Morita Equivalence, Picard Groupoids and Noncommutative Field Theories

Waldmann, Stefan

doi:10.1007/11342786_8

Cited by 6 publications

(9 citation statements)

References 36 publications

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“…Deformation quantization of vector bundles p : E −→ M is already established and can be put down to deformation quantization of finitely generated projective modules, since the sections Γ ∞ (E) are such a right module over C ∞ (M ). The well-known definitions and results can all be found in [13] and [43]. Altogether, one considers the deformed bimodule (6.1)…”

Section: Associated Vector Bundlesmentioning

confidence: 99%

“…where the right module structure itself is defined as a deformation quantization of the vector bundle E. It is known, confer [43], Thm. 1, that for a given star product ⋆ the deformed bimodule structures • ′ E , • and the algebra structure ⋆ ′ E are unique up to equivalence and that the two deformed algebras (Γ ∞ (End(E))[ [λ]], ⋆ ′ E ) and (C ∞ (M )[ [λ]], ⋆) are mutual commutants.…”

Section: Associated Vector Bundlesmentioning

confidence: 99%

See 1 more Smart Citation

Deformation quantization of surjective submersions and principal fibre bundles

Bordemann¹,

Neumaier²,

Waldmann³

et al. 2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle: the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold. In case of a principal fibre bundle we require in addition invariance under the principal action. We prove existence and uniqueness of such deformations. The commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators. Applications to noncommutative gauge field theories and phase space reduction of star products are discussed. * Martin.Bordemann@uha.fr ‡

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Section: Associated Vector Bundlesmentioning

confidence: 99%

Section: Associated Vector Bundlesmentioning

confidence: 99%

Deformation quantization of surjective submersions and principal fibre bundles

Bordemann¹,

Neumaier²,

Waldmann³

et al. 2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In this particular case we can of course use the canonical fiber metric coming from the canonical inner product on n . We refer to [34,38] for a further discussion.…”

Section: Matter Fields and Deformed Vector Bundlesmentioning

confidence: 99%

Noncommutative Field Theories from a Deformation Point of View

Waldmann

2009

Quantum Field Theory

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In this review we discuss the global geometry of noncommutative field theories from a deformation point of view: The space-times under consideration are deformations of classical space-time manifolds using star products. Then matter fields are encoded in deformation quantizations of vector bundles over the classical space-time. For gauge theories we establish a notion of deformation quantization of a principal fiber bundle and show how the deformation of associated vector bundles can be obtained. (2000). Primary 53D55; Secondary 58B34, 81T75. Mathematics Subject Classification

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“…We have now the following structure for the collection of all Pic str (·, ·), see [37,Sect. 6.1] and [135,136] …”

Section: The Strong Picard Groupoidmentioning

confidence: 99%