Toeplitz Operators on Symplectic Manifolds

Ma, Xiaonan; Marinescu, George

doi:10.1007/s12220-008-9022-2

Cited by 97 publications

(154 citation statements)

References 31 publications

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“…In this section, we are concerned with Berezin-Toeplitz operators (simply called in the sequel Toeplitz operators) and quantization of classical systems given by such operators, see Kostant [24], Souriau [33] and Berezin [3], as well as the book [6] by Boutet de Monvel and Guillemin for the corresponding microlocal analysis. Many well-known results for pseudodifferential operators are now known for Toeplitz operators, see [4,5,9,10,18,25].…”

Section: Preliminariesmentioning

confidence: 99%

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Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators

Pelayo

Polterovich

Ngọc

2014

Proceedings of the London Mathematical Society

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Abstract. We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system. This gives a quick alternative solution to the isospectrality problem for quantum toric systems. If the operators are uniformly bounded, the convergence is uniform. Analogous results hold for non-commuting operators.

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Section: Preliminariesmentioning

confidence: 99%

“…If M is a prequantizable closed symplectic (but not necessarily Kähler) manifold, then there exist several constructions of semiclassical quantizations of M satisfying Definition 4, see [5,6,18,25,32].…”

Section: Preliminariesmentioning

confidence: 99%

Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators

Pelayo

Polterovich

Ngọc

2014

Proceedings of the London Mathematical Society

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“…(Recall that a sigma model containing only twisted chiral superfields is dual to one containing only untwisted ones). The superfield expansion of a twisted chiral superfield reads as 17) JHEP05 (2009)055 cf. section 5.…”

Section: Jhep05(2009)055mentioning

confidence: 99%

Generalized Berezin-Toeplitz quantization of Kähler supermanifolds

Lazaroiu

McNamee

Sämann

2009

J. High Energy Phys.

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Abstract:We extend the construction of generalized Berezin and Berezin-Toeplitz quantization to the case of compact Hodge supermanifolds. Our approach is based on certain super-analogues of Rawnsley's coherent states. As applications, we discuss the quantization of affine and projective superspaces. Furthermore, we propose a definition of supersymmetric sigma-models on quantized Hodge supermanifolds. The corresponding quantum field theories are finite and thus yield supersymmetry-preserving regularizations for QFTs defined on flat superspace.

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“…In this paper, we consider the Berezin-Toeplitz quantization for spinor bundles [11,12]. This method can be defined on a compact Riemannian spin-C manifold equipped with a topologically nontrivial gauge field configuration.…”

Section: Introductionmentioning

confidence: 99%

Information metric, Berry connection, and Berezin-Toeplitz quantization for matrix geometry

2018

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We consider the information metric and Berry connection in the context of noncommutative matrix geometry. We propose that these objects give a new method of characterizing the fuzzy geometry of matrices. We first give formal definitions of these geometric objects and then explicitly calculate them for the well-known matrix configurations of fuzzy S 2 and fuzzy S 4. We find that the information metrics are given by the usual round metrics for both examples, while the Berry connections coincide with the configurations of the Wu-Yang monopole and the Yang monopole for fuzzy S 2 and fuzzy S 4 , respectively. Then, we demonstrate that the matrix configurations of fuzzy S n (n ¼ 2, 4) can be understood as images of the embedding functions S n → R nþ1 under the Berezin-Toeplitz quantization map. Based on this result, we also obtain a mapping rule for the Laplacian on fuzzy S 4 .

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Toeplitz Operators on Symplectic Manifolds

Cited by 97 publications

References 31 publications

Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators

Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators

Generalized Berezin-Toeplitz quantization of Kähler supermanifolds

Information metric, Berry connection, and Berezin-Toeplitz quantization for matrix geometry

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