Survey on quantization methods on Kähler manifolds.
problem None explicitly stated; focuses on methods.
method Deformation quantization, geometric quantization, Berezin-Toeplitz quantization, BV quantization.
result New relationships among quantization methods on Kähler manifolds.
Smooth approximations of Kähler-Ricci solitons found using quantized metrics and Futaki invariants.
problem Finding smooth approximations of Kähler-Ricci solitons on Fano manifolds.
method Using semiclassical estimates and quantized Futaki invariants to extend a strategy from Donaldson and Tian-Zhu.
result Smooth approximations of Kähler-Ricci solitons can be found as quantized metrics.
Quantizes geodesics in Kähler and Sasaki geometry.
problem Quantize geodesics in Kähler and Sasaki spaces.
method Classical Fubini-Study map and quantization procedure.
result Proves conditions for geodesics in Kähler potentials.
Study quantization schemes on Kähler manifolds linking star products and BV quantizations.
problem Quantization of structures on Kähler manifolds.
method Construct Fedosov's star products and Batalin-Vilkovisky (BV) quantizations.
result One-loop exactness of BV quantizations, leading to a cochain level formula.
Geometric quantization shows compatibility of symmetries on coadjoint orbits and Kähler-Einstein manifolds.
problem Compatibility of symmetries in geometric quantization.
method Deformation and geometric quantization on Kähler manifolds, Hamiltonian actions.
result Strict compatibility of symmetries on coadjoint orbits and Kähler-Einstein manifolds.
Quantizes functions on Kähler manifolds without formal deformation.
problem Deforming smooth functions on Kähler manifolds to non-formal quantization.
method Using Fedosov connections and prequantum line bundles.
result Quantizable functions form a sheaf of twisted differential operators.
Quantizes Kähler-Ricci flow for Fano manifolds.
problem Optimal degeneration for Fano manifolds.
method Geometric quantization of Kähler-Ricci flow and entropy functional.
result Established convergence to original flow and entropy.
Sharp estimates for Bergman metrics derived from Kähler quantization.
problem Estimating Bergman metrics in Kähler quantization.
method Upper and lower bounds on the Bergman metric expressed in terms of φ \varphi φ . result Optimal C 1 , 1 ˉ C^{1,\bar1} C 1 , 1 ˉ -convergence for quantization of Kähler currents. The paper quantizes the Hitchin system using geometric quantization.
problem Quantizing the Hitchin system using geometric methods.
method Geometric quantization of the Hitchin moduli space by modifying the Quillen metric.
result The modified Kahler form is integral and descends as a prequantum line bundle.
Quantizes Kähler manifolds using sheaves and differential operators.
problem Quantizing Kähler manifolds with sheaves and differential operators.
method Constructing a category enriched over sheaves of modules, defining quantizable morphisms, and showing equivalence to differential operator categories.
result Equivalence of quantized categories under certain conditions.
The paper quantizes vortex moduli spaces on compact Kahler surfaces using determinant bundles.
problem Quantizing vortex moduli spaces on compact Kahler surfaces.
method Developed holomorphic determinant bundles and geometric quantization for vortex moduli spaces.
result Quantized vortex moduli spaces on compact Kahler surfaces using determinant bundles.
We discuss the quantization of mechanical systems for which the Hamiltonian vector fields of observables form the deformation of n n n -dimensional oscilator algebra. Because of this fact these systems can be considered as "deformations" of the harmonic oscillator. The set of abovementioned mechanical systems are realized…
We prove the convergence of geodesic distance during the quantization of the space of Kähler potentials. As applications, this provides alternative proofs of certain inequalities about the K-energy functional in the projective case.
New bound on partition function proves Kähler-Einstein stability.
problem Proving Kähler-Einstein metrics on complex manifolds.
method Quantitative bound on partition function, connecting probabilistic and quantization approaches.
result Direct analytic proof of Kähler-Einstein stability for uniformly Gibbs stable manifolds.
It is known that holomorphic Poisson structures are closely related to theories of generalized Kähler geometry and bi-Hermitian structures. In this article, we introduce quantization of holomorphic Poisson structures which are closely related to generalized Kähler structures /bi-Hermitian structures. By resulting nonco…
Quantizes Kähler metrics using Finsler structures.
problem Approximating Kähler metrics by algebraic metrics.
method Quantization of Finsler structures on Kähler potentials.
result Metric completions of Finsler structures on Kähler potentials are recovered.
