Abstract
We give a new construction of symbols of the differential operators on the sections of a quantum line bundle L over a Kähler manifold M using the natural contravariant connection on L. These symbols are the functions on the tangent bundle TM polynomial on fibres. For high tensor powers of L, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on TM corresponding to some pseudo-Kähler structure on TM. Surprisingly, this star product is intimately related to the formal symplectic groupoid with separation of variables over M. We extend the star product on TM to generalized functions supported on the zero section of TM. The resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the Bergman projection operator. Using this idempotent, we define an algebra of Toeplitz elements and show that it is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on M.
Similar content being viewed by others
References
Bayen F., Flato M., Fronsdal C., Lichnerowicz A. and Sternheimer D. (1978). Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Physics 111(1): 61–110
Berezin F.A. (1974). Quantization. Math. USSR-Izv. 8: 1109–1165
Berezin F.A. (1975). Quantization in complex symmetric spaces. Math. USSR-Izv. 9: 341–379
Bertelson M., Cahen M. and Gutt S. (1997). Equivalence of star products. Geometry and physics. Class. Quantum Grav. 14(1A): A93–A107
Bordemann M., Meinrenken E. and Schlichenmaier M. (1995). Toeplitz quantization of Kähler manifolds and gl(n), n → ∞ limits. Commun. Math. Phys. 165: 281–296
Bordemann M. and Waldmann S. (1997). A Fedosov star product of the Wick type for Kähler manifolds. Lett. Math. Phys. 41(3): 243–253
Cahen M., Gutt S. and Rawnsley J. (1990). Quantization of Kähler manifolds I: Geometric interpretation of Berezin’s quantization. JGP 7: 45–62
Cahen M., Gutt S. and Rawnsley J. (1993). Quantization of Kähler manifolds II. Trans. Amer. Math. Soc. 337: 73–98
Connes A., Flato M. and Sternheimer D. (1992). Closed star-products and cyclic cohomology. Lett. Math. Phys. 24: 1–12
De Wilde M. and Lecomte P.B.A. (1983). Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7(6): 487–496
Deligne P. (1995). Déformations de l’algébre des fonctions d’une variété symplectique: comparison entre Fedosov et De Wilde, Lecomte. Selecta Math. (N.S.) 1(4): 667–697
Fedosov B. (1994). A simple geometrical construction of deformation quantization. J. Diff. Geom. 40(2): 213–238
Fedosov, B.: Deformation quantization and index theory. Mathematical Topics, 9. Berlin: Akademie Verlag (1996)
Guillemin V. (1995). Star products on pre-quantizable symplectic manifolds. Lett. Math. Phys. 35: 85–89
Gutt S. and Rawnsley J. (2003). Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66: 123–139
Karabegov A. (1996). Deformation quantizations with separation of variables on a Kähler manifold. Commun. Math. Phys. 180: 745–755
Karabegov A (1998). On the canonical normalization of a trace density of deformation quantization. Lett. Math. Phys. 45: 217–228
Karabegov A. (1999). Pseudo-Kähler quantization on flag manifolds. Commun. Math. Phys. 200: 355–379
Karabegov A. (2003). On the dequantization of Fedosov’s deformation quantization. Lett. Math. Phys. 65: 133–146
Karabegov A. (2005). Formal symplectic groupoid of a deformation quantization. Commun. Math. Phys. 258: 223–256
Karabegov A. and Schlichenmaier M. (2001). Identification of Berezin-Toeplitz deformation quantization. J. reine angew. Math. 540: 49–76
Kontsevich M. (2003). Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66: 157–216
Moreno C. (1986). *-products on some Kähler manifolds. Lett. Math. Phys. 11: 361–372
Nest R. and Tsygan B. (1995). Algebraic index theorem. Commun. Math. Phys. 172(2): 223–262
Neumaier N. (2003). Universality of Fedosov’s construction for star products of Wick type on Pseudo-Kähler manifolds. Rep. Math. Phys. 52: 43–80
Omori H., Maeda Y. and Yoshioka A. (1991). Weyl manifolds and deformation quantization. Adv. Math. 85: 224–255
Reshetikhin, N., Takhtajan, L.: Deformation quantization of Kähler manifolds. L. D. Faddeev’s Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 201, Providence, RI: Amer. Math. Soc., 2000, pp. 257–276
Schlichenmaier, M.: Berezin-Toeplitz quantization of compact Kähler manifolds. In: Quantization, Coherent States and Poisson Structures, Proc. XIVth Workshop on Geometric Methods in Physics (Bialowieza, Poland, 9–15 July 1995), A. Strasburger, S. T. Ali, J.-P. Antoine, J.-P. Gazeau, A. Odzijewicz (eds.) Warsaw:Polish Scientific Publisher PWN, 1998, pp. 101–115
Xu P. (1998). Fedosov *-products and quantum momentum maps. Commun. Math. Phys. 197(1): 167–197
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Takhtajan.
Rights and permissions
About this article
Cite this article
Karabegov, A.V. A Formal Model of Berezin-Toeplitz Quantization. Commun. Math. Phys. 274, 659–689 (2007). https://doi.org/10.1007/s00220-007-0291-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-007-0291-6