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Morita equivalence bimodules for Wick type star products. (English) Zbl 1091.53063
Deformation quantization is a deformation algebraic structures from classical physics to quantum physics. On Kähler manifolds \(M\), one has a compatible complex structure which carries holomorphic and anti-holomorphic bi-differential operators. There are globally well-defined operators \[ P = g^{k\overline{l}}i_s(z_k) \otimes i_s(\overline{z}_l), \;\;\;\overline{P} = g^{k\overline{l}}i_s(\overline{z}_k)\otimes i_s(z_l), \] for the Kähler metric \(g=(g_{kl})\) and the fiber-wise products \[ a \circ_k b = \mu \circ e^{(k+1)\lambda P + (k-1)\lambda \overline{P}} a\otimes b \] on a complex bundle \(E \to M\) through the Fedosov construction. Note that \(k=0\) the product gives the Weyl product \(\star_{\text{Weyl}}\), \(k=1\) the Wick product \(\star_{\text{Wick}}\) and \(k=-1\) the anti-Wick product \(\star_{\text{Wick}^{-1}}\) on the \(C[[\lambda]]\)-module \(\prod_{s=0}^{\infty} \Gamma^{\infty}(V^sT^*M \otimes \wedge^{\bullet}T*M \otimes F)[[\lambda]]\) for \(F = E\) or \(\text{End}(E)\).
On Kähler manifolds, the three canonical star products are related by their characteristic classes \[ c(\star_{\text{Wick}^{\pm}}) = c(\star_{\text{Weyl}} \pm \pi i c_1(\wedge^{(n, 0)}T^*M). \] A deformation quantization of \(E\) is a deformation right module structure \(\circ_k\) for \(\Gamma^{\infty}(E)[[\lambda]]\) with respect to \(\star_k\), where \(\circ_k\) always exists uniquely up to equivalence, and a deformation left module structure \(\circ_k^{'}\) for \(\Gamma^{\infty}(\text{End}(E))[[\lambda]]\) with respect to \(\star_k^{'}\). The bi-module structure gives a Morita equivalence bi-module \((\Gamma^{\infty}(\text{End}(E))[[\lambda]], \circ_k^{'}, \circ_k)\) for the deformed algebras \((\Gamma^{\infty}(\text{End}(E))[[\lambda]] , \circ_k^{'})\) and \((\Gamma^{\infty}(E)[[\lambda]], \circ_k)\).
In a symplectic case with almost complex structure, two stars \(\star_k\) and \(\star_k^{'}\) are Morita equivalent if and only if there exists a symplectic diffeomorphism \(\psi \) of \(M\) such that \(\psi^* c(\star_k^{'}) - c(\star_k)\) is an integral de Rham class. In B. V. Fedosov [J. Differ. Geom. 40, 213–238 (1994; Zbl 0812.53034)] and S. Waldmann [Lett. Math. Phys. 60, 157–170 (2002; Zbl 1009.53063)], the concrete bi-module structure necessarily exists and depends on non-canonical choices.
The main result of the paper under review shows that using the Kähler geometry there is a canonical construction of a Morita equivalence structure for the canonical line bundle on the Kähler manifold in Theorem 4 and Corollary 3. The local expressions for the bi-module multiplications \(\star_{\text{Wick}^{\pm}} = \star_{\pm 1}\) are computed through the Fedosov construction (section 2) and the star products (section 3), hence explicit existence of bi-module structure. The 1-parameter family of star products links the Weyl product and (anti)-Wick products. The canonical Morita equivalence is contained in section 7.

MSC:
53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods
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References:
[1] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. phys., 111, 61-151, (1978) · Zbl 0377.53025
[2] Berezin, F.A., General concept of quantization, Commun. math. phys., 40, 153-174, (1975) · Zbl 1272.53082
[3] Berezin, F.A., Quantization, Math. USSR izvestija, 8, 5, 1109-1165, (1975) · Zbl 0312.53049
[4] Bertelson, M.; Cahen, M.; Gutt, S., Equivalence of star products, Class. quant. grav., 14, A93-A107, (1997) · Zbl 0881.58021
[5] Bordemann, M.; Neumaier, N.; Waldmann, S., Homogeneous Fedosov star products on cotangent bundles. I. Weyl and standard ordering with differential operator representation, Commun. math. phys., 198, 363-396, (1998) · Zbl 0968.53056
[6] Bordemann, M.; Waldmann, S., A Fedosov star product of Wick type for Kähler manifolds, Lett. math. phys., 41, 243-253, (1997) · Zbl 0892.53028
[7] H. Bursztyn, S. Waldmann, Bimodule deformations: Picard groups and contravariant connections, Preprint Freiburg FR-THEP 2002/10 (July 2002), math.QA/0207255. · Zbl 1054.53101
[8] Bursztyn, H.; Waldmann, S., Deformation quantization of Hermitian vector bundles, Lett. math. phys., 53, 349-365, (2000) · Zbl 0982.53073
[9] Bursztyn, H.; Waldmann, S., The characteristic classes of Morita equivalent star products on symplectic manifolds, Commun. math. phys., 228, 103-121, (2002) · Zbl 1036.53068
[10] Cahen, M.; Gutt, S.; Rawnsley, J., Quantization of Kähler manifolds. I. geometric interpretation of berezin’s quantization, J. geom. phys., 7, 45-62, (1990) · Zbl 0719.53044
[11] Cahen, M.; Gutt, S.; Rawnsley, J., Quantization of Kähler manifolds. II, Trans. am. math. soc., 337, 1, 73-98, (1993) · Zbl 0788.53062
