Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results

DOI10.1155/2010/927280zbMath1207.81049arXiv1003.2523OpenAlexW3098874025WikidataQ58650086 ScholiaQ58650086MaRDI QIDQ606162

Martin Schlichenmaier

Publication date: 16 November 2010

Published in: Advances in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1003.2523

Mathematics Subject Classification ID

Research exposition (monographs, survey articles) pertaining to quantum theory (81-02) Kähler manifolds (32Q15) Geometry and quantization, symplectic methods (81S10) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20) Deformation quantization, star products (53D55)

Related Items

Berezin-Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian, Toeplitz Operators with Discontinuous Symbols on the Sphere, A Star Product for the Volume Form Nambu-Poisson Structure on a Kähler Manifold, Symplectic rigidity and quantum mechanics, Spectral asymptotics of semiclassical unitary operators, Dequantization via quantum channels, The Berezin transform and its applications, Local scaling asymptotics for the Gutzwiller trace formula in Berezin-Toeplitz quantization, Asymptotic expansion of the Bergman kernel via semi-classical symbolic calculus, Commutators of spectral projections of spin operators, Quantum unsharpness and symplectic rigidity, Twist star products and Morita equivalence, Bounds for fidelity of semiclassical Lagrangian states in Kähler quantization, Berezin-Toeplitz quantization for eigenstates of the Bochner Laplacian on symplectic manifolds, Helton-Howe trace, Connes-Chern characters and Toeplitz quantization of Bergman spaces, Remarks on star products for non-compact CR manifolds with non-degenerate Levi form, Semi-classical asymptotics of Bergman and spectral kernels for $(0,q)$-forms, On a graph theoretic formula of Gammelgaard for Berezin-Toeplitz quantization, Almost representations of algebras and quantization, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, Geometric quantization results for semi-positive line bundles on a Riemann surface, Fock representations and deformation quantization of Kähler manifolds, Euler-Maclaurin formulas via differential operators, The classical limit of Schrödinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as an emergent phenomenon, Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells, A Gutzwiller trace formula for large hermitian matrices, BEREZIN–TOEPLITZ QUANTIZATION, HYPERKÄHLER MANIFOLDS, AND MULTISYMPLECTIC MANIFOLDS, Commutative \(C^*\)-algebras of Toeplitz operators on complex projective spaces, Symplectic geometry of quantum noise, Symplectic invariants of semitoric systems and the inverse problem for quantum systems, Spectral aspects of the Berezin transform, Kähler quantization and entanglement, On polynomials in spectral projections of spin operators, Berezin-Toeplitz quantization and naturally defined star products for Kähler manifolds, An explicit formula for the Berezin star product, Geometric quantization via cotangent models, The classical limit of mean-field quantum spin systems, Twisted Fock representations of noncommutative Kähler manifolds, Formal global perturbative quantization of the Rozansky-Witten model in the BV-BFV formalism, Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry, Toeplitz operators on CR manifolds and group actions, Explicit formulas for noncommutative deformations of ${\mathbb C}{P^N}$CPN and ${\mathbb C}{H^N}$CHN, Boundary singularity of Poisson and harmonic Bergman kernels, Noncommutative coherent states and related aspects of Berezin–Toeplitz quantization, Inverse spectral theory for semiclassical Jaynes-Cummings systems

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Generalized Berezin quantization, Bergman metrics and fuzzy Laplacians
• Monopole bundles over fuzzy complex projective spaces
• Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds
• Fuzzy complex projective spaces and their star-products
• *-products on \(D^ 1(C)\), \(S^ 2\) and related spectral analysis
• Toeplitz \(C^*\)-algebras on bounded symmetric domains
• Weyl manifolds and deformation quantization
• Finite gauge theory on fuzzy \(\mathbb CP^2\)
• On a set of polarized Kähler metrics on algebraic manifolds
• Toeplitz operators on symplectic manifolds
• Toeplitz operators on symmetric Siegel domains
• Deformation quantization and asymptotic operator representation
• Quantum Riemann surfaces. I: The unit disc
• Quantum Riemann surfaces. II: The discrete series
• Deformation estimates for the Berezin-Toeplitz quantization
• Deformation theory and quantization. I: Deformations of symplectic structures
• On the canonical normalization of a trace density of deformation quantization
• Semiclassical spectral estimates for Toeplitz operators
• Equivalence of star products on a symplectic manifold; an introduction to Deligne's Čech cohomology classes
• Quantization of Kähler manifolds. III
• Toeplitz quantization of Kähler manifolds and \(gl(N)\), \(N\to \infty\) limits
• A simple geometrical construction of deformation quantization
• A Fedosov star product of the Wick type for Kähler manifolds
• Cohomological classification of deformation quantizations with separation of variables
• Coherent state embeddings, polar divisors and Cauchy formulas
• Existence of a closed star product
• \(gl(\infty{})\) and geometric quantization
• *-products on some Kähler manifolds
• Weighted Bergman kernels and quantization
• Deformation quantization of Poisson manifolds
• Quantization of Kähler manifolds. IV
• Asymptotics of the Berezin transform and quantization on planar domains
• Algebraic index theorem for families
• Toeplitz operators and index theory in several complex variables
• Algebraic index theorem
• The Berezin transform and invariant differential operators
• Deformations of the algebra of functions of a symplectic manifold. Comparison between Fedosov and de Wilde, Lecomte
• Deformation quantizations with separation of variables on a Kähler manifold
• Holomorphic Morse inequalities and Bergman kernels
• Berezin-Toeplitz quantization of the moduli space of flat SU\((N)\) connections
• Toeplitz operators and Hamiltonian torus actions
• Asymptotic faithfulness of the quantum \(\text{SU}(n)\) representations of the mapping class groups
• Deformation quantization and geometric quantization of Abelian moduli spaces
• Quantization of Kähler manifolds. I: Geometric interpretation of Berezin's quantization
• Identification of Berezin-Toeplitz deformation quantization
• Hitchin's Projectively Flat Connection, Toeplitz Operators and the Asymptotic Expansion of TQFT Curve Operators
• GEOMETRIC CONSTRUCTION OF MODULAR FUNCTORS FROM CONFORMAL FIELD THEORY
• The Spectral Theory of Toeplitz Operators. (AM-99)
• The fuzzy sphere
• QUANTIZATION
• QUANTIZATION IN COMPLEX SYMMETRIC SPACES
• COHERENT STATES AND KAHLER MANIFOLDS
• Equivalence of star products
• The asymptotics of a Laplace integral on a Kähler manifold
• FUZZY INSTANTONS
• Quantization of Kahler Manifolds. II
• Fredholm Indices for Toeplitz Operators on Bounded Symmetric Domains
• Generalized Bergman kernels and geometric quantization
• Unified theories from fuzzy extra dimensions
• Berezin Quantization and Reproducing Kernels on Complex Domains
• ABELIAN CONFORMAL FIELD THEORY AND DETERMINANT BUNDLES
• QUANTIZATION METHODS: A GUIDE FOR PHYSICISTS AND ANALYSTS
• Toeplitz operators and quantum mechanics