Twist star products and Morita equivalence

D'Andrea, Francesco; Weber, Thomas

Comptes Rendus. Mathématique, Tome 355 (2017) no. 11, pp. 1178-1184.

Nous exposons un théorème de non-existence concernant la quantification par déformation d'un espace homogène M, induite par un twist de Drinfel'd : nous montrons qu'un fibré en droites équivariant sur M avec une classe de Chern non triviale et un produit étoile symplectique ne peuvent coexister sur une même variété M. Ceci implique, par exemple, qu'il n'y a pas de produit étoile symplectique sur l'espace projectif complexe induit par un twist basé sur U(gln(C))〚h〛, ou sur toute sous-algébre, pour tout n≥2.

We present a simple no-go theorem for the existence of a deformation quantization of a homogeneous space M induced by a Drinfel'd twist: we argue that equivariant line bundles on M with non-trivial Chern class and symplectic twist star products cannot both exist on the same manifold M. This implies, for example, that there is no symplectic star product on the projective space CPn−1 induced by a twist based on U(gln(C))〚h〛 or any sub-bialgebra, for every n≥2.

[1] Aschieri, P.; Dimitrijevic, M.; Meyer, F.; Wess, J. Noncommutative geometry and gravity, Class. Quantum Gravity, Volume 23 (2006), pp. 1883-1912 | arXiv

[2] Aschieri, P.; Lizzi, F.; Vitale, P. Twisting all the way: from classical mechanics to quantum fields, Phys. Rev. D, Volume 77, 2008 | arXiv

[3] Aschieri, P. Star product geometries, Russ. J. Math. Phys., Volume 16 (2009), pp. 371-383 | arXiv

[4] Aschieri, P. Twisting all the way: from algebras to morphisms and connections, Int. J. Mod. Phys. Conf. Ser., Volume 13, 2012, pp. 1-19 | arXiv

[5] Bieliavsky, P.; Esposito, C.; Waldmann, S.; Weber, T. Obstructions for twist star products | arXiv

[6] Bieliavsky, P.; Tang, X.; Yao, Y. Rankin–Cohen brackets and formal quantization, Adv. Math., Volume 212 (2007), pp. 293-314 | arXiv

[7] Bordemann, M.; Meinrenken, E.; Schlichenmaier, M. Toeplitz quantization of Kähler manifolds and gl(N), N→∞ limits, Commun. Math. Phys., Volume 165 (1994), pp. 281-296 | arXiv

[8] Bordemann, M.; Neumaier, N.; Waldmann, S.; Weiss, S. Deformation quantization of surjective submersions and principal fibre bundles, J. Reine Angew. Math., Volume 639 (2010), pp. 1-38 | arXiv

[9] Bursztyn, H. Semiclassical geometry of quantum line bundles and Morita equivalence of star products, Int. Math. Res. Not., Volume 16 (2002), pp. 821-846 | arXiv

[10] Bursztyn, H.; Waldmann, S. Deformation quantization of Hermitian vector bundles, Lett. Math. Phys., Volume 53 (2000), pp. 349-365 | arXiv

[11] Bursztyn, H.; Waldmann, S. The characteristic classes of Morita equivalent star products on symplectic manifolds, Commun. Math. Phys., Volume 228 (2002), pp. 103-121 | arXiv

[12] Chari, V.; Pressley, A.N. A Guide to Quantum Groups, Cambridge University Press, 1994

[13] D'Andrea, F. Topics in noncommutative geometry, Würzburg, Germany (2015) | arXiv

[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., Volume 7 (1983), pp. 487-496

[15] Drinfeld, V.G. Quasi-Hopf algebras, Leningr. Math. J., Volume 1 (1990), pp. 1419-1457

[16] Esposito, C.; Schnitzer, J.; Waldmann, S. An universal construction of universal deformation formulas, Drinfel'd twists and their positivity | arXiv

[17] Etingof, P.I.; Schiffmann, O. Lectures on Quantum Groups, International Press, 2001

[18] Fedosov, B.V. A simple geometrical construction of deformation quantization, J. Differ. Geom., Volume 40 (1994), pp. 213-238

[19] Fiore, G. On second quantization on noncommutative spaces with twisted symmetries, J. Phys. A, Volume 43 (2010) | arXiv

[20] Giaquinto, A.; Zhang, J.J. Bialgebra actions, twists, and universal deformation formulas, J. Pure Appl. Algebra, Volume 128 (1998), pp. 133-151 | arXiv

[21] Gutt, S.; Rawnsley, J. Equivalence of star products on a symplectic manifold: an introduction to Deligne's Čech cohomology classes, J. Geom. Phys., Volume 29 (1999), pp. 347-392

[22] Kontsevich, M. Deformation quantization of Poisson manifolds, I, Lett. Math. Phys., Volume 66 (2003), pp. 157-216 | arXiv

[23] Lam, T.Y. Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, Springer, 1999

[24] Majid, S. Foundations of Quantum Group Theory, Cambridge University Press, 1995

[25] Omori, H.; Maeda, Y.; Yoshioka, A. Weyl manifolds and deformation quantization, Adv. Math., Volume 85 (1991), pp. 224-255

[26] Rieffel, M.A. Matrix algebras converge to the sphere for quantum Gromov–Hausdorff distance, Mem. Amer. Math. Soc., Volume 168 (2004), pp. 67-91 | arXiv

[27] Rieffel, M.A. Dirac operators for coadjoint orbits of compact Lie groups, Münster J. Math., Volume 2 (2009), pp. 265-298 | arXiv

[28] Schlichenmaier, M. Berezin–Toeplitz quantization for compact Kähler manifolds. A review of results, Adv. Math. Phys., Volume 2010 (2010) | arXiv

[29] Schlichenmaier, M. Deformation quantization of compact Kähler manifolds by Berezin–Toeplitz quantization (Dito, G.; Sternheimer, D., eds.), Proc. Conference Moshe Flato 1999, Kluwer, 2000, pp. 289-306 | arXiv

[30] Schlichenmaier, M. Zwei Anwendungen algebraisch-geometrischer Methoden in der theoretischen Physik: Berezin–Toeplitz-Quantisierung und globale Algebren der zweidimensionalen konformen Feldtheorie, University of Mannheim, Germany, 1996 (Habilitation Thesis)

[31] Waldmann, S. Recent developments in deformation quantization, Quantum Mathematical Physics: A Bridge Between Mathematics and Physics, Springer, 2016, pp. 421-439 | arXiv

[32] Xu, P. Quantum groupoids, Commun. Math. Phys., Volume 216 (2001), pp. 539-581 | arXiv