An Explicit Formula for the Berezin Star Product

Xu, Hao

doi:10.1007/s11005-012-0552-y

Cited by 18 publications

(21 citation statements)

References 57 publications

(120 reference statements)

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“…By way of application, motivated by the standard Berezin-Toeplitz quantization of a classical observable (see [2,6,10,13,17,[26][27][28]31] and [1]), let us consider the scaling asymptotics of the equivariant components of certain Toeplitz operators (we will consider Toeplitz operators in the sense of [4]). Given f ∈ C ∞ (M) and assuming for simplicity that f is invariant under the action of the product group P = G × T , we can consider the Toeplitz operators…”

Section: Applications To Toeplitz Operator Kernelsmentioning

confidence: 99%

Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions

Camosso

2016

Annali di Matematica

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Let M be a connected d M -dimensional complex projective manifold, and let A be a holomorphic positive Hermitian line bundle on M, with normalized curvature ω. Let G be a compact and connected Lie group of dimension d G , and let T be a compact torus of dimension d T . Suppose that both G and T act on M in a holomorphic and Hamiltonian manner, that the actions commute, and linearize to A. If X is the principal S 1 -bundle associated to A, then this setup determines commuting unitary representations of G and T on the Hardy space H (X ) of X , which may then be decomposed over the irreducible representations of the two groups. If the moment map for the T -action is nowhere zero, all isotypical components for the torus are finite dimensional, and thus provide a collection of finite-dimensional G-modules. Given a nonzero integral weight ν T for T , we consider the isotypical components associated to the multiples kν T , k → +∞, and focus on how their structure as G-modules is reflected by certain local scaling asymptotics on X (and M). More precisely, given a fixed irreducible character ν G of G, we study the local scaling asymptotics of the equivariant Szegö projectors associated to ν G and kν T , for k → +∞, investigating their asymptotic concentration along certain loci defined by the moment maps.

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Section: Applications To Toeplitz Operator Kernelsmentioning

confidence: 99%

Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions

Camosso

2016

Annali di Matematica

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“…In view of (26), it is always a simple matter to pass from T (ǫ) to T (ǫ) or vice versa. In the formula (4), the reproducing kernel K h (x, y) is the integral kernel of the orthogonal projection P h onto L 2 hol,h , i.e.…”

Section: Toeplitz-type Operatorsmentioning

confidence: 99%

Hermite polynomials and quasi-classical asymptotics

Ali

Engliš

2014

Journal of Mathematical Physics

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“…Feynman diagrams or directed graphs are effective tools in the construction and calculation of star products on Kähler manifolds. See [14,26,18,32,33]. Inspired by work of Reshetikhin and Takhtajan [26], Gammelgaard [14] obtained a remarkable universal formula in terms of acyclic graphs for a star product with separation of variables once a classifying Karabegov form is given.…”

Section: Introductionmentioning

confidence: 99%

Weyl invariant polynomial and deformation quantization on Kähler manifolds

Xu¹

2014

Journal of Geometry and Physics

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Given a polynomial P of partial derivatives of the Kähler metric, expressed as a linear combination of directed multigraphs, we prove a simple criterion in terms of the coefficients for P 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 Kähler manifolds.

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An Explicit Formula for the Berezin Star Product

Cited by 18 publications

References 57 publications

Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions

Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions

Hermite polynomials and quasi-classical asymptotics

Weyl invariant polynomial and deformation quantization on Kähler manifolds

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