Paper discusses star products and Kähler metrics, linking deformation quantization and constant curvature metrics.
problem Existence of Kähler metrics with constant scalar curvature.
method Analyzes Fedosov and Berezin-Toeplitz star products, and studies K-stability conditions.
result Formulates a cohomology formula for K-stability conditions on Kähler metrics.
The paper classifies quantizable functions and explores symmetry in quantization methods.
problem Classifying quantizable functions and understanding symmetry in quantization methods.
method Deformation quantization and geometric quantization methods are compared and classified.
result Formal quantizable functions are of a specific form and relate to Hamiltonian Killing vector fields.
Geometrically constructs representations for quantization on Kähler manifolds.
problem Quantization of Kähler manifolds using Berezin-Toeplitz method.
method Using peak sections to localize Hilbert spaces around points in the large volume limit.
result Geometric construction of representations for Berezin-Toeplitz quantization.
We give an explicit local formula for any formal deformation quantization, with separation of variables, on a Kähler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.
Griffiths extremal metrics solve complex Finsler equations and quantify Kähler geometry.
problem Interpolation of norms and complex Finsler geometry.
method Introduced Griffiths extremal Finsler metrics and solved their Dirichlet problem.
result Griffiths extremal Finsler metrics quantize solutions to a PDE in Kähler geometry.
Study proves convergence of quantized geodesics to Mabuchi geodesics.
problem Convergence of quantized geodesics to Mabuchi geodesics in short time.
method Real-analytic initial data and convergence proof.
result Proves convergence of quantized Bergman geodesics to Mabuchi geodesics.
Let P P P be a Delzant polytope. We show that the quantization of the corresponding toric manifold X P X_{P} X P in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time t = − 1 s t = \sqrt{-1} s t = − 1 s . We relate the quantization of X P X_{P} X P in two different …
Study Mabuchi rays on toric Kähler manifolds to understand quantization.
problem Understanding quantization in toric Kähler manifolds.
method Analyzing Mabuchi rays associated with test configurations and moment polytopes.
result Quantization in limit polarizations corresponds to restrictions of monomial sections.
Researchers extend geometric quantization to complex Abelian Lie supergroups.
problem Quantization of super Kähler structures on complex Abelian Lie supergroups.
method Extended geometric quantization scheme to super Kähler setting, constructed unitary representation.
result Irreducible subrepresentations of the constructed representation are determined by the moment map.
New method shows unitarity in quantization for toric manifolds.
problem Unitarity in quantization commutes with reduction for toric manifolds.
method Generalized coherent state transform (gCST) and geodesic rays of toric Kähler polarizations.
result Quantization commutes unitarily with reduction for the new mixed polarization.
New approach to geometric quantization for symplectic manifolds.
problem Quantization of symplectic manifolds with non-singular Lagrangian fibrations.
method Using spectral convergence of metric measure spaces, the authors develop a new geometric quantization approach.
result Spectral and quantum Hilbert space convergence results for Kähler and almost Kähler quantizations.
Study geometric quantization on K3 surfaces, showing spectral convergence.
problem Quantization of K3 surfaces from spectral perspective.
method Special Lagrangian fibrations and hyper-Kähler structures.
result Spectral convergence of ∂ ˉ \bar{\partial} ∂ ˉ -Laplacians on prequantum line bundles. Calculates the Hilbert space dimension for vortex moduli spaces.
problem Determining the dimension of the Hilbert space for vortex moduli spaces.
method Kahler quantization of the moduli space of vortices on a Riemann surface.
result The dimension is given by the holomorphic Euler characteristic of the quantum line bundle.
Study provides explicit formula for complex 2D Kähler manifold quantization.
problem Quantization of complex 2D locally symmetric Kähler manifolds.
method Deformation quantization with separation of variables, solving recurrence relations.
result Explicit formula for star product on complex 2D locally symmetric Kähler manifolds.
Study entanglement in complex manifold sections.
problem Entanglement properties of complex manifold sections.
method Kähler quantization for ample line bundles.
result Characterized entanglement in tensor products of sections.
We suggest a way to quantize, using Berezin-Toeplitz quantization, a compact hyperkahler manifold (equipped with a natural 3-plectic form), or a compact integral Kahler manifold of complex dimension n regarded as a (2n-1)-plectic manifold. We show that quantization has reasonable semiclassical properties.
This paper connects symplectic and Kähler manifolds via brane quantization.
problem Quantizing Kähler manifolds using brane techniques.
method Using physical proposals and geometric quantization, the authors relate A-model morphism spaces to quantizations of symplectic and Kähler manifolds.
result Chan-Leung-Li's work provides a mathematical realization of the action of A-branes on B-branes, linking deformation quantizations of symplectic and Kähler manifolds.
