A Homological Approach to Singular Reduction in Deformation Quantization

Bordemann, Martin; Herbig, Hans-Christian; Pflaum, Markus J.

doi:10.1142/9789812707499_0016

Cited by 16 publications

(34 citation statements)

References 19 publications

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“…The usage of invariant star products (and even better: invariant star products with a quantum momentum map) will hopefully allow also to treat the Morita theory of star products on singular quotients, see e.g. [6,29].…”

Section: G-actions On Symplectic Manifoldsmentioning

confidence: 99%

Morita theory in deformation quantization

Waldmann

2011

Bull Braz Math Soc, New Series

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Various aspects of Morita theory of deformed algebras and in particular of star product algebras on general Poisson manifolds are discussed. We relate the three flavours ring-theoretic Morita equivalence, * -Morita equivalence, and strong Morita equivalence and exemplify their properties for star product algebras. The complete classification of Morita equivalent star products on general Poisson manifolds is discussed as well as the complete classification of covariantly Morita equivalent star products on a symplectic manifold with respect to some Lie algebra action preserving a connection.

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Section: G-actions On Symplectic Manifoldsmentioning

confidence: 99%

Morita theory in deformation quantization

Waldmann

2011

Bull Braz Math Soc, New Series

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“…In this paper, we apply the homological approach to singular reduction in deformation quantization developed in [13] to a model of gauge theory obtained via lattice approximation of Yang-Mills theory within the Hamiltonian approach. We refer to the classical paper [49] for the formulation of the full model (including matter fields) on a finite lattice and for its canonical quantization.…”

Section: References 50 1 Introductionmentioning

confidence: 99%

“…They proved that, under appropriate regularity properties of the group action, the BRST-procedure induces a star product on the reduced phase space. In [13], Bordemann, Herbig and Pflaum showed that this method may be extended to singular symplectic reduction, provided the following assumptions are fulfilled:…”

Section: References 50 1 Introductionmentioning

confidence: 99%

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

Pflaum

Rudolph

Schmidt

2021

Commun. Math. Phys.

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For a compact Lie group G we consider a lattice gauge model given by the G-Hamiltonian system which consists of the cotangent bundle of a power of G with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-Civita connection of the Killing metric on G. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for G = SU(2) the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the-in general singular-symplectically reduced phase space of the corresponding lattice gauge model carries a star product.

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“…Batalin-Vilkovisky-Fradkin's method [11], [12] was proposed for gauge systems. In [8] the BRST method was developed based on the rather complicated homological construction including ghosts fields.…”

Section: Introductionmentioning

confidence: 99%

Remarks on Singular Symplectic Reduction and Quantization of the Angular Moment

Palamodov

2012

Geometric Methods in Physics

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The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular Poisson spaces. For some examples of singular Poisson spaces the deformation quantization is explicitly constructed. It is shown that for the flat phase space with the classical moment map and the orthogonal group action the deformation quantization converges for the class of entire functions.Non free MSC (2010) Primary 53D20; Secondary 53D55

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A Homological Approach to Singular Reduction in Deformation Quantization

Cited by 16 publications

References 19 publications

Morita theory in deformation quantization

Morita theory in deformation quantization

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

Remarks on Singular Symplectic Reduction and Quantization of the Angular Moment

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