scholarly journals Comparison and continuity of Wick-type star products on certain coadjoint orbits

2019 ◽  
Vol 31 (5) ◽  
pp. 1203-1223
Author(s):  
Chiara Esposito ◽  
Philipp Schmitt ◽  
Stefan Waldmann
Keyword(s):  
Star Product ◽  
Coadjoint Orbit ◽  
Type Star ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Continuity Properties ◽  
Fréchet Algebra

AbstractIn this paper, we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev–Lachowska and Karabegov, and we prove that they agree in general. In the case of the 2-sphere, we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

scholarly journals Traces for Star Products on the Dual of a Lie Algebra

2003 ◽  
Vol 15 (05) ◽  
pp. 425-445 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Simone Gutt ◽  
Martin Bordemann ◽  
Stefan Waldmann
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Star Product ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Compact Lie Group ◽  
Regular Orbit ◽  
Elementary Argument

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

scholarly journals Hermitian realizations of κ-Minkowski space–time

2015 ◽  
Vol 30 (03) ◽  
pp. 1550019 ◽  
Author(s):  
Domagoj Kovačević ◽  
Stjepan Meljanac ◽  
Andjelo Samsarov ◽  
Zoran Škoda
Keyword(s):  
Minkowski Space ◽  
Star Product ◽  
Plane Waves ◽  
Space Time ◽  
Star Products ◽  
Systematic Construction ◽  
Minkowski Space Time ◽  
Noncommutative Plane

General realizations, star products and plane waves for κ-Minkowski space–time are considered. Systematic construction of general Hermitian realization is presented, with special emphasis on noncommutative plane waves and Hermitian star product. Few examples are elaborated and possible physical applications are mentioned.

MOYAL STAR PRODUCT OF μ-HOLOMORPHIC j-DIFFERENTIALS

2008 ◽  
Vol 05 (03) ◽  
pp. 363-373
Author(s):  
M. KACHKACHI
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Quantum Effect ◽  
Star Product ◽  
Complex Structure ◽  
Complex Structures ◽  
Star Products ◽  
First Order ◽  
Conformal Fields ◽  
Conformal Covariance

It was shown in [1], only for scalar conformal fields, that the Moyal–Weyl star product can introduce the quantum effect as the phase factor to the ordinary product. In this paper we show that, even on the same complex structure, the Moyal–Weyl star product of two j-differentials (conformal fields of weights (j, 0)) does not vanish but it generates the quantum effect at the first order of its perturbative series. More generally, we get the explicit expression of the Moyal–Weyl star product of j-differentials defined on any complex structure of a bi-dimensional Riemann surface Σ. We show that the star product of two j-differentials is not a j-differential and does not preserve the conformal covariance character. This can shed some light on the Moyal–Weyl deformation quantization procedure connection's with the deformation of complex structures on a Riemann surface. Hence, the situation might relate the star products to the Moduli and Teichmüller spaces of Riemann surfaces.

THE METHOD OF COADJOINT ORBITS: AN ALGORITHM FOR THE CONSTRUCTION OF INVARIANT ACTIONS

1990 ◽  
Vol 05 (20) ◽  
pp. 3943-3983 ◽  
Author(s):  
GUSTAV W. DELIUS ◽  
PETER VAN NIEUWENHUIZEN ◽  
V. G. J. RODGERS
Keyword(s):  
Quantum Gravity ◽  
Central Extension ◽  
Central Charge ◽  
Virasoro Algebra ◽  
Coadjoint Orbit ◽  
The Other ◽  
Coadjoint Orbits ◽  
Two Dimensional ◽  
Supersymmetric Extension ◽  
Infinite Dimensional

The method of coadjoint orbits produces for any infinite dimensional Lie (super) algebra A with nontrivial central charge an action for scalar (super) fields which has at least the symmetry A. In this article, we try to make this method accessible to a larger audience by analyzing several examples in more detail than in the literature. After working through the Kac-Moody and Virasoro cases, we apply the method to the super Virasoro algebra and reobtain the supersymmetric extension of Polyakov's local nonpolynomial action for two-dimensional quantum gravity. As in the Virasoro case this action corresponds to the coadjoint orbit of a pure central extension. We further consider the actions corresponding to the other orbits of the super Virasoro algebra. As a new result we construct the actions for the N = 2 super Virasoro algebra.