[12] Cahen, M.; Gutt, S.; Rawnsley, J., Quantization of Kähler manifolds. III, Lett. math. phys., 30, 291-305, (1994) · Zbl 0826.53052
[13] Cahen, M.; Gutt, S.; Rawnsley, J., Quantization of Kähler manifolds. IV, Lett. math. phys., 34, 159-168, (1995) · Zbl 0831.58026
[14] DeWilde, M.; Lecomte, P.B.A., Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. math. phys., 7, 487-496, (1983) · Zbl 0526.58023
[15] G. Dito, D. Sternheimer, Deformation quantization: genesis, developments and metamorphoses, in: G. Halbout (Ed.), Deformation Quantization, vol. 1 in IRMA Lectures in Mathematics and Theoretical Physics, Walter de Gruyter, Berlin, 2002, pp. 9-54. · Zbl 1014.53054
[16] Fedosov, B.V., A simple geometrical construction of deformation quantization, J. diff. geom., 40, 213-238, (1994) · Zbl 0812.53034
[17] B.V. Fedosov, Deformation Quantization and Index Theory, Akademie Verlag, Berlin, 1996. · Zbl 0867.58061
[18] S. Gutt, Variations on deformation quantization, in: G. Dito, D. Sternheimer (Eds.), Conférence Moshé Flato 1999, Quantization, Deformations, and Symmetries, Mathematical Physics Studies no. 21, Kluwer Academic Publishers, Dordrecht, 2000, pp. 217-254. · Zbl 0997.53068
[19] Gutt, S.; Rawnsley, J., Equivalence of star products on a symplectic manifold: an introduction to deligne’s čech cohomology classes, J. geom. phys., 29, 347-392, (1999) · Zbl 1024.53057
[20] G. Halbout (Ed.), Deformation Quantization, vol. 1 in IRMA Lectures in Mathematics and Theoretical Physics, Walter de Gruyter, Berlin, 2002. · Zbl 1015.53060
[21] Hawkins, E., Geometric quantization of vector bundles and the correspondence with deformation quantization, Commun. math. phys., 215, 409-432, (2000) · Zbl 1031.53119
[22] Karabegov, A.V., Deformation quantization with separation of variables on a Kähler manifold, Commun. math. phys., 180, 745-755, (1996) · Zbl 0866.58037
[23] Karabegov, A.V., Cohomological classification of deformation quantizations with separation of variables, Lett. math. phys., 43, 347-357, (1998) · Zbl 0938.53049
[24] A.V. Karabegov, On Fedosov’s approach to deformation quantization with separation of variables, in: G. Dito, D. Sternheimer (Eds.), Conférence Moshé Flato 1999, Quantization, Deformations, and Symmetries, Mathematical Physics Studies no. 22, Kluwer Academic Publishers, Dordrecht, 2000. · Zbl 0988.53042
[25] Karabegov, A.V.; Schlichenmaier, M., Almost-Kähler deformation quantization, Lett. math. phys., 57, 135-148, (2001) · Zbl 1044.53061
[26] Karabegov, A.V.; Schlichenmaier, M., Identification of berezin – toeplitz deformation quantization, J. reine angew. math., 540, 49-76, (2001) · Zbl 0997.53067
[27] S. Kobayashi, K. Nomizu, Foundations of differential geometry. II, Interscience Tracts in Pure and Applied Mathematics no. 15, Wiley, New York, 1969. · Zbl 0175.48504
[28] M. Kontsevich, Deformation Quantization of Poisson Manifolds, I (September 1997). q-alg/9709040. · Zbl 1058.53065
[29] N.P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics, Springer, Berlin, 1998. · Zbl 0923.00008
[30] Nest, R.; Tsygan, B., Algebraic index theorem, Commun. math. phys., 172, 223-262, (1995) · Zbl 0887.58050
[31] N. Neumaier, Local ν-Euler Derivations and Deligne’s Characteristic Class of Fedosov star products and star products of special type, Commun. Math. Phys. 230 (2002) 271-288. · Zbl 1035.53124
[32] N. Neumaier, Universality of Fedosov’s construction for star products of Wick type on semi-Kähler manifolds, Preprint Freiburg FR-THEP-2002/07 (April 2002). math.QA/0204031.
[33] Omori, H.; Maeda, Y.; Yoshioka, A., Weyl manifolds and deformation quantization, Adv. math., 85, 224-255, (1991) · Zbl 0734.58011
[34] M. Schlichenmaier, Deformation quantization of compact Kähler manifolds via Berezin-Toeplitz operators, in: H.-D. Doebner, P. Nattermann, W. Scherer (Eds.), Proceedings of the XXI International Colloquium on Group Theoretical Methods in Physics, Group 21, Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras, World Scientific, Singapore, 1997, pp. 396-400.
[35] Waldmann, S., Morita equivalence of Fedosov star products and deformed Hermitian vector bundles, Lett. math. phys., 60, 157-170, (2002) · Zbl 1009.53063
[36] S. Waldmann, On the representation theory of deformation quantization, in: G. Halbout (Ed.), Deformation Quantization, vol. 1 in IRMA Lectures in Mathematics and Theoretical Physics, Walter de Gruyter, Berlin, 2002, pp. 107-133. · Zbl 1014.53055
[37] A. Weinstein, P. Xu, Hochschild cohomology and characteristic classes for star-products, in: A. Khovanskij, A. Varchenko, V. Vassiliev (Eds.), Geometry of Differential Equations, American Mathematical Society, Providence, RI, 1998, pp. 177-194 (dedicated to V.I. Arnold on the occasion of his 60th birthday). · Zbl 0956.53055
[38] R.O. Wells, Differential analysis on complex manifolds, Graduate Texts in Mathematics, vol. 65, Springer, New York, 1980. · Zbl 0435.32004
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