14 research outputs found
Weyl invariant polynomial and deformation quantization on Kahler manifolds
Given a polynomial P of partial derivatives of the Kahler metric, expressed
as a linear combination of directed multigraphs, we prove a simple criterion in
terms of the coefficients for to be an invariant polynomial, i.e. invariant
under the transformation of coordinates. As applications, we prove an explicit
composition formula for covariant differential operators under a canonical
basis, also known as invariant differential operators in the case of bounded
symmetric domains. We also prove a general explicit formula of star products on
Kahler manifolds.Comment: 17 page
On a formula of Gammelgaard for Berezin-Toeplitz quantization
We give a proof of a slightly refined version of Gammelgaard's graph
theoretic formula for Berezin-Toeplitz quantization on (pseudo-)Kaehler
manifolds. Our proof has the merit of giving an alternative approach to
Karabegov-Schlichenmaier's identification theorem. We also identify the dual
Karabegov-Bordemann-Waldmann star product.Comment: 18 page
An explicit formula for the Berezin star product
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.Comment: 19 pages, to appear in Lett. Math. Phy
Generalized Bergman kernels on symplectic manifolds
We study the near diagonal asymptotic expansion of the generalized Bergman
kernel of the renormalized Bochner-Laplacian on high tensor powers of a
positive line bundle over a compact symplectic manifold. We show how to compute
the coefficients of the expansion by recurrence and give a closed formula for
the first two of them. As consequence, we calculate the density of states
function of the Bochner-Laplacian and establish a symplectic version of the
convergence of the induced Fubini-Study metric. We also discuss generalizations
of the asymptotic expansion for non-compact or singular manifolds as well as
their applications. Our approach is inspired by the analytic localization
techniques of Bismut-Lebeau.Comment: 48 pages. Add two references on the Hermitian scalar curvatur
Toeplitz operators on symplectic manifolds
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 non-compact manifolds and orbifolds are also established.Comment: 40 page
Felix Alexandrovich Berezin and his work
This is a survey of Berezin's work focused on three topics: representation
theory, general concept of quantization, and supermathematics.Comment: LaTeX, 27 page
Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization
For arbitrary compact quantizable Kähler 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 manner. It is shown that the star product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given