Berezin-Toeplitz Quantization Research Articles

Berezin-type quantization on even-dimensional compact manifolds

In this article we show that a Berezin-type quantization can be achieved on a compact even dimensional manifold M 2d by removing a skeleton M 0 of lower dimension such that what remains is diffeomorphic to R 2d (cell decomposition) which we identify with Cd and embed in CPd . A local Poisson structure and Berezin-type quantization are induced from CPd . Thus we have a Hilbert space with a reproducing kernel. The symbols of bounded linear operators on the Hilbert space have a star product which satisfies the correspondence principle outside a set of measure zero. This construction depends on the diffeomorphism. One needs to keep track of the global holonomy and hence the cell decomposition of the manifold. As an example, we illustrate this type of quanitzation of the torus. We exhibit Berezin-Toeplitz quantization of a complex manifold in the same spirit as above.

Quantization of symplectic fibrations and canonical metrics

We relate Berezin–Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including the spectral gap of the Berezin transform and the convergence rate of Donaldson’s iterations toward balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson’s iterations toward balanced metrics on Kähler manifolds with constant scalar curvature.

Vector bundles on fuzzy Kähler manifolds

Abstract We propose a matrix regularization of vector bundles over a general closed Kähler manifold. This matrix regularization is given as a natural generalization of the Berezin–Toeplitz quantization and gives a map from sections of a vector bundle to matrices. We examine the asymptotic behaviors of the map in the large-N limit. For vector bundles with algebraic structure, we derive a beautiful correspondence of the algebra of sections and the algebra of corresponding matrices in the large-N limit. We give two explicit examples for monopole bundles over a complex projective space CPn and a torus T2n.

Deformation Quantization via Toeplitz Operators on Geometric Quantization in Real Polarizations

In this paper, we study quantization on a compact integral symplectic manifold X with transversal real polarizations. In the case of complex polarizations, namely X is Kähler equipped with transversal complex polarizations $$T^{1, 0}X, T^{0, 1}X$$ , geometric quantization gives $$H^0(X, L^{\otimes k})$$ ’s. They are acted upon by $${\mathcal {C}}^\infty (X, {\mathbb {C}})$$ via Toeplitz operators as $$\hbar = \tfrac{1}{k} \rightarrow 0^+$$ , determining a deformation quantization $$({\mathcal {C}}^\infty (X, {\mathbb {C}})[[\hbar ]], \star )$$ of X. We investigate the real analogue to these, comparing deformation quantization, geometric quantization and Berezin-Toeplitz quantization. The techniques used are different from the complex case as distributional sections supported on Bohr-Sommerfeld fibres are involved. By switching the roles of the two real polarizations, we obtain Fourier-type transforms for both deformation quantization and geometric quantization, and they are compatible asymptotically as $$\hbar \rightarrow 0^+$$ . We also show that the asymptotic expansion of traces of Toeplitz operators realizes a trace map on deformation quantization.

Berezin–Toeplitz Quantization on Symplectic Manifolds of Bounded Geometry

We establish the theory of Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry. The quantum space of this quantization is the spectral subspace of the renormalized Bochner Laplacian associated with some interval near zero. We show that this quantization has the correct semiclassical limit.

Symmetry reduction of states II: A non-commutative Positivstellensatz for [formula omitted

We give a non-commutative Positivstellensatz for CPn: The (commutative) ⁎-algebra of polynomials on the real algebraic set CPn with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on C1+n, restricted to the real (2n+1)-sphere inside C1+n, and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on C1+n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative ⁎-algebra of polynomials on C1+n, the Weyl algebra, and give an algebraic description of the real-valued U(1)-invariant polynomials that are positive in certain ⁎-representations on Hilbert spaces of holomorphic sections of line bundles over CPn. It is especially noteworthy that the non-commutative result applies not only to strictly positive, but to all positive (semidefinite) elements. As an application, all ⁎-representations of the quantization of the polynomial ⁎-algebra on CPn, obtained e.g. through phase space reduction or Berezin–Toeplitz quantization, are determined.

Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian

In this paper, we construct a family of Berezin–Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers $L^p$ of the prequantum line bundle $L$ asymptotically splits into clusters of size ${\\mathcal O}(p^{3/4})$ around the points $p\\Lambda$, where $\\Lambda$ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigenvalues from the cluster.We show that it provides a Berezin–Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kähler quantization.

Anticanonically balanced metrics on Fano manifolds

We show that if a Fano manifold has discrete automorphism group and admits a polarized Kähler–Einstein metric, then there exists a sequence of anticanonically balanced metrics converging smoothly to the Kähler–Einstein metric. Our proof is based on a simplification of Donaldson’s proof of the analogous result for balanced metrics, replacing a delicate geometric argument by the use of Berezin–Toeplitz quantization. We then apply this result to compute the asymptotics of the optimal rate of convergence to the fixed point of Donaldson’s iterations in the anticanonical setting.

The Twistor Space of $${\mathbb {R}}^{4n}$$ and Berezin–Toeplitz Operators

A hyperkähler manifold M has a family of induced complex structures indexed by a two-dimensional sphere $$S^2 \cong \mathbb {CP}^1$$ . The twistor space of M is a complex manifold $${{\,\mathrm{Tw}\,}}(M)$$ together with a natural holomorphic projection $${{\,\mathrm{Tw}\,}}(M) \rightarrow \mathbb {CP}^1$$ , whose fiber over each point of $$\mathbb {CP}^1$$ is a copy of M with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space), and for the case of $$M = {\mathbb {R}}^{4n}$$ , $$n\in {{\mathbb {N}}}$$ , equipped with the standard hyperkähler structure, we construct one quantization that replaces the family of Berezin–Toeplitz quantizations parametrized by $$S^2-\{ pt\}$$ . We provide semiclassical asymptotics for this quantization.

