Abstract
In this chapter we consider Toeplitz operators in two cases: on a hyperkähler manifold and on a Kähler manifold regarded as a (2n − 1)-plectic manifold. We give a treatment of the general procedure of assigning an operator to a function and discuss a higher order version of the correspondence between a bracket of functions and the commutator of operators. We do explicit computations on the 4-torus and provide examples of the asymptotics that appear in the theorems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Artin. Algebra. Prentice Hall, 1991.
A. Avez. Représentation de l’algèbre de Lie des symplectomorphismes par des opérateurs bornés. C. R. Acad. Sci. Paris Sér. A 279 (1974), 785–787.
W. Baily, Jr. Classical theory of Θ-functions. In Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) pp. 306–311, Amer. Math. Soc., Providence, R.I., 1966.
T. Barron, X. Ma, G. Marinescu, M. Pinsonnault. Semi-classical properties of Berezin-Toeplitz operators with Ck-symbol. J. Math. Phys. 55, issue 4, 042108 (2014). 25 pages.
T. Barron, B. Serajelahi. Berezin-Toeplitz quantization, hyperkähler manifolds, and multisymplectic manifolds. Glasgow Math. J. 59 (2017), issue 1, 167–187.
A. Besse. Einstein manifolds. Springer-Verlag, Berlin, 1987.
M. Bordemann, E. Meinrenken, M. Schlichenmaier. Toeplitz quantization of Kähler manifolds and gl(N), N →∞ limits. Comm. Math. Phys. 165 (1994), no. 2, 281–296.
F. Cantrijn, A. Ibort, M. de León. On the geometry of multisymplectic manifolds. J. Austral. Math. Soc. Ser. A 66 (1999), no. 3, 303–330.
H. Castejón Díaz. Berezin-Toeplitz quantization on K3 surfaces and hyperkähler Berezin-Toeplitz quantization. Ph.D. thesis, Université du Luxembourg, 2016.
J.A. de Azcárraga, A. M. Perelomov, J. C. Pérez Bueno. The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures. J. Phys. A 29 (1996), no. 24, 7993–8009.
P. Gautheron. Some remarks concerning Nambu mechanics. Lett. Math. Phys. 37 (1996), no. 1, 103–116.
V. Ginzburg, R. Montgomery. Geometric quantization and no-go theorems. In Poisson geometry (Warsaw, 1998), 69–77, Banach Center Publ., 51, Polish Acad. Sci. Inst. Math., Warsaw, 2000.
R. Ibàñez, M. de León, J. Marrero, D. Martín de Diego. Dynamics of generalized Poisson and Nambu-Poisson brackets. J. Math. Phys. 38 (1997), no. 5, 2332–2344.
R. Ibàñez, M. de León, J. Marrero, D. Martín de Diego. Reduction of generalized Poisson and Nambu-Poisson manifolds. Rep. Math. Phys. 42 (1998), no. 1–2, 71–90.
L. Takhtajan. On foundation of the generalized Nambu mechanics. Comm. Math. Phys. 160 (1994), no. 2, 295–315.
I. Vaisman. Nambu-Lie groups. J. Lie Theory 10 (2000), no. 1, 181–194.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 The Author(s), under exclusive licence to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Barron, T. (2018). Toeplitz Operators on Multisymplectic and Hyperkähler Manifolds. In: Toeplitz Operators on Kähler Manifolds. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-94292-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-94292-6_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94291-9
Online ISBN: 978-3-319-94292-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)