Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 048, 25 pages      arXiv:2410.03322      https://doi.org/10.3842/SIGMA.2025.048

Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions

Andrea Galasso
Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via R. Cozzi 55, 20125 Milano, Italy

Received January 29, 2025, in final form June 17, 2025; Published online June 25, 2025

Abstract
The asymptotic results for Berezin-Toeplitz operators yield a strict quantization for the algebra of smooth functions on a given Hodge manifold. It seems natural to generalize this picture for quantizable pseudo-Kähler manifolds in presence of a group action. Thus, in this setting we introduce a Berezin transform which has a complete asymptotic expansion on the preimage of the zero set of the moment map. It leads in a natural way to prove that certain quantization maps are strict.

Key words: CR manifolds; Toeplitz operators; star products; group actions.

pdf (540 kb)   tex (37 kb)  

References

  1. Bordemann M., Meinrenken E., Schlichenmaier M., Toeplitz quantization of Kähler manifolds and ${\rm gl}(N)$, $N\to\infty$ limits, Comm. Math. Phys. 165 (1994), 281-296, arXiv:hep-th/9309134.
  2. Boutet de Monvel L., Guillemin V., The spectral theory of Toeplitz operators, Ann. of Math. Stud., Vol. 99, Princeton University Press, Princeton, NJ, 1981.
  3. Burbea J., Masani P., Banach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy, Res. Notes Math., Vol. 90, Pitman, Boston, MA, 1984.
  4. Carmeli C., De Vito E., Toigo A., Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem, Anal. Appl. (Singap.) 4 (2006), 377-408.
  5. Folland G.B., Harmonic analysis in phase space, Ann. of Math. Stud., Vol. 122, Princeton University Press, Princeton, NJ, 1989.
  6. Folland G.B., Stein E.M., Estimates for the $\bar \partial_{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522.
  7. Galasso A., Hsiao C.-Y., Toeplitz operators on CR manifolds and group actions, J. Geom. Anal. 33 (2023), 21, 55 pages, arXiv:2108.11061.
  8. Galasso A., Hsiao C.-Y., Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds, Math. Z. 308 (2024), 5, 40 pages, arXiv:2112.11257.
  9. Guillemin V., Sternberg S., Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515-538.
  10. Hsiao C.-Y., Projections in several complex variables, Mém. Soc. Math. Fr. 123 (2010), viii+136 pages, arXiv:0810.4083.
  11. Hsiao C.-Y., Huang R.-T., $G$-invariant SzegHo kernel asymptotics and CR reduction, Calc. Var. Partial Differential Equations 60 (2021), 47, 48 pages, arXiv:1702.05012.
  12. Hsiao C.-Y., Ma X., Marinescu G., Geometric quantization on CR manifolds, Commun. Contemp. Math. 25 (2023), 2250074, 73 pages, arXiv:1906.05627.
  13. Hsiao C.-Y., Marinescu G., Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles, Comm. Anal. Geom. 22 (2014), 1-108, arXiv:1112.5464.
  14. Hsiao C.-Y., Marinescu G., Berezin-Toeplitz quantization for lower energy forms, Comm. Partial Differential Equations 42 (2017), 895-942, arXiv:1411.6654.
  15. Karabegov A.V., Schlichenmaier M., Identification of Berezin-Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49-76, arXiv:math.QA/0006063.
  16. Kobayashi S., Transformation groups in differential geometry, Class. Math., Springer, Berlin, 1995.
  17. Kostant B., Quantization and unitary representations. I. Prequantization, in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, 87-208.
  18. Landsman N.P., Mathematical topics between classical and quantum mechanics, Springer Monogr. Math., Springer, New York, 1998.
  19. Le Floch Y., A brief introduction to Berezin-Toeplitz operators on compact Kähler manifolds, CRM Short Courses, Springer, Cham, 2018.
  20. Lee J.M., Introduction to smooth manifolds, 2nd ed., Grad. Texts in Math., Vol. 218, Springer, New York, 2013.
  21. Ma X., Quantization commutes with reduction, a survey, Acta Math. Sci. Ser. B (Engl. Ed.) 41 (2021), 1859-1872.
  22. Ma X., Marinescu G., The first coefficients of the asymptotic expansion of the Bergman kernel of the ${\rm Spin}^c$ Dirac operator, Internat. J. Math. 17 (2006), 737-759, arXiv:math.CV/0511395.
  23. Ma X., Marinescu G., Holomorphic Morse inequalities and Bergman kernels, Progr. Math., Vol. 254, Birkhäuser, Basel, 2007.
  24. Ma X., Marinescu G., Toeplitz operators on symplectic manifolds, J. Geom. Anal. 18 (2008), 565-611, arXiv:0806.2370.
  25. McDuff D., Salamon D., Introduction to symplectic topology, 3rd ed., Oxf. Grad. Texts Math., Oxford University Press, Oxford, 2017.
  26. Meinrenken E., On Riemann-Roch formulas for multiplicities, J. Amer. Math. Soc. 9 (1996), 373-389, arXiv:alg-geom/9405014.
  27. Meinrenken E., Sjamaar R., Singular reduction and quantization, Topology 38 (1999), 699-762, arXiv:dg-ga/9707023.
  28. Pedrick G., Theory of reproducing kernels of Hilbert spaces of vector valued functions, Ph.D. Thesis, University of Kansas, Lawrence, Kansas, 1957.
  29. Pham H., The Lie group of isometries of a pseudo-Riemannian manifold, Ann. Math. Sci. Appl. 8 (2023), 223-238.
  30. Rawnsley J.H., Coherent states and Kähler manifolds, Quart. J. Math. Oxford Ser. (2) 28 (1977), 403-415.
  31. Rungi N., Pseudo-Kähler geometry of Hitchin representations and convex projective structures, Ph.D. Thesis, SISSA, 2023, available at https://iris.sissa.it/handle/20.500.11767/134510.
  32. Schlichenmaier M., Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization, in Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., Vol. 22, Kluwer, Dordrecht, 2000, 289-306, arXiv:math.QA/9910137.
  33. Schlichenmaier M., Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results, Adv. Math. Phys. 2010 (2010), 927280, 38 pages, arXiv:1003.2523.
  34. Shen W.C., Semi-classical spectral asymptotics of Toeplitz operators for lower energy forms on non-degenerate compact CR manifolds, Ph.D. Thesis, University of Cologne, 2024.
  35. Tian Y., Zhang W., An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math. 132 (1998), 229-259.
  36. Vergne M., Multiplicities formula for geometric quantization, Part I, Duke Math. J. 82 (1996), 143-179.
  37. Vergne M., Multiplicities formula for geometric quantization, Part II, Duke Math. J. 82 (1996), 181-194.
  38. Waldmann S., Convergence of star products: from examples to a general framework, EMS Surv. Math. Sci. 6 (2019), 1-31, arXiv:1901.11327.
  39. Zelditch S., Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I, in Spectral Geometry, Proc. Sympos. Pure Math., Vol. 84, American Mathematical Society, Providence, RI, 2012, 299-339, arXiv:1107.0463.

Previous article  Next article  Contents of Volume 21 (2025)