scholarly journals ON RELATION BETWEEN WEYL AND KONTSEVICH QUANTUM PRODUCTS: DIRECT EVALUATION UP TO THE ℏ3-ORDER

2001 ◽  
Vol 16 (10) ◽  
pp. 615-625 ◽  
Author(s):  
A. ZOTOV
Keyword(s):  
Deformation Quantization ◽  
Jacobi Identity ◽  
Star Product ◽  
Star Products ◽  
Moyal Product ◽  
Third Order ◽  
Direct Evaluation ◽  
The Third ◽  
Arbitrary Change ◽  
Poisson Bivector

In his celebrated paper Kontsevich has proved a theorem which manifestly gives a quantum product (deformation quantization formula) and states that changing coordinates leads to gauge equivalent star products. To illuminate his procedure, we make an arbitrary change of coordinates in the Weyl (Moyal) product and obtain the deformation quantization formula up to the third order. In this way, the Poisson bivector is shown to depend on ℏ and not to satisfy the Jacobi identity. It is also shown that the values of coefficients in the formula obtained follow from associativity of the star product.

scholarly journals A COHOMOLOGICAL CONSTRUCTION OF MODULES OVER FEDOSOV DEFORMATION QUANTIZATION ALGEBRA

2008 ◽  
Vol 05 (04) ◽  
pp. 547-556
Author(s):  
S. A. POL'SHIN
Keyword(s):  
Left Ideal ◽  
Symplectic Manifold ◽  
Commutative Algebra ◽  
Deformation Quantization ◽  
Separation Of Variables ◽  
Star Product ◽  
Arbitrary Point ◽  
Type Star ◽  
Vanishing Ideal ◽  
Deformed Algebra

In certain neighborhood U of an arbitrary point of a symplectic manifold M we construct a Fedosov-type star-product *L such that for an arbitrary leaf ℘ of a given polarization [Formula: see text] the vanishing ideal of ℘ ∩ U in the commutative algebra C∞(U)[[h]] is a left ideal in the deformed algebra (C∞(U)[[h]],*L). With certain additional assumptions on M, *L becomes a so-called star-product with separation of variables.

scholarly journals Wick rotations in deformation quantization

2021 ◽  
pp. 2150035
Author(s):  
Philipp Schmitt ◽  
Matthias Schötz
Keyword(s):  
Analytic Functions ◽  
Star Product ◽  
Holomorphic Extension ◽  
Star Products ◽  
Formal Deformation ◽  
Space Reduction ◽  
Function Algebras ◽  
Special Cases ◽  
Complex Projective ◽  
Holomorphic Extensions

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from [Formula: see text] with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space [Formula: see text] and the complex hyperbolic disc [Formula: see text]. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on [Formula: see text] and [Formula: see text]. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the [Formula: see text]-involution of pointwise complex conjugation.

scholarly journals Star products on coadjoint orbits

2001 ◽  
Vol 64 (12) ◽  
pp. 2136-2138
Author(s):  
M. A. Lledó
Keyword(s):  
Coadjoint Orbits ◽  
Star Products

COSET CONSTRUCTION OF SUPERSTRINGS VIA THE COADJOINT ORBIT METHOD

1990 ◽  
Vol 05 (31) ◽  
pp. 2615-2624
Author(s):  
H. ARATYN ◽  
E. NISSIMOV ◽  
S. PACHEVA
Keyword(s):  
Coadjoint Orbit ◽  
Coadjoint Orbits ◽  
Alternative Formulation ◽  
Orbit Method ◽  
Coset Space ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Coset Construction ◽  
Left And Right ◽  
Geometric Action

The previously proposed general construction of geometric actions on infinite-dimensional group coadjoint orbits in terms of fundamental group one-cocycles is applied to provide an alternative formulation of the Green-Schwarz superstring. It is shown that the latter model can be consistently constructed as geometric action on a certain infinite-dimensional coset space of the semi-direct product of left- and right-handed Virasoro groups with a Kac-Moody group based on an appropriate modification of the super-Heisenberg-Weyl group.

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