arXiv research

We prove an explicit formula of the Berezin star product on Kaehler manifolds. The formula is expressed as a summation over certain strongly connected digraphs. The proof relies on a combinatorial interpretation of Englis' work on the asymptotic expansion of the Laplace integral.

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.

We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…

Researchers describe a new Thom form for mapping cones.

We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free repesentation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes's theorem is derived - here, th…

For phase-space manifolds which are compact Kaehler manifolds relations between the Berezin-Toeplitz quantization and the quantization with the help of Berezin's coherent states and symbols are studied. First the results on the Berezin-Toeplitz quantization of arbitrary compact Kaehler manifolds due to Bordemann, Meinr…

Study finds modular classes help in proving Berezin volumes for supersymmetric theories.

We study the Berezin-Toeplitz quantization on Kaehler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz operators. As application…

The paper applies Poincaré duality to supergravity, proving its equivalence to other formulations.

Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.

In 1974, Berezin proposed a quantum theory for dynamical systems having a Kähler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. For any homogeneous Kähler manifold, the Lie algebra of its group of motions may be represented either by holomorphic differential …

We prove that a homogeneous bounded domain admits a Berezin quantization.

The paper defines and analyzes coherent and squeezed states on manifolds and their quantization.

We prove the existence of a Berezin-Engliš quantization for Cartan-Hartogs domains.

We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin-Toeplitz quantization obtained recently by Ma-Marinescu, and of the composition of two Berezin-Toeplitz quantizations. Our main tool is the stationary phase formula of Melin-Sjöstrand.

In this lecture results on the Berezin-Toeplitz quantization of arbitrary compact quantizable Kaehler manifolds are presented. These results are obtained in joint work with M. Bordemann and E. Meinrenken. The existence of the Berezin-Toeplitz deformation quantization is also covered. Recent results obtained in joint wo…

The article defines and compares two types of quantizations on compact manifolds.

This is a survey of Berezin's work focused on three topics: representation theory, general concept of quantization, and supermathematics.

The transformation formula of the Berezin integral holds, in the non-compact case, only up to boundary integrals, which have recently been quantified by Alldridge-Hilgert-Palzer. We establish divergence theorems in semi-Riemannian supergeometry by means of the flow of vector fields and these boundary integrals, and sho…

Quantum propagation studied for Berezin-Toeplitz operators.

Quantization on even-dimensional compact manifolds using cell decomposition.

We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.

Develops a dynamical method to prove the sharp Berezin-Li-Yau inequality.

We compute the second coefficient of the composition of two Berezin-Toeplitz operators associated with the $\text{spin}^c$ Dirac operator on a symplectic manifold, making use of the full-off diagonal expansion of the Bergman kernel.

Survey on quantization methods on Kähler manifolds.

We introduce new tools for analytic microlocal analysis on Kähler manifolds. As an application, we prove that the space of Berezin-Toeplitz operators with analytic contravariant symbol is an algebra. We also give a short proof of the Bergman kernel asymptotics up to an exponentially small error.

We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.

We study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-pro…

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for no…

This talk reports on results on the deformation quantization (star products) and on approximative operator representations for quantizable compact K"ahler manifolds obtained via Berezin-Toeplitz operators. After choosing a holomorphic quantum line bundle the Berezin-Toeplitz operator associated to a differentiable func…

The paper quantizes Kähler manifolds using differential operators.

Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.

The relations between the infinite dimensional geometry of $q_R$-conformal symmetries at $q_R\to\infty$, Berezin quantization of the Lobachevskii plane and Karasev-Maslov asymptotic quantization are explicated. Some aspects of the ``approximate'' representation theory are discussed.

Let G be a Lie supergroup and H a closed subsupergroup. We study the unimodularity of the homogeneous supermanifold G/H, i.e. the existence of G-invariant sections of its Berezinian line bundle. To that end, we express this line bundle as a G-equivariant associated bundle of the principal H-bundle G over G/H. We also s…

Let $M$ be an arbitrary complex manifold and let $L$ be a Hermitian holomorphic line bundle over $M$. We introduce the Berezin-Toeplitz quantization of the open set of $M$ where the curvature on $L$ is non-degenerate. The quantum spaces are the spectral spaces corresponding to $[0,k^{-N}]$ ($N>1$ fixed), of the Kodaira…

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint up to an error w…

The consequences for Berezin's quantization on symmetric spaces of the identity of the set of coherent vectors orthogonal to a fixed one with the cut locus are stated precisely. It is shown that functions expressing the coherent states, the covariant symbols of operators, the diastasis function, the characteristic and …

The paper classifies quantizable functions and explores symmetry in quantization methods.

For a Kähler 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…

For arbitrary compact quantizable Kaehler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential man…

Quantizes the standard hyperkähler space R^(4n) without a point.

Invited lecture at the XIV-th workshop on geometric methods in physics, Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global Toeplitz operator…

Study asymptotics of unitary matrix elements in quantum mechanics.

We show that the Wei-Norman method applied to describe the evolution on the Siegel-Jacobi disk $\mathcal{D}^J_1=\mathcal{D}_1\times\mathbb{C}^1$, where $\mathcal{D}_1$ denotes the Siegel disk, determined by a hermitian Hamiltonian linear in the generators of the Jacobi group $G^J_1$ and Berezin's scheme using coherent …

The standard (Berezin-Toeplitz) geometric quantization of a compact Kaehler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal covering space, provided that the lift of the symplectic form is exact. I relate this construction to the Baum-Connes assembly …

Quantizes symplectic fibrations to analyze vector bundles and metrics.

New algebraic structures on manifolds generalize supergeometry concepts.

In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the $(2,2)$-dimensional supersphere $S^{2,2}$ by means of global projectors $p$ via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding `rank 1' supervector bundle ove…