1,328 research outputs found
Berezin-Toeplitz Quantization and Star Products for Compact Kaehler Manifolds
For compact quantizable K\"ahler manifolds certain naturally defined star
products and their constructions are reviewed. The presentation centers around
the Berezin-Toeplitz quantization scheme which is explained. As star products
the Berezin-Toeplitz, Berezin, and star product of geometric quantization are
treated in detail. It is shown that all three are equivalent. A prominent role
is played by the Berezin transform and its asymptotic expansion. A few ideas on
two general constructions of star products of separation of variables type by
Karabegov and by Bordemann--Waldmann respectively are given. Some of the
results presented is work of the author partly joint with Martin Bordemann,
Eckhard Meinrenken and Alexander Karabegov. At the end some works which make
use of graphs in the construction and calculation of these star productsComment: 39 pages, Based on a talk presented in the frame of the Thematic
Program on Quantization, Spring 2011, at the University of Notre Dame, USA.
In the revised version some additional references are given in relation to
the role of the metaplectic correction and quotients. Also now there is an
additional section about applications and related reference
Some results on the invertibility of Toeplitz plus Hankel operators
The paper deals with the invertibility of Toeplitz plus Hankel operators
T(a)+H(b) acting on classical Hardy spaces on the unit circle T. It is supposed
that the generating functions a and b satisfy the condition
a(t)a(1/t)=b(t)b(1/t). Special attention is paid to the case of piecewise
continuous generating functions. In some cases the dimensions of null spaces of
the operator and its adjoint are described
String Scattering from Decaying Branes
We develop the general formalism of string scattering from decaying D-branes
in bosonic string theory. In worldsheet perturbation theory, amplitudes can be
written as a sum of correlators in a grand canonical ensemble of unitary random
matrix models, with time setting the fugacity. An approach employed in the past
for computing amplitudes in this theory involves an unjustified analytic
continuation from special integer momenta. We give an alternative formulation
which is well-defined for general momenta. We study the emission of closed
strings from a decaying D-brane with initial conditions perturbed by the
addition of an open string vertex operator. Using an integral formula due to
Selberg, the relevant amplitude is expressed in closed form in terms of zeta
functions. Perturbing the initial state can suppress or enhance the emission of
high energy closed strings for extended branes, but enhances it for D0-branes.
The closed string two point function is expressed as a sum of Toeplitz
determinants of certain hypergeometric functions. A large N limit theorem due
to Szego, and its extension due to Borodin and Okounkov, permits us to compute
approximate results showing that previous naive analytic continuations amount
to the large N approximation of the full result. We also give a free fermion
formulation of scattering from decaying D-branes and describe the relation to a
grand canonical ensemble for a 2d Coulomb gas.Comment: 36 pages, harvmac; v2: references added; v3: references adde
Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface
We generalize the results of Montgomery for the Bochner Laplacian on high
tensor powers of a line bundle. When specialized to Riemann surfaces, this
leads to the Bergman kernel expansion and geometric quantization results for
semi-positive line bundles whose curvature vanishes at finite order. The proof
exploits the relation of the Bochner Laplacian on tensor powers with the
sub-Riemannian (sR) Laplacian
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