The abstract discusses embedding theorems for pseudo-Kähler manifolds.
problem Embedding theorems for pseudo-Kähler manifolds.
method Using quantizable pseudo-Kähler manifolds and Hermitian line bundles, the asymptotic expansion of Bergman kernels is analyzed.
result The asymptotic expansion of Bergman kernels implies analogues of Kodaira embedding theorem and Tian's almost-isometry theorem.
Study quantization of Kähler-Einstein metrics using balanced metrics.
problem Approximating Kähler-Einstein metrics by balanced metrics.
method Use canonical Bergman metrics and introduce algebro-geometric obstructions.
result Existence and weak convergence of balanced metrics for CKE manifolds.
Quantizes vortex moduli space using modified Quillen metric.
problem Quantizing symplectic form on vortex moduli space.
method Modifying Quillen metric of determinant line bundle.
result Geometric quantization achieved.
Constructs a sheaf of Bargmann-Fock modules on Kähler manifolds.
problem Deformation quantization on Kähler manifolds.
method Extends Fedosov's method to Kähler manifolds with compatible Fedosov abelian connections.
result Explicit construction of sheaf of flat sections as a module over deformation quantization algebras.
The paper studies geometric quantization and coherent state transforms on Lie groups.
problem Quantization of cotangent bundles of compact Lie groups with new invariant structures.
method Analytic continuation of Hamiltonian flows on G i m e s T G imes T G im es T -invariant Kähler structures. result Partial coherent state transforms and new Kähler structures on T ∗ G T^*G T ∗ G . In this paper we show how Einstein metrics are naturally described using the quantization of the algebra of functions on a Kahler manifold M. In this setup one interprets M as the phase space itself, equipped with the Poisson brackets inherited from the Kahler 2-form. We compare the geometric quantization framework wit…
The study bounds entanglement entropy for coherent states on Kähler manifolds.
problem Bounding entanglement entropy for coherent states on Kähler manifolds.
method Geometric quantization and Hilbert spaces of Kähler manifolds.
result Positive lower bounds for entanglement entropy are provided asymptotically.
Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.
problem Existence of Kähler-Einstein metrics with arbitrary polarizations.
method Quantization techniques and pluripotential theory.
result Uniform Yau-Tian-Donaldson theorem for Kähler-Einstein metrics.
Study quantized extremal Kähler metrics for algebro-geometric stability.
problem Stability of extremal Kähler metrics and manifolds.
method Quantized extremal Kähler metrics and equivariant Riemann-Roch theorem.
result Proves weak relative Chow polystability and K-semistability.
The paper quantizes Kähler manifolds using differential operators.
problem Quantizing classical observables on Kähler manifolds as differential operators.
method Constructing higher-order differential operators using Fedosov-type constructions and proving asymptotic equivalence to Berezin-Toeplitz operators.
result Holomorphic differential operators are precisely those that arise as Berezin-Toeplitz operators for quantizable functions.
The paper defines and analyzes coherent and squeezed states on manifolds and their quantization.
problem Defining and characterizing coherent and squeezed states on various manifolds.
method Definition and analysis of Rawnsley-type coherent and squeezed states, Berezin quantization.
result Properties and quantization of coherent and squeezed states on manifolds.
Study quantizes topological numbers on degenerating Einstein manifolds.
problem Quantizing topological numbers on non-collapsed degenerating Einstein manifolds.
method Compactness theory of bubbles, classical vanishing theorems, and Hirzebruch-Riemann-Roch theorems.
result Established quantization results for various topological numbers.
Study shows convergence of anticanonically balanced metrics to Kähler-Einstein metrics on Fano manifolds.
problem Finding anticanonically balanced metrics on Fano manifolds.
method Simplification of Donaldson's proof using Berezin-Toeplitz quantization.
result Sequence of anticanonically balanced metrics converges to Kähler-Einstein metric.
Quantizes symplectic manifolds with toric singularities using Toeplitz operators.
problem Quantize symplectic manifolds with toric singularities.
method Establishes quantization for compact toric symplectic manifolds with transversal singular real polarizations using Toeplitz operators.
result Toeplitz operators determine a star product on compact toric symplectic manifolds with toric singularities as ℏ o 0 + \hbar o 0^+ ℏ o 0 + . The paper constructs quantizations for symplectic manifolds with specific Laplacian properties.
problem Quantization of compact symplectic manifolds with higher Landau levels.
method Develops Berezin-Toeplitz quantization using a Bochner Laplacian with specific spectral properties.
result The quantization provides a formal star-product for the lowest Landau level.