Results 11  20
of
49
Geometric quantization on Kähler and symplectic manifolds
 International Congress of Mathematicians
, 2010
"... We explain various results on the asymptotic expansion of the Bergman kernel on Kähler manifolds and also on symplectic manifolds. We also review the “quantization commutes with reduction ” phenomenon for a compact Lie group action, and its relation to the Bergman kernel. ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
We explain various results on the asymptotic expansion of the Bergman kernel on Kähler manifolds and also on symplectic manifolds. We also review the “quantization commutes with reduction ” phenomenon for a compact Lie group action, and its relation to the Bergman kernel.
Geometric approaches to computing Kostka numbers and LittlewoodRichardson coefficients
 PH.D. DEGREE IN MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY (MIT
, 2004
"... ..."
A symplectic proof of Verlinde factorization
 J. Diff. Geom
"... Abstract. We prove a multiplicity formula for RiemannRoch numbers of reductions of Hamiltonian actions of loop groups. This includes as a special case the factorization formula for the quantum dimension of the moduli space of flat connections over a Riemann surface. Contents ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
Abstract. We prove a multiplicity formula for RiemannRoch numbers of reductions of Hamiltonian actions of loop groups. This includes as a special case the factorization formula for the quantum dimension of the moduli space of flat connections over a Riemann surface. Contents
On the instanton complex of holomorphic Morse theory
, 2003
"... Consider a holomorphic torus action on a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixedpoint set form a partially ordered set, we construct, using sheaftheoretical techniques, two spectral sequences that converges to the twisted Dolbeault co ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Consider a holomorphic torus action on a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixedpoint set form a partially ordered set, we construct, using sheaftheoretical techniques, two spectral sequences that converges to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences are the holomorphic counterparts of the instanton complex in standard Morse theory. The results proved imply holomorphic Morse inequalities and fixedpoint formulas on a possibly noncompact manifold. Finally, examples and applications are given.
Holomorphic Morse inequalities in singular reduction
 Math. Res. Letters
, 1998
"... Abstract. We extend our Morse type inequalities for holomorphic symplectic reductions [TZ1, 2] to the case of singular reductions. 0. Introduction and the statement of main results Let (M, ω) be a closed symplectic manifold. We make the assumption that there is a Hermitian line bundle L over M admit ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Abstract. We extend our Morse type inequalities for holomorphic symplectic reductions [TZ1, 2] to the case of singular reductions. 0. Introduction and the statement of main results Let (M, ω) be a closed symplectic manifold. We make the assumption that there is a Hermitian line bundle L over M admitting a Hermitian connection ∇L with the property that √−1
Quantum surfaces, special functions, and the tunneling effect
, 2001
"... The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with nonLie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with nonLie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommutative product on the leaves incorporates a closed 2form and a measure which (in general) are different from the classical symplectic form and the Liouville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structures could even occur to be exponentially small with respect to the deformation parameter. That is interpreted as a tunneling effect in the quantum geometry. Dedicated to the memory of Professor M. Flato 1
Spin cQuantization and the Kmultiplicities of the discrete series
, 2002
"... We express the Kmultiplicities of a representation of the discrete series associated to a coadjoint orbit O in terms of Spin cindex on symplectic reductions of O. ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We express the Kmultiplicities of a representation of the discrete series associated to a coadjoint orbit O in terms of Spin cindex on symplectic reductions of O.
SYMPLECTIC REDUCTION AND A WEIGHTED MULTIPLICITY FORMULA FOR TWISTED SPIN CDIRAC OPERATORS
, 1999
"... We extend our earlier work in [TZ1], where an analytic approach to the GuilleminSternberg conjecture [GS] was developed, to cases where the Spin ccomplex under consideration is allowed to be further twisted by certain exterior power bundles of the cotangent bundle. The main result is a weighted qu ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
We extend our earlier work in [TZ1], where an analytic approach to the GuilleminSternberg conjecture [GS] was developed, to cases where the Spin ccomplex under consideration is allowed to be further twisted by certain exterior power bundles of the cotangent bundle. The main result is a weighted quantization formula in the presence of commuting Hamiltonian actions. The corresponding Morsetype inequalities in holomorphic situations are also established.
Formal geometric quantization II
, 2009
"... In this paper we pursue the study of formal geometric quantization of noncompact Hamiltonian manifolds. Our main result is the proof that two quantization process coincide. This fact was obtained by Ma and Zhang in the preprint Arxiv:0812.3989 by completely different means. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In this paper we pursue the study of formal geometric quantization of noncompact Hamiltonian manifolds. Our main result is the proof that two quantization process coincide. This fact was obtained by Ma and Zhang in the preprint Arxiv:0812.3989 by completely different means.