Berezin–Toeplitz quantization in real polarizations with toric singularities

On a compact Kahler manifold $X$, Toeplitz operators determine a deformation quantization $(\operatorname{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1, 0}X, T^{0, 1}X$ as $\hbar \to 0^+$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $\hbar \to 0^+$.

Berezin-Toeplitz Quantization on Symplectic Manifolds of Bounded Geometry

Мы развиваем теорию квантования по Березину-Теплицу на симплектических многообразиях ограниченной геометрии. Пространством квантования является подходяшее собственное подпространство ренормализованного оператора Бохнера, ассоциированное с некоторой окрестностью нуля. Доказывано, что квантование имеет корректный квазиклассический предел. Библиография: 38 названий.

A geometric construction of representations of the Berezin–Toeplitz quantization

For a K\"ahler manifold $X$ equipped with a prequantum line bundle $L$, we give a geometric construction of a family of representations of the Berezin-Toeplitz deformation quantization algebra $(C^\infty(X)[[\hbar]],\star_{BT})$ parametrized by points $z_0 \in X$. The key idea is to use peak sections to suitably localize the Hilbert spaces $H^{0}\left(X,L^{\otimes m}\right)$ around $z_{0}$ in the large volume limit.

Quantization and Isotropic Submanifolds

We introduce the notion of an isotropic quantum state associated with a Bohr–Sommerfeld manifold in the context of Berezin–Toeplitz quantization of general prequantized symplectic manifolds, and we study its semiclassical properties using the off-diagonal expansion of the Bergman kernel. We then show how these results extend to the case of noncompact orbifolds and give an application to relative Poincaré series in the theory of automorphic forms.

Spectral aspects of the Berezin transform

We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin–Toeplitz quantization, Donaldson’s dynamical system on the space of Hermitian inner products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the Q-operator on Kähler manifolds with constant scalar curvature, and yield exponential convergence of Donaldson’s iterations to the fixed point. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance.

Berezin–Toeplitz Quantization and the Least Unsharpness Principle

Abstract We show that compatible almost-complex structures on symplectic manifolds correspond to optimal positive quantizations.

Landau-Hall states and Berezin-Toeplitz quantization of matrix algebras

We argue that the Landau-Hall states provide a suitable framework for formulating the Berezin-Toeplitz quantization of classical functions on a K\"ahler phase space. We derive the star-products for such functions in this framework and generalize the Berezin-Toeplitz quantization to matrix-valued classical functions. We also comment on how this is related to different calculations of the effective action for Hall systems.

Matrix regularization for Riemann surfaces with magnetic fluxes

We consider the matrix regularization of fields on a Riemann surface which couple to gauge fields with a nonvanishing magnetic flux. We show that such fields are described as rectangular matrices in the matrix regularization. We construct the matrix regularization explicitly for the case of the sphere and torus based on the Berezin-Toeplitz quantization, and also discuss a possible generalization to cases with higher genera. We also discuss the matrix version of the Laplacian acting on the rectangular matrices.

Toeplitz Operators with Analytic Symbols

We develop a new semiclassical calculus in analytic regularity, and apply these techniques to the study of Berezin–Toeplitz quantization in real-analytic regularity. We provide asymptotic formulas for the Bergman projector and Berezin–Toeplitz operators on a compact Kahler manifold. These objects depend on an integer N and we study, in the limit $$N\rightarrow +\infty $$ , situations in which one can control them up to an error $$O(e^{-cN})$$ for some $$c>0$$ . We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to Kahler manifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to $$O(e^{-cN})$$ on any real-analytic Kahler manifold as $$N\rightarrow +\infty $$ . We also prove that Toeplitz operators with analytic symbols can be composed and inverted up to $$O(e^{-cN})$$ . As an application, we study eigenfunction concentration for Toeplitz operators if both the manifold and the symbol are real-analytic. In this case we prove exponential decay in the classically forbidden region.

SO(5) Landau models and nested Nambu matrix geometry

The SO(5) Landau model is the mathematical platform of the 4D quantum Hall effect and provide a rare opportunity for a physical realization of the fuzzy four-sphere. We present an integrated analysis of the SO(5) Landau models and the associated matrix geometries through the Landau level projection. With the SO(5) monopole harmonics, we explicitly derive matrix geometry of a four-sphere in any Landau level: In the lowest Landau level the matrix coordinates are given by the generalized SO(5) gamma matrices of the fuzzy four-sphere satisfying the quantum Nambu algebra, while in higher Landau level the matrix geometry becomes a nested fuzzy structure realizing a pure quantum geometry with no counterpart in classical geometry. The internal fuzzy geometry structure is discussed in the view of an SO(4) Pauli-Schrödinger model and the SO(4) Landau model, where we unveil a hidden singular gauge transformation between their background non-Abelian field configurations. Relativistic versions of the SO(5) Landau model are also investigated and relationship to the Berezin-Toeplitz quantization is clarified. We finally discuss the matrix geometry of the Landau models in even higher dimensions.

Matrix geometry for ellipsoids

Abstract We study description of ellipsoids by matrices to gain insights into emergence of space in matrix models. We apply the coherent state method to the fuzzy ellipsoids and perform the Berezin-Toeplitz quantization for ellipsoids. We see that the manifold described by the coherent state method is an ellipsoid. We find that the Toeplitz operators obtained in the Berezin-Toeplitz quantization agree with the matrices for the fuzzy ellipsoids used in the coherent state method and that the wave functions in the above two methods coincide.