klimek and A. Lesniewski, “Quantum Riemann surfaces. I. The unit disc,” Communications in Mathematical Physics, vol. 146, no. 1, pp. 103-122, 1992. · Zbl 0771.46036
[45] S. Klimek and A. Lesniewski, “Quantum Riemann surfaces. II. The discrete series,” Communications in Mathematical Physics, vol. 24, pp. 125-139, 1992. · Zbl 0777.46039
[46] M. Engli\vs, “Weighted Bergman kernels and quantization,” Communications in Mathematical Physics, vol. 227, no. 2, pp. 211-241, 2002. · Zbl 1010.32002
[47] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, “Quantum mechanics as a deformation of classical mechanics-part I,” Letters in Mathematical Physics, vol. 1, no. 6, pp. 521-530, 1977.
[48] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, “Deformation theory and quantization-parts II and III,” Annals of Physics, vol. 111, pp. 61-110, 111-115, 1978. · Zbl 0377.53024
[49] F. A. Berezin, “Quantization,” Mathematics of the USSR-Izvestiya, vol. 8, pp. 1109-1165, 1974. · Zbl 0312.53049
[50] J. E. Moyal, “Quantum mechanics as a statistical theory,” vol. 45, pp. 99-124, 1949. · Zbl 0031.33601
[51] H. Weyl, Gruppentheorie und Quantenmechanik, Wissenschaftliche Buchgesellschaft, Darmstadt, Germany, 2nd edition, 1977. · JFM 57.1579.01
[52] Marc De Wilde and P. B. A. Lecomte, “Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds,” Letters in Mathematical Physics, vol. 7, no. 6, pp. 487-496, 1983. · Zbl 0526.58023
[53] H. Omori, Y. Maeda, and A. Yoshioka, “Weyl manifolds and deformation quantization,” Advances in Mathematics, vol. 85, no. 2, pp. 224-255, 1991. · Zbl 0734.58011
[54] H. Omori, Y. Maeda, and A. Yoshioka, “Existence of closed star-products,” Letters in Mathematical Physics, vol. 26, pp. 284-294, 1992. · Zbl 0771.58017
[55] B. V. Fedosov, “Deformation quantization and asymptotic operator representation,” Funktional Anal i Prilozhen, vol. 25, no. 2, pp. 184-194, 1990. · Zbl 0737.47042
[56] B. V. Fedosov, “A simple geometrical construction of deformation quantization,” Journal of Differential Geometry, vol. 40, no. 2, pp. 213-238, 1994. · Zbl 0812.53034
[57] M. Kontsevich, “Deformation quantization of Poisson manifolds,” Letters in Mathematical Physics, vol. 66, no. 3, pp. 157-216, 2003. · Zbl 1058.53065
[58] P. Deligne, “Déformations de l/algèbre des fonctions d/une variètè symplectique: comparaison entre Fedosov et De Wilde, Lecomte,” Selecta Mathematica. New Series, vol. 1, no. 4, pp. 667-697, 1995. · Zbl 0852.58033
[59] S. Gutt and J. Rawnsley, “Equivalence of star products on a symplectic manifold; an introduction to Deligne/s \vCech cohomology classes,” Journal of Geometry and Physics, vol. 29, no. 4, pp. 347-392, 1999. · Zbl 1024.53057
[60] M. Bertelson, M. Cahen, and S. Gutt, “Equivalence of star products,” Classical and Quantum Gravity, vol. 14, no. 1, pp. A93-A107, 1997. · Zbl 0881.58021
[61] R. Nest and B. Tsygan, “Algebraic index theorem,” Communications in Mathematical Physics, vol. 172, no. 2, pp. 223-262, 1995. · Zbl 0887.58050
[62] R. Nest and B. Tsygan, “Algebraic index theory for families,” Advances in Mathematics, vol. 113, pp. 151-205, 1995. · Zbl 0837.58029
[63] A. V. Karabegov, “Deformation quantizations with separation of variables on a Kähler manifold,” Communications in Mathematical Physics, vol. 180, no. 3, pp. 745-755, 1996. · Zbl 0866.58037
[64] M. Bordemann and St. Waldmann, “A Fedosov star product of the Wick type for Kähler manifolds,” Letters in Mathematical Physics, vol. 41, no. 3, pp. 243-253, 1997. · Zbl 0892.53028
[65] N. Reshetikhin and L. A. Takhtajan, “Deformation quantization of Kähler manifolds,” in L. D. Faddeev/s Seminar on Mathematical Physics, vol. 201 of American Mathematical Society Translations: Series 2, pp. 257-276, American Mathematical Society, Providence, RI, USA, 2000. · Zbl 1003.53061
[66] M. Schlichenmaier, “Berezin-Toeplitz quantization of compact Kähler manifolds,” in Quantization, Coherent States and Poisson Structures, A. Strasburger, S. T. Ali, J.-P. Antoine, J.-P. Gazeau, and A. Odzijewicz, Eds., pp. 101-115, Polish Scientific Publisher PWN, 1998, Proceedings of the 14th Workshop on Geometric Methods in Physics (Białowie\Dza, Poland, July 1995).
[67] M. Schlichenmaier, “Deformation quantization of compact Kähler manifolds via Berezin-Toeplitz operators,” in Proceedings of the 21st International Colloquium on Group Theoretical Methods in Physics, H.-D. Doebner, P. Nattermann, and W. Scherer, Eds., pp. 369-400, World Scientific, Goslar, Germany, 1996.
[68] E. Hawkins, “The correspondence between geometric quantization and formal deformation quantization,” http://arxiv.org/abs/physics/9811049.
[69] D. Borthwick, T. Paul, and A. Uribe, “Semiclassical spectral estimates for Toeplitz operators,” Annales de l’Institut Fourier, vol. 48, no. 4, pp. 1189-1229, 1998. · Zbl 0920.58059
[70] A. V. Karabegov, “Cohomological classification of deformation quantizations with separation of variables,” Letters in Mathematical Physics, vol. 43, no. 4, pp. 347-357, 1998. · Zbl 0938.53049
[71] A. V. Karabegov, “On the canonical normalization of a trace density of deformation quantization,” Letters in Mathematical Physics, vol. 45, no. 3, pp. 217-228, 1998. · Zbl 0943.53052
[72] V. Guillemin, “Some classical theorems in spectral theory revisited,” in Seminar on Singularities of Solutions of Linear Partial Differential Equations, L. Hörmander, Ed., vol. 91 of Annals of Mathematics Studies, pp. 219-259, Princeton University Press, Princeton, NJ, USA, 1979. · Zbl 0452.35093
[73] J. H. Rawnsley, “Coherent states and Kähler manifolds,” The Quarterly Journal of Mathematics. Oxford. Second Series, vol. 28, no. 112, pp. 403-415, 1977. · Zbl 0387.58002
[74] J. Rawnsley, M. Cahen, and S. Gutt, “Quantization of Kähler manifolds. I. Geometric interpretation of Berezin/s quantization,” Journal of Geometry and Physics, vol. 7, no. 1, pp. 45-62, 1990. · Zbl 0719.53044
[75] M. Cahen, S. Gutt, and J. Rawnsley, “Quantization of Kähler manifolds. II,” Transactions of the American Mathematical Society, vol. 337, no. 1, pp. 73-98, 1993. · Zbl 0788.53062
[76] M. Cahen, S. Gutt, and J. Rawnsley, “Quantization of Kähler manifolds. III,” Letters in Mathematical Physics, vol. 30, no. 4, pp. 291-305, 1994. · Zbl 0826.53052
[77] M. Cahen, S. Gutt, and J. Rawnsley, “Quantization of Kähler manifolds. IV,” Letters in Mathematical Physics, vol. 34, no. 2, pp. 159-168, 1995. · Zbl 0831.58026
[78] S. Berceanu and M. Schlichenmaier, “Coherent state embeddings, polar divisors and Cauchy formulas,” Journal of Geometry and Physics, vol. 34, no. 3-4, pp. 336-358, 2000. · Zbl 1002.81027
[79] C. Moreno and P. Ortega-Navarro, “\ast -products on D1(\Bbb C), S2 and related spectral analysis,” Letters in Mathematical Physics, vol. 7, no. 3, pp. 181-193, 1983. · Zbl 0528.58014
[80] C. Moreno, “\ast -products on some Kähler manifolds,” Letters in Mathematical Physics, vol. 11, no. 4, pp. 361-372, 1986. · Zbl 0618.53049
[81] M. Engli\vs, “Berezin quantization and reproducing kernels on complex domains,” Transactions of the American Mathematical Society, vol. 348, no. 2, pp. 411-479, 1996. · Zbl 0842.46053
[82] M. Engli\vs, “Asymptotics of the Berezin transform and quantization on planar domains,” Duke Mathematical Journal, vol. 79, no. 1, pp. 57-76, 1995. · Zbl 0848.30028
[83] M. Engli\vs, “The asymptotics of a Laplace integral on a Kähler manifold,” Journal für die Reine und Angewandte Mathematik, vol. 528, pp. 1-39, 2000. · Zbl 0965.32012
[84] A. Unterberger and H. Upmeier, “The Berezin transform and invariant differential operators,” Communications in Mathematical Physics, vol. 164, no. 3, pp. 563-597, 1994. · Zbl 0843.32019
[85] M. Engli\vs and J. Peetre, “On the correspondence principle for the quantized annulus,” Mathematica Scandinavica, vol. 78, no. 2, pp. 183-206, 1996. · Zbl 0865.30005
[86] M. Schlichenmaier, “Berezin-Toeplitz quantization and Berezin transform,” in Long Time Behaviour of Classical and Quantum Systems, vol. 1, pp. 271-287, World Scientific, River Edge, NJ, USA, 2001. · Zbl 0983.81035
[87] M. Schlichenmaier, “Berezin-Toeplitz quantization and Berezin symbols for arbitrary compact Kähler manifolds,” in Proceedings of the 17th Workshop on Geometric Methods in Physics, M. Schlichenmaier, et al., Ed., pp. 45-56, Warsaw University Press, Białowie\Dza, Poland, 1998.
[88] L. Boutet de Monvel and J. Sjöstrand, “Sur la singularité des noyaux de Bergman et de Szegö,” in Journées: Équations aux Dérivées Partielles de Rennes, Asterisque, pp. 123-164, Société Mathématique de France, Paris, France, 1976. · Zbl 0344.32010
[89] S. Zelditch, “Szegö kernels and a theorem of Tian,” International Mathematics Research Notices, no. 6, pp. 317-331, 1998. · Zbl 0922.58082
[90] G. Tian, “On a set of polarized Kähler metrics on algebraic manifolds,” Journal of Differential Geometry, vol. 32, no. 1, pp. 99-130, 1990. · Zbl 0706.53036
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